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The last few weeks I've been exploring Grothendieck topologies, as well as the different notions of topological-esque space, which has been very informative. In the last topic, we discussed ways of characterizing space-like objects. But in category theory, there's two ways of going about abstracting a mathematical property. One can do it at the object level, like how in the previous topic we abstracted elements of topological spaces to understand topological-esque spaces. But the other way is to do it at the category level, where we take an entire category of some mathematical objects and abstract its properties, getting a class of categories with some extra property or structure equipped. These definitions put the focus more on the categories than their objects and as such their objects may - but also might not- be concretely realized as "sets with extra structure".
There's many ways I found to do this when it comes to defining what it means for a category to be a "category of spaces". Here are some that I noticed. As we discussed before, Grothendieck topologies are the first that come to mind, enabling an abstraction of the notion of "open" from topological spaces to general categories. There's also categorical homotopy theory including things like model categories and certain infinity categories, which abstract homotopy theory from topological spaces to general categories. Slightly more concretely, we also have Grothendieck "gros topos", which under the philosophy of "generalized spaces" have objects which can be interpreted as, well, generalized spaces. We also have categories equipped with a topological forgetful functor, AKA topologically concrete categories, which all the categories of topological-esque spaces possess. Lastly, we have "synthetic categories" of spaces, such as smooth topos or cohesive topos, where we specify axioms these categories are supposed to possess.
My question here is similar to my opener for the last topic. Is there a notion of the "most general" kind of "category of space", such that such a category with structure is induced by all other cases? Are there any interesting connections or intersections between these kinds of "category of space"? (Well, other than the obvious one that a category with a Grothendieck topology gives rise to a Grothendieck topos via the sheaf category construction. A good example of a connection would be if one can prove that all smooth topos or cohesive topos are Grothendieck and thus can all be thought of as consisting of sheaves on some site.) Again, the main goal here is to determine a notion of "locality", but unlike in the last topic where we considered this within an object, I now want to consider this on the level of a category to determine what a "category with locality" is like. Thanks for any help you can provide!
Again, the answer will be "I doubt it", because once you zoom out from objects, there are usually multiple choices of morphism available preserving different amounts of "spatial" structure. Think of metric spaces: one has continuous maps, but also multiple different choices of map interacting with the metric in other ways. The resulting categories (for the respective morphism types) have some things in common, but it may not be clear without a particular goal in mind which of them deserves to be enshrined in a "most general" notion of category of spaces.
On the other hand, a more interesting answer would be to point you to some of the categorical properties that arise in doing what you're describing. I think [[topological category]] may be a good place to start?
Lawvere was interested in a pursuit of categories which capture the important features of spaces, so [[cohesive topos]] is also a key word you can look into.
I've been trying to dig up the recent discussion of "best generalization" that occurred either here on Zulip or on MO before, but my search has failed. To repeat the point I surely made in that discussion, identifying common features of examples is surely generalization, but there is no a priori criterion for which categories are included in the collection to be generalized: you need a goal or justification for what exactly it is about spaces that you're trying to capture. The weaker the criterion for being a category of spaces, the weaker the properties that satisfying that criterion will impose, consequently restricting what you can actually do at that level of generality. On the other hand, if you succeed in finding a very general unifying feature and actually want to try lifting results which apply to special cases (the motivating examples), you will be forced to identify what further special features/hypotheses underpin that result, which can eventually be very illuminating when compared with the usual trajectory of using whatever structure one has to hand in studying a special case directly. (I can elaborate on that last point if it's too vague to be meaningful).
This one? :) https://philosophy.stackexchange.com/a/111165/72812
That's the one, thanks!
Morgan Rogers (he/him) said:
On the other hand, a more interesting answer would be to point you to some of the categorical properties that arise in doing what you're describing. I think [[topological category]] may be a good place to start?
Lawvere was interested in a pursuit of categories which capture the important features of spaces, so [[cohesive topos]] is also a key word you can look into.
Right, those are actually two of the notion of "categories of space" I discussed above!
Morgan Rogers (he/him) said:
you need a goal or justification for what exactly it is about spaces that you're trying to capture. The weaker the criterion for being a category of spaces, the weaker the properties that satisfying that criterion will impose, consequently restricting what you can actually do at that level of generality
My goal would be to have a category with sufficient structure to define a notion of "locality" internal to it. But as I've learned on a previous topic here, "locality" might be a vague term with many possible meanings. For instance, it might refer to objects that have a defined notion of "closeness" or "apartness" for their parts given by a relation of some kind, but it might also refer to operations where one "glues together" parts into a whole, which is given by a gluing operation instead.
"Schemes", I believe, are geometric things, made from algebraic things. You can start with any commutative ring and make a scheme from it. So, from this angle, the category of commutative rings is also a geometric setting: each object contains the data needed to construct a corresponding geometric thing. I don't know if this would count as a category of "spaces" for you, though.
I'm reminded of this picture, from "The Geometry of Schemes":
picture
That book explains:
An affine scheme is an object made from a commutative ring. The relationship is modeled on and generalizes the relationship between an affine variety and its coordinate ring...Scheme theory arises if we adopt the opposite point of view: if we do not accept the restrictions “finitely generated,” “nilpotent-free” or “K-algebra” and insist that the right-hand side include all commutative rings, what sort of geometric object should we put on the left? The answer is “affine schemes”...
John Onstead said:
Right, those are actually two of the notion of "categories of space" I discussed above!
It has been revealed that I didn't carefully read your post before replying to it :eyes: whoops!
Morgan Rogers (he/him) said:
It has been revealed that I didn't carefully read your post before replying to it :eyes: whoops!
No problem!
David Egolf said:
"Schemes", I believe, are geometric things, made from algebraic things. You can start with any commutative ring and make a scheme from it. So, from this angle, the category of commutative rings is also a geometric setting: each object contains the data needed to construct a corresponding geometric thing. I don't know if this would count as a category of "spaces" for you, though.
That's a great example I didn't list above! It's part of a more general scheme called the "duality between space and algebra". So under this another notion of "category of spaces" could be "categories dual to an algebraic category in a similar way to how affine schemes are dual to commutative rings"
This brings me to a new question: how exactly are "space categories dual to an algebraic category" and "categories of generalized spaces" related? I know there's strong connections between these two ideas (IE, both the philosophy of "space and quantity" and their duality, and the philosophy of "generalized spaces"). But while schemes in general are indeed "generalized spaces" in the sense of being sheaves on some site (actually, sheaves on affine schemes), the same isn't true for affine schemes. That's because the category of affine schemes is not a topos and so its objects are not sheaves on some site and thus are not generalized spaces (nor is the category of commutative rings, meaning they can't be cosheaves/generalized quantity on some site either). So how do these two ideas connect, especially in this example of schemes?
John Onstead said:
It's part of a more general scheme called the "duality between space and algebra".
The most general form I am aware of in this direction is Isbell duality. Unfortunately, I am not knowledgeable enough to unfold the concept (this is the kind of concept I keep chewing through more low-brow instances).
There is also this short explanation on the arxiv by @John Baez
According to that short explanation, an Isbell duality is an adjunction between and . So for the duality between affine schemes and commutative rings to be seen as a special case of this, it must be possible to see the category of affine schemes as for some , and dually see the category of commutative rings as for that very same . However this isn't possible since neither are toposes while both the categories involved in the Isbell duality are toposes.
In addition the Isbell duality is an adjunction, while the relationship between affine schemes and the opposite category of commutative rings is an equivalence, which is even stronger than an adjunction!
Mmh, I don't know a lot about algebraic geometry, but, if I recall correctly, schemes are "locally made up of affine schemes".
Looking at the nlab, there is a definition of a scheme as a sheaf on a category of rings.
The passage from the usual definition (locally ringed space with an open cover whose parts are isomorphic to affine schemes) to the one above is detailed further down the page. There is a full and faithful functor from the category of schemes to the category of presheaves on (commutative rings)
So maybe we should try to state Isbell's duality with , and see what happens to schemes?
Hmm I tried working a few things out... Under the page of Isbell duality, it says a case of the duality (somehow) is that, given a Lawvere theory, there's an adjunction . So let's try your suggestion and let be the theory of commutative rings. Then and is a fully faithful embedding on all product preserving functors. I have no idea what this means but just thought I'd put that out there!
Let me take a break from Isbell duality for the time being since I'm just confusing myself more and more... What I want to do next is organize the "topos-y" categories of space together. First, I want to understand something. Oftentimes, when people are discussing toposes in geometric terms, they use the word "topos" to mean specifically Grothendieck toposes- that is, those of sheaves on some site. In geometric contexts, the notion of an "elementary topos" is therefore wholly irrelevant. Is this the case when discussing smooth and cohesive toposes? For instance, I briefly reviewed Urs Schreiber's book "Differential cohomology in a differential infinity-topos" where he defines an "-topos" to be a category with an accessible left exact fully faithful embedding into a presheaf category (valued in infinity groupoids), which seems to be precisely the definition of a sheaf topos, not an elementary topos. But at the same time, it also states that this topos is only an "-topos of -sheaves" only if this localization is a "topological localization". Which leaves me extremely confused: are -topos sheaf toposes, or only when there's a topological localization? But if it's the latter, then why does that not appear for the 1-category case wherein all left exact reflections of a presheaf category are sheaf topos? Any help in clearing this confusion is appreciated!
For instance, I briefly reviewed Urs Schreiber's book "Differential cohomology in a differential infinity-topos" where he defines an "(∞,1)-topos" to be a category with an accessible left exact fully faithful embedding into a presheaf category (valued in infinity groupoids), which seems to be precisely the definition of a sheaf topos, not an elementary topos.
I'm no expert, but I think this would reduce to the definition of sheaf topos if you changed the phrase "valued in infinity groupoids" to the phrase "valued in sets", and perhaps turned some other knobs, not clearly emphasized in what you wrote, from their "(∞,1)" setting down to their "1" setting.
For example, saying an (∞,1)-topos is a category sounds weird to me: it's a kind of (∞,1)-category. So beware: in some literature people get so sick of saying "(∞,1)-category" that they abbreviate this as "category", and similarly for "(∞,1)-functor" and so on.
But on the question of "sheaf topos", in the -world it depends on how expansively you want to define "sheaf". After Lurie, an -topos is defined as Urs defined it. But unlike in the 1-categorical case, not every -topos in this sense can be defined by a small -category with an "-Grothendieck-topology" on it. This is basically because a Grothendieck topology in the usual sense is, in the world of presheaves, about sieves, i.e. subfunctors of representables. In -categories for finite , you can induct up to say something about all morphisms from monomorphisms, but you can't "get to " that way.
There are fancier kinds of structure with which you can equip a small -category that suffice to present all -toposes, but for historical reasons most people seem resistant to calling those "-Grothendieck-topologies", even though from my point of view that would seem more natural since they are what play the same role, even if they don't look quite as similar.
Mike Shulman said:
There are fancier kinds of structure with which you can equip a small -category that suffice to present all -toposes, but for historical reasons most people seem resistant to calling those "-Grothendieck-topologies", even though from my point of view that would seem more natural since they are what play the same role, even if they don't look quite as similar.
I see, that's very interesting! Would you mind sharing what one of these "fancier" structures are? Is there some resource where I can find it defined?
I found it here. If I'm understanding correctly, a "higher site" is a category equipped with a class of maps in its presheaf topos. A higher sheaf with respect to a higher site is then a local object with respect to the "congruence" generated by this class of map. In addition, every local object with respect to some congruence is a higher sheaf for some higher site. A congruence on the other hand is just a class of maps in the presheaf topos that is closed under composition, colimits, and finite limits, so it's not too difficult to understand! The paper also shows how higher sites reduce to typical ones in certain scenarios- a higher site is the same as a "normal" site when the congruence generated by the higher site's class of morphisms is "topological", meaning that the original class of morphisms consisted of monomorphisms! Anyways I hope I understood this sufficiently!
Yep, that's right! For a while this was an open problem but fortunately it got solved.
Mike Shulman said:
Yep, that's right! For a while this was an open problem but fortunately it got solved.
Glad it was!
I want to continue with this discussion since there's a lot more to unpack, but I'm taking leave for Thanksgiving. Hope to see you all next week and I'd like to wish you an early Happy Thanksgiving!
@John Onstead
Your: "glues together" is exactly how Professor F. William spells out COHESION.
P.S. TIME, on the other hand is characterized by an urge to change.
Hi everyone hope you had a great Thanksgiving. I want to come back by continuing the discussion of toposes and sites, this time in the context of cohesive toposes. On both the nlab page and in Urs Schreiber's book, a "cohesive site" is given as a kind of site whose category of sheaves is guaranteed to be a cohesive topos. However, the converse does not seem to be true- given a cohesive topos, only the ones where "pieces have points" are categories of sheaves on cohesive sites. So my question is: why aren't all cohesive toposes generated by sheaves on cohesive sites? Is this for the same reason as discussed above- the concept of a "sheaf on a site" isn't powerful enough, meaning we should instead consider "higher sheaves on higher sites" (in the sense we discussed above, as a category equipped with a class of morphism in the presheaf category)? In this case, it would be true that, given some generalization of cohesive site to the higher site setting, every cohesive topos is generated by a "cohesive higher site". Or is the problem in the definition of a cohesive site, in which case, is there a weaker notion of "cohesive site" such that sheaves or higher sheaves on that site do generate all cohesive toposes? Thanks!
It's mainly a question of how someone chose to define "cohesive site". Maybe it should have been called a "cohesive site with points".
By the way, when John is saying "topos" in his last post, does this really mean topos or does it mean -topos? Or could it be either?
John Baez said:
By the way, when John is saying "topos" in his last post, does this really mean topos or does it mean -topos? Or could it be either?
I was still talking about -toposes. But I think you bring up an interesting point. Urs Schreiber's paper says that cohesive -toposes are direct and complete generalizations of the theory of cohesive 1-topos, so that means anything you can do with a cohesive 1-topos you can do with some equivalent cohesive -topos. But I could be wrong about this!
The nlab distinguishes between a [[cohesive site]], whose category of 1-sheaves is a cohesive 1-topos, and an [[infinity-cohesive site]], whose -category of -sheaves is a cohesive -topos. However, the definitions are a bit inconsistent, and I (IMHO) wouldn't regard them as fully settled. In particular:
Mike Shulman said:
The definition of -cohesive site is as a property of an ordinary 1-site
That's really odd. Any "infinity construct" should be based on other infinity constructs by definition.
Mike Shulman said:
The definition of (1-)cohesive site includes an axiom guaranteeing that "pieces have points", but that of -cohesive site does not. So neither is actually a special case of the other as stated.
The definition of -cohesive site given by Urs Schrieber's book does guarantee that "pieces have points" (it's proposition 3.4.18 in that book). In any case, even the slightly different definition given on the nlab still gives rise to an extra condition, that of "discrete objects are concrete".
In any case one can directly generalize the case of sheaves on a 1-site to sheaves on an -site by realizing 1-sites as the -sites where all n-morphisms higher than 1 are trivial/discrete. But then one would have to prove the resulting sheaf -topos is still cohesive, which may no longer always be the case. Which might then explain why 1-sites are used in the definition of -cohesive site even when they seem "out of place".
But I think there's another important difference between the 1-categorical and -categorical approach that needs to be discussed. As we covered above, -topos are always Grothendieck as they are always lex reflective subcategories of a presheaf topos. In addition, there's no such thing as an elementary -topos (well, there's not much agreement on how to define them, but it's still possible they might exist in some capacity). However, there is a such thing as an elementary 1-topos, which raises the question of if the axioms of a cohesive topos could still work, at least in some capacity, in the elementary case as opposed to the more specific Grothendieck case.
If this is the case, then it's certainly problematic. Lawvere imagined cohesive toposes to be an axiomatization of a "gros topos". Already there's plenty counterexamples of gros toposes that are not cohesive (like the topological topos). But this would provide evidence that the converse also isn't true, and that there are cohesive toposes that are not gros toposes. This is because, by definition, a gros topos is defined (and thus is required) to be a Grothendieck topos, otherwise it doesn't make sense to think of the objects as "generalized spaces".
But as Urs Schreiber covered in his paper, it's an objective fact that cohesive -toposes are a generalization of cohesive 1-toposes. This raises a few possibilities of how to deal with both these pieces of information. The first possibility is that the forgetful functor from the category of cohesive toposes to all toposes lands in the subcategory of toposes on Grothendieck toposes. In other words, there simply can't be a cohesive topos whose underlying topos isn't Grothendieck.
But let's say it is possible. If that's true, then the fact that cohesive -toposes generalize cohesive 1-toposes means that a cohesive 1-topos which isn't Grothendieck can become Grothendieck when passing to the setting. That is, there exists an embedding of some sort from the category of cohesive toposes (even including the non-Grothendieck ones) to the category of cohesive -toposes (which are all Grothendieck) that preserves all the important information.
I hope I'm understanding correctly, but it's my understanding that under the Homotopy Hypothesis the category of topological spaces, which isn't a topos at all in the 1-categorical setting, becomes one in the setting. If this is true it means it might be possible for a non-Grothendieck topos to become Grothendieck in a similar way when passing from 1-categories to -categories.
So my question is: what is the correct answer? Can cohesive non-Grothendieck toposes exist? And if so, is my guess for how cohesive -toposes still act as generalizations of them correct? Thanks!
There's an emerging consensus on the outlines of what an elementary -topos is, although it's not clear that there will ever be a single standard definition of them the way there is for an elementary 1-topos.
I'm not sure what you are referring to with the statement that "cohesive -toposes generalize cohesive 1-toposes". The notion of cohesive -topos is an analogue for higher toposes of the notion of cohesive 1-topos, but it's not true that every cohesive 1-topos is a cohesive -topos (obviously, since no 1-topos can itself also be an -topos), nor is it the case even that every cohesive 1-topos gives rise to a cohesive -topos in any canonical or straightforward way.
The category of topological spaces and continuous maps is neither a 1-topos nor an -topos. The -category of CW-complexes, continuous maps, homotopies, and higher homotopies is an -topos, but I regard this as a sort of accident: it's only because it happens to be equivalent to the -category of -groupoids, which is an -topos.
As for whether a non-Grothendieck topos can be cohesive, the property of cohesiveness of a Grothendieck topos (like many other properties of a topos, such as local connectedness and compactness) should really be regarded as a property of its geometric morphism to Set (or to in the -topos case). It just so happens that a Grothendieck topos admits a unique such geometric morphism. But in general, for any geometric morphism , we can regard as analogous to Set and as "a topos defined in the universe of " (a.k.a. an "-topos"), and then ask whether it is cohesive as such. In this sense, plenty of elementary toposes can be cohesive over some base topos, e.g. if you start from an arbitrary elementary topos then I would expect you can mimic the construction of some of the standard cohesive Grothendieck toposes to get other elementary toposes that are cohesive over .
The subtlest question is about elementary toposes that aren't Grothendieck but do still admit a
(still necessarily unique) geometric morphism to Set; or in the general case of geometric morphisms that aren't [[bounded geometric morphisms]]. I don't know offhand whether any of these are cohesive, but I don't know a reason why they couldn't be. And unbounded geometric morphisms can in principle exist for -toposes too, although we understand even less about them.
Mike Shulman said:
it's not true that every cohesive 1-topos is a cohesive -topos (obviously, since no 1-topos can itself also be an -topos)
How not? An infinity category generalizes a 1-category because a 1-category is an infinity category where all the higher morphisms are identities. It should be true that an infinity topos with all higher morphisms as identities should be equivalent to a 1-topos, right?
Mike Shulman said:
The category of topological spaces and continuous maps is neither a 1-topos nor an -topos. The -category of CW-complexes, continuous maps, homotopies, and higher homotopies is an -topos, but I regard this as a sort of accident: it's only because it happens to be equivalent to the -category of -groupoids, which is an -topos.
Then why does the nlab state that the Homotopy Hypothesis gives an equivalence of categories between and -? In addition, isn't it true that any topological space is equivalent to an groupoid, so much so that we call infinity groupoids "spaces"?
John Onstead said:
It should be true that an infinity topos with all higher morphisms as identities should be equivalent to a 1-topos, right?
No. It is impossible for an -category with all higher morphisms as identities to be an -topos; the axioms of an -topos force there to be nontrivial higher morphisms.
John Onstead said:
Then why does the nlab state that the Homotopy Hypothesis gives an equivalence of categories between and -?
That's wrong, it shouldn't.
In addition, isn't it true that any topological space is equivalent to an groupoid, so much so that we call infinity groupoids "spaces"?
Any topological space is weakly equivalent to a CW-complex, in that there's a continuous map from the CW-complex to it that induces an isomorphism on all homotopy groups. The use of "space" for -groupoid does come historically from this.
I fixed the nLab.
Mike Shulman said:
It is impossible for an -category with all higher morphisms as identities to be an -topos
(except in the trivial case of the [[terminal category]])
Mike Shulman said:
There's an emerging consensus on the outlines of what an elementary -topos is, although it's not clear that there will ever be a single standard definition of them the way there is for an elementary 1-topos.
Maybe very unrelated, so feel free to delete or move it: was it settled in such consensus whether -finite spaces should form a higher elementary topos?
I mean they are usually the correct higher analogue of finite sets, but everybody seems to insist on the existence of higher inductive types and a natural numbers object.
Mike Shulman said:
I fixed the nLab.
I see, so somehow I guess this implies that knowing all the path information about a topological space is only sufficient for a space that is a CW complex. Maybe this is analogous to how the lattice of opens is only sufficient if the space is sober?
In any case I think categorical homotopy theory is interesting and it's one of the things I wanted to discuss here anyways since its relevant to finding "categories of spaces"!
Where "sufficient" means "sufficient to determine it up to homotopy equivalence", which is much weaker than up to homeomorphism.
Going back to relating cohesive toposes and gros toposes. A gros topos is a category of generalized spaces since it's a category of sheaves on a site where the objects are spaces, and so can be thought of as a category of spaces "locally modeled" on the spaces in the site. In a sense one can think of generalized spaces as consisting of "patching together" the spaces you are modeling on. Now one can see exactly how this leads into Lawvere's intuition of cohesive toposes as being categories where objects are collections of points with a notion of how those points "hang together". Once you can "patch together" spaces and thus their points, it's not hard to see how adding a bit of extra structure can allow one to extend this "patching together" of spaces to "hanging together" of points.
However, I'm running into a conceptual/philosophical problem with trying to understand what exactly the nature of objects in a non-Grothendieck cohesive topos are. They can't be "points with a notion of how they hang together" since "hanging together" requires some notion of "patching together", which brings you right back to sheaves. So if that's not it, then what exactly are they? And should they even be considered "spaces", or should an object of a cohesive topos only be considered a "space" if it's also Grothendieck?
Given that we don't have any examples of non-Grothendieck cohesive toposes, I wouldn't spend a lot of time worrying about that. (-:
(By which I mean, of course, unbounded cohesive geometric morphisms.)
Indeed, finite directed graphs are an example of a cohesive elementary 1-topos over finite sets
Mike Shulman said:
Given that we don't have any examples of non-Grothendieck cohesive toposes, I wouldn't spend a lot of time worrying about that. (-:
That's understandable, but you know this possibility will be constantly annoying me in the back of my head!
Mike Shulman said:
No. It is impossible for an -category with all higher morphisms as identities to be an -topos; the axioms of an -topos force there to be nontrivial higher morphisms.
This makes sense to me, though I'm still a little confused about why Schrieber mentioned that the infinity case generalized the lower dimensional case. Let me share one of his examples for why. Consider the site of cartesian spaces and smooth maps between them. The category of sheaves on this site is the quintessential smooth topos of smooth sets, and includes all smooth manifolds as well as diffeological spaces. On the other hand, the category of -sheaves on this site is the category of smooth -groupoids.
Apparently, this not only contains the smooth sets, but also things like orbifolds and essentially most if not all kinds of "manifold-y" things. The infinity case is then said to "generalize" the lower case since the 1-truncated objects in the category of smooth -groupoids are the smooth sets from the original sheaf 1-topos. So unlike what I previously thought, it's not that every Grothendieck topos is also an -topos, it's that every Grothendieck 1-topos embeds into an -topos.
Here's some of my questions about this. First, is this a reasonable approach to answering the question of how the infinity topos case "generalizes" the 1-topos case? Second, it's clear that all Grothendieck 1-toposes embed into some -topos given a choice of site of definition, just because embeds into - (and so 1-sheaves on the site embed into -sheaves on that same site). However, is it always true that one can "go back" and recover the original 1-topos by taking the subcategory on all truncated objects in the -topos, as we did with the category of smooth sets? Lastly, is category of truncated objects of an -topos (a general one, so not required to be a category of sheaves on a site but rather just a lex reflective subcategory of a presheaf topos) necessarily a Grothendieck 1-topos, or can it be something more general?
Proposition 6.4.5.7. in HTT tells us that the (m+1)-category of m-toposes embeds fully faithfully into the (n+1)-category of n-toposes for
Morgan Rogers (he/him) said:
Indeed, finite directed graphs are an example of a cohesive elementary 1-topos over finite sets
Right, so that's an example of a bounded cohesive geometric morphism over a non-Grothendieck base.
Sorry, new here, how do I get inline maths equations?
Use double dollar signs.
Look what an amazing thread here :rolling_on_the_floor_laughing: ! Well done guys ...
Thanks!
Mike Shulman said:
Use double dollar signs.
Adrian Clough said:
Proposition 6.4.5.7. in HTT tells us that the (m+1)-category of m-toposes embeds fully faithfully into the (n+1)-category for
This proposition is a little difficult to parse, but what it is saying that the functor from n-toposes to m-toposes given by sending any n-topos to its full subcategory of (m-1)-truncated objects admits a fully faithful right adjoint.
So it is true that if you send an m-topos to its associated n-topos , then you can recover as the subcategory of spanned by the (m-1)-truncated objects.
However, if is given by the m-category of (m-1)-groupoid valued sheaves on some site , then it is not necessarily true that is given by the n-category of (n-1)-groupoid valued sheaves on when
This is explored in https://youtu.be/PpfacHtBX8U?si=VD5_mmuRKPdhvJ4x.
(However, it is true when or is finitely complete; the latter case is discussed in HTT)
It's also important to note that the functor from -toposes to -toposes (where, as before, ) that takes the -truncated objects is not fully faithful or even conservative: a given -topos can occurr as the -truncated objects in many different -toposes. Given an -topos, there's a "universal" choice of such an -topos (given by the right adjoint), but there are generally other non-universal choices too, even in very simple cases like .
And when we specialize to cohesive toposes, it's not generally true that the -topos associated to a cohesive 1-topos is a cohesive -topos. Roughly, the objects of a cohesive 1-topos are built out of locally connected pieces, while the objects of a cohesive -topos are built out of locally contractible pieces, which is a stronger condition.
Does the concept of a cohesive (-)pretopos make sense as well, or are the power objects in an elementary topos required for the cohesion?
Actually Lawvere's original definition of cohesion didn't require any topos-ness at all.
Mike Shulman said:
a given -topos can occurr as the -truncated objects in many different -toposes.
Out of curiosity, is there any work attempting to characterize e.g., the -topoi whose 0-truncated objects correspond to ? (More generally, one can of course ask about any 1-topos, but seems particularly natural as some sort of generalization of boolean topoi)
I don't know of any.
So we can take any base category such as or spectra and try to define cohesive categories over said base category?
Emphasis on "try"...
Adrian Clough said:
However, if is given by the m-category of (m-1)-groupoid valued sheaves on some site , then it is not necessarily true that is given by the n-category of (n-1)-groupoid valued sheaves on when
- n = and
- is not finitely complete.
That's really interesting! After what I was told above I didn't think there was a canonical way to associate a lower topos with a higher topos. It's interesting that there is a way after all to do this! So applying to my example, given some site and the 1-topos of 1-sheaves on this site, the "canonical" -topos corresponding to this isn't necessarily equivalent to the -topos of -sheaves on this same site (but it is when the site is finitely complete or if we don't use infinity toposes). Though, as mentioned by Mike Shulman, multiple -topos can have the same 1-topos as their category of truncated objects. This means it's still possible that if we start with the -topos of -sheaves on some site, the 1-topos of truncated objects will still always correspond to the 1-topos of 1-sheaves on that site. Is that the case or also wrong?
This also makes me really curious about the properties of the adjunction between 1-toposes and -toposes. For instance, what even are the involved categories? As mentioned before, there's not a standard notion of an elementary -topos, so most likely we are working with Grothendieck -toposes. But then, is the other category in the adjunction the category of elementary 1-toposes, or the category of Grothendieck 1-toposes? (I guess, in other words, this relates to my question of if the 1-topos of truncated objects of a -topos has to be Grothendieck or not).
All Grothendieck.
The category of 0-truncated objects in an elementary -topos is an elementary 1-topos, but in that case there's no obvious way to go the other direction.
And yes, the 0-truncated objects in an -topos of -sheaves on a site is the 1-topos of 1-sheaves on that site.
(Outside of a small circle of people, most of whom are connected in some way with homotopy type theory, if you hear someone say "-topos" they always mean the Grothendieck version. In particular, whenever anyone mentions Lurie they're talking about the Grothendieck version.)
Mike Shulman said:
And yes, the 0-truncated objects in an -topos of -sheaves on a site is the 1-topos of 1-sheaves on that site.
Thank goodness! I guess that does indeed confirm how -topos theory can be a "generalization" of 1-topos theory, at least when it comes to Grothendieck toposes.
While I'm reviewing, I had a related question that I wanted to ask here. Recall that the category of sheaves on the site of Euclidean topological spaces includes topological manifolds, the category of sheaves on the site of Euclidean spaces with differentiable maps includes differentiable manifolds, and the category of sheaves on the site of Euclidean spaces with smooth maps includes smooth manifolds. There's one other case I wanted to consider.
On another thread, I learned that every finite real vector space has a unique topological structure. This gives a functor of the form . From this, it seems reasonable that can inherit a site structure from the coverings in (correct me if I'm wrong!) My question is then obvious: what is the category of sheaves on this site of finite real vector spaces, and how does it relate to smooth manifolds and similar concepts?
The topology on FinVect induced from open covers in Top will be trivial, since a proper subspace of a finite-dimensional vector space is never open.
So the category of sheaves on it coincides with the category of presheaves.
Mike Shulman said:
So the category of sheaves on it coincides with the category of presheaves.
I see, so probably nothing too interesting...
Above I mentioned I want to understand more about categorical homotopy theory; after all that's the main motivation for introducing infinity categories (and thus infinity toposes) in the first place. The central object of focus in categorical homotopy theory is a "model category", which is a 1-category that "stands in" for some infinity category. Just as a Grothendieck topology gives an abstraction of an "open" and "open cover" from topology, the higher morphisms in the infinity category for a model category give an abstraction of a "higher homotopy" from topology.
First, is my basic understanding here correct? And if so, I'd like to start off by asking: on a conceptual/ philosophical level, to what extent can the objects of a model category (or the resulting infinity category) be considered "spaces"? For instance, is there any notion of "locality" induced by studying homotopies in this abstract of a way?
I think that's mostly right. But insofar as the objects of a model/ category are like "spaces" they are like "spaces up to homotopy", which already don't have a notion of "locality" in the sense of open subsets.
Mike Shulman said:
which already don't have a notion of "locality" in the sense of open subsets.
I guess I meant "locality" in a loose term, as in some abstract notion of describing how things are "close" or "stitched" together.
So above, we covered how the category of infinity groupoids is not equivalent to topological spaces, but rather CW complexes. But I know for sure that as a whole does give rise to an infinity category- one where the objects are topological spaces, morphisms are continuous functions, and higher morphisms are homotopies between any continuous map of a topological space. Is this the infinity category corresponding to the "classical" model structure on ? What are the properties of this infinity category? Does have any interesting properties as an infinity category when compared with its 1-category incarnation?
No, it corresponds to the [[Strøm model structure]]. The classical or "Quillen" model structure uses weak homotopy equivalences, so every space is equivalent to a CW complex and thus it presents the -category of -groupoids.
The -category coming from the Strøm model structure isn't very well-studied. It's not locally presentable or an -topos.
That's interesting it isn't well-studied. I guess in the context of homotopy theory CW complexes just "work a lot better" than topological spaces.
I have some follow-ups about this. First, are topological spaces the most general setting of "space-like object" to do a traditional homotopy theory? That is, is it possible to do homotopy theory with pretopological, pseudotopological, approach, convergence, etc. (all the "generalized spaces" we went over on the previous topic) spaces, or are topological spaces the "most general" kind this is possible for? (and if they are, why?)
Secondly, when I hear about homotopy theory, I think of algebraic topology with fundamental groups (and more generally fundamental n-groupoids). Is there a model-theoretic analogue of the fundamental group construction- that is, given a model category, can I find a functor into the category of groups or n-groupoids that "acts like" the fundamental group/groupoid functor from normal topology? If not, then what is the use of extending homotopy theory outside of topology and into model categories if this fundamental use of homotopy is not possible?
You can certainly talk about homotopies and homotopy theory for any of those kinds of space. I don't know which of them admit a full Quillen model structure.
To the second question, every -category has hom--groupoids, so for any objects and you have an "-parametrized fundamental -groupoid of " given by . This is the most you should expect, along the lines of how for a general category you have to consider [[generalized elements]] rather than ordinary ones.
Mike Shulman said:
This is the most you should expect, along the lines of how for a general category you have to consider [[generalized elements]] rather than ordinary ones.
Ah, that makes sense! So I'm guessing that for usual homotopy theory the object you parameterize by is the unit interval.
Well, the point, but that's homotopy equivalent to the unit interval. (-:
The role of the unit interval is to define homotopies to pass from the 1-category of topological spaces to the -category.
Once you're in the -category, you don't need an interval any more.
Ah, I see!
I now want to try and link together categorical homotopy theory with gros toposes to see if they can produce a unified notion of "category of space". As we covered, every Grothendieck or gros 1-topos embeds into some infinity topos, so infinity topos theory generalizes the theory of gros toposes. However, infinity toposes are just one of a special case of locally presentable infinity categories, which are precisely those that come from combinatorial simplicial model categories. Even more generally, infinity categories with certain limits and colimits come from model categories, and most generally it seems any infinity category comes from a "category with weak equivalences". (Let me know if I got anything wrong here)
An even more concrete link between the two that I noticed and want to understand more about is the following quote from locally presentable infinity category: "The canonical example is the presentation of the infinity topos of infinity sheaves on a 1-site S by the simplicial model category of simplicial presheaves on S". What I'm wondering about is what exactly does this mean- does every site determine some model structure somewhere, and if so, how?
Yes, every site determines a [[model structure on simplicial presheaves]] that presents the corresponding -topos.
I've been going over through more basic homotopy theory, but I'm stumbling on some very elementary stuff! My first question is: given a category with weak equivalences, are the path space and cylinder objects for some given object unique? Based on the definition the answer seems pretty clearly "no", but then that raises the question of which path or cylinder object do I use to define the homotopies?
Secondly, a cylinder object is defined by a factorization of the codiagonal into where the map is one of the designated weak equivalences. However, the definition of a left homotopy between two morphisms as a map requires the definition of two maps of the form . But this is clearly the "wrong direction": we have only defined a map in the direction as the weak equivalence above. So what are these maps? I thought it might have something to do with the universal property of the coproduct, but I don't think so. The universal property states that if we know we have two maps from to , then we can conclude there's a universal morphism . But we don't care if such a morphism exists, and we are trying to find the two maps into in the first place!
In a model category, it doesn't matter which cylinder object you use (up to homotopy).
The two maps are the two components of the map that's the first factor of the factorization.
(Also you have the universal property backwards.)
Mike Shulman said:
(Also you have the universal property backwards.)
Oh! Oops... I knew I had done something wrong somewhere... thanks for the help!
Madeleine Birchfield said:
So we can take any base category such as Ab or spectra and try to define cohesive categories over said base category?
Yes, there's no problem with the concept of cohesion relative to any base category. To see this idea appearing in "real mathematics", take a look at this MO discussion in the context of condensed mathematics.
For more such discussion, there's a further MO question.
So I reviewed more about the model structure corresponding to a site. Here's my basic understanding of what is going on. Given a site, which is just a set of covering families , there's a way to convert each covering family in the site into a single morphism in the category of simplicial presheaves. The way to do so is to take a sort of "Cech nerve" which then yields a morphism (a "Cech cover"), where, I guess, X is the representable (so the image of X under the Yoneda embedding followed by the embedding of presheaves into simplicial presheaves).
Then, the model structure for the site is when you start with the usual global model structure on simplicial presheaves and then, in a particularly way, freely add all morphisms of the form (the Cech covers corresponding to all the covering families in the site) to the class of weak morphisms in the model structure. In a sense, this is saying that the cover over an object is weakly equivalent to that object, which I guess is a sensible property for a cover to satisfy! The procedure for freely adding the weak equivalences is like a localization but instead of turning morphisms into equivalences it turns them into weak equivalences (which in the corresponding infinity category does end up turning them into equivalences). Hopefully I got everything right, let me know if I didn't!
What I'm interested about most here is the connection between model structures and site structures- this relationship between them seems to suggest that one can view the latter concepts as appropriate cases of the former concepts! For this to be the case, one must be able to go backwards and recover the original site from the model structure. I think this is possible, so long as a particular condition is satisfied: that the notion of "category equipped with a site" is fully equivalent to "category equipped with a collection of Cech covers (morphisms of the form ) in its category of simplicial presheaves". Obviously, as we have seen, the former induces the latter, and the weak localization is done on the Cech covers in that collection. I'm curious to know if given any collection of Cech covers if there's a way to go backwards to the original site.
Recall that model structures provide an abstract account of homotopy concepts, like homotopy, weak equivalence, and so on. A site structure provides an abstract account of open concepts, like open subset and open cover. If the above is true, it could imply many things, such as realizing the abstract notion of "open" as a special case of the abstract notion of "weak equivalence". In this case, an "open cover" is the special case of a weak equivalence, where a weak equivalence is an open cover precisely when it is also a Cech cover (and hopefully there's methods to determine if some morphism in, at the very least a category of simplicial presheaves, is a Cech cover without having to know which "family" it might have been generated from). I'm wondering if there's any interesting implications of this being true- let me know!
Hmm maybe it's controversial to assert that model structures are "more general" than site structures... But it's in the spirit of this thread to find the "most general" kind of category of space and I think that model categories and their variants are probably the best candidate so far!
Yes, a Grothendieck topology on a category can be recovered from the induced model structure on the category of simplicial presheaves on in the way you suggest: define a family to be covering if the Cech morphism it induces is a weak equivalence in the model structure. However, not every model structure on simplicial presheaves arises from a topology -- not every every model structure that presents an -topos! -- and if you start from an arbitrary model structure, this definition doesn't always produce a topology.
Interesting!
Mike Shulman said:
However, not every model structure on simplicial presheaves arises from a topology -- not every every model structure that presents an -topos!
That makes sense, as we covered earlier there are -topos that do not arise as categories of sheaves on sites. For instance, there are also model structures corresponding to "hypercovers". And even more generally, I don't see a reason there can't be a model category friendly version of the "higher site" notion that generates any -topos (IE, making a class of the "congruences" the weak equivalences)
Yes, certainly there is. Indeed, every -topos is presented by some model structure on a category of simplicial presheaves (as long as you allow the domain of the simplicial presheaves to also be a simplicial category).
I'm extremely confused about something. The nlab page "fibrant object" seems to imply that fibrant objects are those for which any object in a category has a weak equivalence to. But this doesn't make sense for a number of reasons. First, a category with the same class of weak equivalences can have multiple model structures with different fibrant objects. But obviously, given a class of weak equivalences, there's only one class of objects that all other objects have a weak equivalence to. Secondly, by the axioms of a model category and category with weak equivalences, every object is trivially weakly equivalent to itself, and thus for any category all objects should be fibrant at all times, which obviously is not the case.
This is also leading me to get confused about how the localization of topological spaces at weak equivalences in the classical model structure leads to a category equivalent to CW complexes. This is because every CW complex is weakly equivalent to a simplicial complex; therefore, since every space is homotopy equivalent to a CW complex and weak equivalence are closed under composition, every space should be weakly equivalent to a simplicial complex. Yet the localization does not lead to a category equivalent to that of simplicial complexes. A similar argument goes the other direction: every CW complex is homotopy equivalent to a "space with good cover", therefore every space should be homotopy equivalent to a "space with good cover". So why doesn't the localization happen at that subcategory? Any help in untangling this mess is greatly appreciated!
But obviously, given a class of weak equivalences, there's only one class of objects that all other objects have a weak equivalence to. Secondly, by the axioms of a model category and category with weak equivalences, every object is trivially weakly equivalent to itself
The second sentence here shows why the first sentence is false.
John Onstead said:
I'm extremely confused about something. The nlab page "fibrant object" seems to imply that fibrant objects are those for which any object in a category has a weak equivalence to.
No, that's not any sort of definition of fibrant objects. Maybe someone was trying to say that in a model category any object is weakly equivalent to some fibrant object. (Which is true.)
Ok, I think I see... so a subcategory of fibrant objects for some model structure would have the property that every object is weakly equivalent to some object in this subcategory. But depending on your choice of fibration this subcategory can include anything up to and including all objects in the category by the trivial weak equivalence. In Top I think this is the case where in the classical model structure all objects are fibrant. But there's other model structures on Top where different other classes of object are the fibrant ones.
But I'm still confused about why the homotopy category of the classical model structure on Top is equivalent to CW complexes and not simplicial complexes if every CW complex is weakly equivalent (in the form of homotopy equivalent) to a simplicial complex.
John Onstead said:
But I'm still confused about why the homotopy category of the classical model structure on Top is equivalent to CW complexes and not simplicial complexes if every CW complex is weakly equivalent (in the form of homotopy equivalent) to a simplicial complex.
Who said that? I'd like to see the precise statement. It sounds wrong to me.
(When you say "the homotopy category of .... is equivalent to CW complexes", you're being a bit abbreviated in a way that's potentially confusing. For example, it's not true that the homotopy category of the classical model structure on Top is equivalent to the category of CW complexes and continuous maps. You probably didn't mean that, so I'm left hoping you meant something true.)
John Baez said:
Who said that? I'd like to see the precise statement. It sounds wrong to me.
(When you say "the homotopy category of .... is equivalent to CW complexes", you're being a bit abbreviated in a way that's potentially confusing. For example, it's not true that the homotopy category of the classical model structure on Top is equivalent to the category of CW complexes and continuous maps. You probably didn't mean that, so I'm left hoping you meant something true.)
It says it right on the nlab page for Ho(Top): "its full subcategory on those topological spaces homeomorphic to a CW-complex. The latter is technically the homotopy category obtained by localizing the category of topological spaces at those continuous functions that are weak homotopy equivalences, hence it is also the homotopy category of a model category of the classical model structure on topological spaces."
Okay: yes, it's certainly true that Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a CW complex.
But I'm pretty sure it's also true that Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to (the geometric realization of) a simplicial complex. I was asking: where did you read that was false? What exactly was the claim there?
John Baez said:
Okay: yes, it's certainly true that Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a CW complex.
Well above it was discussed that groupoids are equivalent to CW complexes. This is because the infinity category of infinity groupoids is equivalent to the infinity category generated by the classical model structure on Top. If the infinity category generated by the classical model structure on Top is the infinity category of CW complexes, this makes sense. But if it's some smaller category like the infinity category of simplicial complexes, then it's no longer accurate to say that groupoids are equivalent to CW complexes, just as it's not accurate to say that groupoids are equivalent to topological spaces as a whole (since the infinity category of topological spaces where we make all weak equivalences into actual equivalences is equivalent to the infinity category of CW complexes, symbolizing how we "lose" information about topological spaces until the only remaining information is that invariant to weak equivalences)
Okay, sorry, I mistakenly thought you read, somewhere on the nLab, that
the homotopy category of the classical model structure on Top is equivalent to CW complexes and not simplicial complexes
I'm still trying to understand the negative claim here. Maybe you already said it, but can you just say which category, or infinity-category, with simplicial complexes as objects is not equivalent to some other category, or infinity-category? I don't want to know the reasoning; I just want to know what the claim is, so I can think about whether it's correct.
John Baez said:
Maybe you already said it, but can you just say which category, or infinity-category, with simplicial complexes as objects is not equivalent to some other category, or infinity-category? I don't need to know the reasoning; I just want to know what the claim is, so I can think about whether it's correct.
Sure. So I know for sure that the infinity category generated by the classical model structure on Top is equivalent to the infinity category of infinity groupoids. So whatever the objects of that infinity category are, they contain "the same information" as infinity groupoids.
My claim is that the infinity category generated by the classical model structure on Top is equivalent to the infinity category of CW complexes, but NOT to the infinity category of simplicial constructs. That way, we can say for certain that infinity groupoids are equivalent to CW complexes.
My claim is that the infinity category generated by the classical model structure on Top is equivalent to the infinity category of CW complexes, but NOT to the infinity category of simplicial constructs.
I assume you mean "simplicial complexes". Which infinity category of simplicial complexes are you talking about?
I just want to get a precise statement of your negative claim. You made precise the positive claim in one way (using homotopy categories, not infinity categories: Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a CW complex), and indeed its true. But I'm very suspicious of this negative claim, so I want to see it made precise.
John Baez said:
Which infinity category of simplicial complexes are you talking about?
The infinity category where objects are (spaces homeomorphic to) simplicial complexes (or more generally triangulable spaces), morphisms are continuous maps, and higher morphisms are higher homotopies.
John Baez said:
using homotopy categories, not infinity categories
Right, but that was just to simplify things a bit; the homotopy category is directly generated from some infinity category.
I like simplifying things a bit: it's easier for me to work with homotopy categories. Here's something I believe is true:
"Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a simplicial complex".
John Baez said:
"Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a simplicial complex".
If that's true, then you are also making the claim: "an infinity groupoid is not equivalent to a CW complex". You are making the counterclaim: "an infinity groupoid is equivalent to a triangulable space". Would you agree with this assessment?
@John Onstead not sure if this is the answer you are looking for, but it is a general fact about model categories that one can present the homotopy category by only considering the bifibrant i.e. both fibrant and cofibrant objects as its objects. In the classical model structure on topological spaces the CW complexes are the bifibrant objects (in fact they are the cofibrant objects, and all objects are fibrant). More in general, cofibrations are "cellular extensions". Now, these have the crucial property that they are closed under pushouts along arbitrary maps, which must be true of the class of cofibrations in any model category. But "(simplicial) cellular extensions of simplicial complexes" are not closed under pushouts so you cannot replace CW complexes with simplicial complexes in constructing a model structure.
It can be (and is) true both that every -groupoid can be presented by a CW complex and that every -groupoid can be presented by a simplicial complex.
Amar Hadzihasanovic said:
not sure if this is the answer you are looking for, but it is a general fact about model categories that one can present the homotopy category by only considering the bifibrant i.e. both fibrant and cofibrant objects as its objects
Ah! That is quite helpful. So given a model category, its homotopy category is equivalent to its subcategory on bifibrant objects. So under this definition, given CW complexes are the bifibrations in the classical model structure, the homotopy category would be equivalent to the category of CW complexes.
But I'm still confused by a few things. This still doesn't answer my original question: why does localizing by inverting every weak equivalence in Top only bring you down to the category of CW complexes, when every CW complex admits an additional weak equivalence to some simplicial complex? That is, why doesn't the localization process invert the weak equivalences that exist between every CW complex and some simplicial complex? I get that simplicial complexes are not closed under pushouts which disqualifies them from being cofibrant objects, that makes sense. But whether or not something is closed under pushouts has absolutely no bearing whatsoever on whether you can freely invert some morphism via localization.
Here's a syllogism: "Every top space is weakly equivalent to some CW complex. Every CW complex is weakly equivalent to some simplicial complex. Therefore, every top space is weakly equivalent to some simplicial complex". What is wrong about this syllogism? The transitive property holds since weak equivalences are closed under composition, so it can't be logically false.
Sorry I'm having such a tough time understanding this...
Morgan Rogers (he/him) said:
Indeed, finite directed graphs are an example of a cohesive elementary 1-topos over finite sets
Do you mean reflexive graphs, since irreflexive directed graphs don't necessarily satisfy one of the axioms of cohesion: pi0 (A x B) = pi0 (A) x pi0 (B)?
Mike Shulman said:
It can be (and is) true both that every -groupoid can be presented by a CW complex and that every -groupoid can be presented by a simplicial complex.
That makes sense, but here's something similar to what I'm asking: "can it be true both that every simplicial complex is presented by some -groupoid and that every CW complex is presented by some -groupoid"? If the answer is no, then definitively an infinity groupoid is NOT equivalent to a CW complex! And I think the answer is no because by definition a space homeomorphic to the geometric realization of an -groupoid is a triangulable space, not a CW complex. And recall the equivalence of infinity categories between infinity groupoids and the infinity category presented by the classical model structure is given in part by the geometric realization, which by the property of equivalence must of course be essentially surjective and have an inverse defined on all objects!
It doesn't make sense to ask whether something like a CW complex or a simplicial complex is "presented by" an -groupoid. There is no such thing as "the geometric realization of an -groupoid", at least not in the defined-up-to-equivalence sense of -groupoid. If you pick a particular way of presenting -groupoids, such as Kan complexes, then they will have realizations, but in that case the spaces you get via such geometric realizations will depend on the choice of such a presentation.
Well, the standard model structure on simplicial sets has Kan complexes as its bifibrant objects, and by the nerve construction every infinity groupoid corresponds to some Kan complex. I believe this is an equivalence; that is, every Kan complex is equivalent to the nerve of some infinity groupoid. So to refine my question: is every CW complex presented by some Kan complex?
John Onstead said:
John Baez said:
"Ho(Top) is equivalent to its full subcategory on the topological spaces homeomorphic to a simplicial complex".
If that's true, then you are also making the claim: "an infinity groupoid is not equivalent to a CW complex".
No. That's a vague claim, but I disagree with the most interesting interpretations of it.
John Baez said:
No. That's a vague claim, but I disagree with the most interesting interpretations of it.
Let's axiomatically (as in- don't think about) assume that the homotopy category of a model category is given by its subcategory on bifibrant objects. Then the bifibrant objects of topological spaces under weak homotopy equivalence are the CW complexes, and so our axiom tells us that the homotopy category is that of CW complexes. But earlier you made the statement that "Ho(Top) is equivalent to topological spaces homeomorphic to a simplicial complex". If we take this at face value to, we now have the following equivalences: and . By composition of equivalences, we get the following equivalence: . This equivalence is false (here is some evidence, more evidence), so one of the assumptions (either that the homotopy category is that of CW complexes, or that the homotopy category is that of simplicial complexes) must be wrong. Can you tell me which one it is?
John Onstead said:
But earlier you made the statement that "Ho(Top) is equivalent to topological spaces homeomorphic to a simplicial complex".
Just to be clear, I said Ho(Top) is equivalent to its full subcategory whose objects are spaces homeomorphic to simplicial complexes. It's not equivalent to the category of topological spaces homeomorphic to simplicial complexes and continuous maps between these. You just said I said something vagur that might mean either of these things, one true and one false.
If we take this at face value to, we now have the following equivalences:
If means the category of CW complexes (or spaces homeomorphic to CW complexes) and continuous maps, this is false.
and .
If means the category of simplicial complexes (or spaces homeomorphic to these) and continuous maps, this is also false.
The links show there are CW complexes not homeomorphic to simplicial complexes, which is interesting but compatible with what I'm claiming.
John Baez said:
If means the category of CW complexes (or spaces homeomorphic to CW complexes) and continuous maps, this is false.
How? Aren't CW complexes the bifibrant objects in topological spaces with respect to weak equivalences? Do you mean that the localization at weak equivalences somehow induce a different kind of map than the usual continuous ones?
Yes, localization always changes the notion of morphism, and the morphisms in Ho(Top) are radically different from the morphisms in Top! This is the main reason homotopy theorists introduced Ho(Top).
In fact Ho(Top) is not equivalent to any concrete category.
John Baez said:
Yes, localization always changes the notion of morphism, and the morphisms in Ho(Top) are radically different from the morphisms in Top! This is the main reason homotopy theorists introduced Ho(Top).
Ok, I guess that makes sense!
John Baez said:
In fact Ho(Top) is not equivalent to any concrete category.
That's odd... a localization of a category (or, for that matter, taking its subcategory on a particular kind of object, say bifibrant objects) embeds into that category. Since an embedding is a fully faithful functor and Top is concrete over Set, the composition of the embedding of Ho(Top) into Top and the forgetful functor into Set should be a faithful functor.
Oh dear! A localization of a category doesn't always embed into the category... I was thinking of reflective localizations but I just found out (doing a bit of re-reading) not every localization is reflective. My bad...
Still, this implies that the subcategory on bifibrant objects isn't actually the same as the localization at weak equivalences, since we know for sure the former embeds while the latter only does so in reflective scenarios. So that leaves me confused again about the true relationship between these.
Maybe it would help to have a map between the following categories: , (localization of Top at weak equivalences), (category of bifibrant objects in Top at weak equivalences), , , , , and so on. To determine what's an embedding, what's an equivalence, and what isn't either.
John Onstead said:
Still, this implies that the subcategory on bifibrant objects isn't actually the same as the localization at weak equivalences,
Right, they're quite different for Top.
I always think it's good to look at examples to develop intuition. In this case it can help you understand why topologists invented Ho(Top).
So, here's an exercise. If I is the closed unit interval, what is the cardinality of hom(I,I) inTop and what is the cardinality of hom(I,I) in Ho(Top)?
Then maybe repeat this exercise using the circle instead of I.
John Baez said:
So, here's an exercise. If I is the closed unit interval, what is the cardinality of hom(I,I) inTop and what is the cardinality of hom(I,I) in Ho(Top)?
That's a good exercise! I did some more reading and I think I know what the morphisms in Ho(Top) are: they are homotopy classes of morphism in Top. Implicitly, when you "collapse" the infinity category down to the homotopy category, you turn all 2-isomorphisms (the homotopies) into equalities, which explains this.
So if I is the unit interval, the cardinality of Hom(I, I) is infinity. Any choice of real number between 0 and 1 corresponds to a path in the unit interval (if the base of the path is fixed at 0), so there's uncountably many options. But as for homotopy classes of maps, and thus morphisms in Ho(Top), the cardinality should be 1. The interval is homotopy equivalent to the point.
That's right! So, you can begin to see how working in Ho(Top) simplifies things.
More precisely, homotopy classes of morphisms between bifibrant objects in Top correspond to morphisms between them in Ho(Top). (There are stronger results along these lines, but this is good enough for these "exercises".)
In fact the closed unit interval is isomorphic to a point in Ho(Top), so it's terminal.
The set of endomorphisms of the circle in Ho(Top) is countably infinite. Can you prove it's infinite? (That may be the easier part.)
John Baez said:
The set of endomorphisms of the circle in Ho(Top) is countably infinite. Can you prove it's infinite? (That may be the easier part.)
It's the integers, right? If you have a path that doesn't close around the circle, it can contract to a point, and we label it "zero". Then there's the path that loops clockwise around the circle and the one that loops counterclockwise, which we call 1 and -1. And then looping 2 times, 3 times, and so on.
Fun fact, it seems there's an entire article on nlab devoted to this!
Yes, it's the integers!
There's a nuance: the well-known "fundamental group" of the circle is the integers - but that's the set of base-point preserving homotopy classes of base-point preserving maps from the circle to itself. I wasn't asking about that. I was asking about the set of homotopy classes of maps from the circle to itself. But this too is the set of integers!
Often the two base-point preserving conditions change the answer, but in this case you can show it doesn't!
It's pretty easy to define the 'winding number' for arbitrary maps from the circle to itself and show it's invariant under homotopy. With a bit more work you can show it's a complete invariant, i.e. two maps with the same winding number are homotopy equivalent.
Ah, that's interesting!
Well, now that I understand things a bit better, I think I'm more prepared to go back to my original concern. It's comparing the categories of CW complexes and homotopy classes of maps between them (which we showed is equivalent to the category Ho(Top) ), and the category of simplicial complexes and homotopy classes of maps between them. But even now I'd still insist that these are not equivalent categories. This is because the objects of the first category are bifibrant objects in the model structure on Top and the objects of the latter are not, so their categories should not be equivalent (otherwise, under the principle of equivalence, we'd have to be able to interpret the simplicial complexes as the bifibrant objects of Top, which we shouldn't be allowed to do). In addition, if they were, then we wouldn't say that objects of Ho(Top) is equivalent to CW complexes, we'd say it's equivalent to simplicial complexes- which we don't do!
Every simplicial complex gives a CW complex by intepreting simplices as balls. We don't get all homeomorphism classes of CW complexes this way - your MathOverflow link gave a 'wild' CW complex that's not homeomorphic to any simplicial complex. But I believe any CW complex is weakly homotopy equivalent to a simplicial complex - if I'm wrong, I hope @Mike Shulman or someone will correct me.
If so, Ho(Top) should be equivalent to its full subcategory with simplicial complexes as objects, just as its equivalent to its full subcategory with CW complexes as objects.
That's what I expect.
I believe every topological space is weakly homotopy equivalent to a simplicial complex. Certainly every topological space is weakly homotopy equivalent to the geometric realization of a simplicial set. And my vague memory is that if you start with a simplicial set and do enough things to it like the category of simplices and the barycentric subdivision, you get a simplicial complex that's weakly homotopy equivalent to it.
Yes, it's a bit quirky for us to be using simplicial complexes, but someone says any -complex gives a simplicial complex when you barycentrically subdivide it twice. What's a -complex? Is it the same as a simplicial set?
John Baez said:
But I believe any CW complex is weakly homotopy equivalent to a simplicial complex
Yes, in fact, it's more than that- every CW complex is homotopy equivalent to a simplicial complex.
Mike Shulman said:
I believe every topological space is weakly homotopy equivalent to a simplicial complex.
Maybe this is what I was trying to demonstrate in my syllogism above: "Every space is weakly equivalent to a CW complex. Every CW complex is weakly equivalent to a simplicial complex (since a homotopy equivalence is a weak equivalence). Therefore, by the property that weak equivalences are closed under composition, every space is weakly equivalent to a simplicial complex". My question at the beginning was: why then does localizing at the weak equivalence not yield simplicial complexes instead of CW complexes? But if Ho(Top) is equivalent to its full subcategory with simplicial complexes as objects, this resolves the problem- it turns out that localizing at weak equivalences does yield simplicial complexes.
This also clearly implies then that the principle of equivalence is violated- the bifibrant objects can be equivalent to a class of objects that are patently not the bifibrant objects (IE, not closed under pushouts). Thus in some ways this concept is "evil". I find that massively uncomfortable.
But this brings me back to the notion of "equivalence" in infinity categories. Under the above observation, we have a three-way equivalence between the simplicial localization of Top (whose objects are ALL topological spaces), the simplicial localization of CW complexes, and the simplicial localization of simplicial complexes. However, in spite of all three of these infinity categories being equivalent, we only ever say that Kan complexes are equivalent to CW complexes- not all topological spaces, but not simplicial complexes either. What is the explanation for this? Why can't we say that Kan complexes are equivalent to all topological spaces when there exists a category of all topological spaces equivalent to that of CW complexes? Likewise, how can we be sure that Kan complexes consist of enough information to fully define any CW complex, thus justifying not saying that Kan complexes and simplicial complexes are equivalent?
I think a -complex is another name for a semi-simplicial set.
we only ever say that Kan complexes are equivalent to CW complexes- not all topological spaces, but not simplicial complexes either.
I don't know who this "we" is that you're referring to. I certainly say that the homotopy category of Kan complexes (or simplicial sets) is equivalent to the homotopy category of topological spaces.
And I would say it about simplicial complexes too, if I ever had any reason to.
John Onstead said:
However, in spite of all three of these infinity categories being equivalent, we only ever say that Kan complexes are equivalent to CW complexes- not all topological spaces, but not simplicial complexes either.
Who is "we"? The [[homotopy hypothesis]] is commonly stated in terms of topological spaces.
You could state it for simplicial complexes; if people don't tend to do that, it's probably because simplicial complexes are less popular nowadays than simplicial sets or Kan complexes, at least in homotopy theory.
I think there are a host of other kinds of spaces that would work too, but they are even less popular than simplicial complexes. For example, I bet Ho(Top) is equivalent to its full subcategory on the cubical complexes.
John Onstead said:
This also clearly implies then that the principle of equivalence is violated- the bifibrant objects can be equivalent to a class of objects that are patently not the bifibrant objects (IE, not closed under pushouts). Thus in some ways this concept is "evil". I find that massively uncomfortable.
Perhaps by "IE" you mean "e.g."? But bifibrant objects are not in general closed under pushouts. They are closed under homotopy pushouts, as is any category that's equivalent to them. I'm not sure what form of equivalence principle you think is being violated.
@John Onstead What you need to keep in mind is that, to define the homotopy category, you only need a class of weak equivalences; this should tell you that the homotopy category has no memory at all of (co)fibrants and (co)fibrations or any of the additional structure of a model category.
What a model structure gives you is (among other things) a particular presentation of the homotopy category (and, with more modern machinery, an -category), as well as better criteria to decide whether two presentations are equivalent in the sense that they present the same homotopy category up to categorical equivalence.
And the point I made about cofibrations and the need for them to be closed under pushouts indicates that you cannot have a model structure on the category of topological spaces and continuous maps whose cofibrations are "simplicial complex cellular extensions", but you can with CW complexes and their cellular extensions.
But in a certain sense the Kan--Quillen model structure on simplicial sets tells you that you can make it work if you restrict to the category of simplicial sets and their morphisms, which you can think of as a non-full subcategory of topological spaces and continuous maps via "geometric realisation", and you do end up having the "same" homotopy category.
(Generalising simplicial complexes to simplicial sets can be seen as a way to "put in the missing colimits")
So again this is about presentations of homotopy categories and you shouldn't expect facts about presentations to be invariant under equivalence of the presented objects!
For example, some presentations of a group have the property that you can "orient equations" in such a way that applying equations always in the same direction solves the "word problem" for that group by reducing everything to a normal form, but some other presentations do not.
So saying "but every object of the homotopy category can be presented by a simplicial complex" is like saying "there is a presentation of a certain group whose generators are taken from the set ". There is no guarantee that any of the nice properties of another presentation will hold for this (or any) presentation with those generators.
Amar Hadzihasanovic said:
So again this is about presentations of homotopy categories and you shouldn't expect facts about presentations to be invariant under equivalence of the presented objects!
So saying "but every object of the homotopy category can be presented by a simplicial complex" is like saying "there is a presentation of a certain group whose generators are taken from the set ". There is no guarantee that any of the nice properties of another presentation will hold for this (or any) presentation with those generators.
I think I see what you are saying. Just as two isomorphic groups can have non-isomorphic presentations, two equivalent infinity categories have non equivalent presentations. I was trying to figure out what the "correct" way to interpret the objects in an infinity category for some model category was, but maybe there is no "correct" way to do this.
Mike Shulman said:
I don't know who this "we" is that you're referring to. I certainly say that the homotopy category of Kan complexes (or simplicial sets) is equivalent to the homotopy category of topological spaces.
John Baez said:
Who is "we"? The [[homotopy hypothesis]] is commonly stated in terms of topological spaces
Yes, I guess I was just thrown off by our discussion a few days ago where it was stated that infinity groupoids (I guess, in the form of Kan complexes) were not equivalent to topological spaces. I understand this now to be in reference to the 1-categorical Top- but the equivalence does hold true if we think about the homotopy category (or infinity category) for Top at the weak equivalences.
I think I was just worried about saying something like "Kan complexes are equivalent to CW complexes" if such a statement didn't also hold in the 1-categorical case. Because I guess it would sound misleading if you didn't already know we were working in the homotopy/infinity category setting. Just as misleading as the original version of the nlab article you corrected which stated that topological spaces were equivalent to Kan complexes!
Mike Shulman said:
I think a -complex is another name for a semi-simplicial set.
Right, I later looked it up.
For anyone who wants proofs of claims in this conversation, here's one:
In Hatcher's Algebraic Topology, Theorem 2C.5:
Every CW complex is homotopy equivalent to a simplicial complex.
I think I was just worried about saying something like "Kan complexes are equivalent to CW complexes"
Good! Unless you're among good friends who always interpret your words charitably, you really need to say
"The category of Kan complexes is equivalent to the category of CW complexes"
or
"The homotopy category of Kan complexes is equivalent to the homotopy category of CW complexes"
or
"The model category of Kan complexes is equivalent to the model category of CW complexes"
or
"The -category of Kan complexes is equivalent to the -category of CW complexes"
or something like that, depending on what you actually mean, and hope the reader knows which (higher) morphisms and/or model structures you're talking about.
(I'm not claiming all the above sentences are true.)
Hi! Thanks for all the help before... I can get frustrated with stuff when I don't understand it so I appreciate your patience! Now, continuing on...
As I was learning more about doing homotopy theory in a category with weak equivalences, I noticed a peculiarity: the definitions took "map space" and "cylinder" objects as fundamental, and nowhere to be seen was the notion of an "interval". This was really odd since in Top map space and cylinder objects are less fundamental than the interval, not to mention that intervals define paths most directly and paths are the central focus of all of homotopy theory and thus its extension to category theory. So I wanted to know: where were the interval objects in the definition of a category with weak equivalences?
Well anyways I looked up "interval object" on nlab and sure enough there was a real article for it. But, strangely, it didn't really mention them much in the context of categorical homotopy theory. There were some mentions of if you already had a category with weak equivalences and an interval object in that, but then it started talking about "Trimble n-categories" and I have no idea what those are.
But that wasn't what I'm interested in: I want to know if it's possible to use interval objects as an alternative to map space and cylinder objects (or perhaps complementary to them if we're in a category with finite products) in "doing" categorical homotopy theory in a category with weak equivalences. If the answer was to be yes, then I'd need to know: given some interval object in some category, what is the canonical weak equivalence structure associated to it such that the homotopy theory generated by that interval object is the same as the categorical homotopy theory of the resulting category of weak equivalences?
The obvious choice is to lift the definition of "weak equivalence" straight out of Top and right into a generic category with an interval object. For axiom 1 of a weak equivalence in Top, it's pretty straightforward: given terminal object and an interval object with its defining maps, define a "set of path components" of to be the coequalizer in Set of . A morphism is then a weak equivalence candidate if the resulting path components functor sends to a bijection.
But axiom 2- that all homotopy groups must be isomorphic- seems harder to lift. As discussed before, you generally first need a weak equivalence structure to define a notion of homotopy and thus homotopy group, but here we have to use some notion of homotopy group to define the weak equivalences in the first place! My question is: how can this be done? And more generally, is this the best way to define a canonical weak equivalence structure for an arbitrary generic ("plain") interval object? Are there any alternatives for doing so? Thanks!
The general notion of cylinder allows for the cylinder to not be obtained by "tensoring" with some interval object, so it is taken to be more fundamental because it is strictly more general, and one can define notions of homotopy relative to any cylinder object. Dually for path objects.
You always get an interval as a cylinder on the terminal object, but it may not be obvious how to get a "good" cylinder from an interval.
Although if you have a monoidal structure (which could be the cartesian monoidal structure), or more generally an enrichment, then you can try to define a cylinder by tensoring with an interval, as you do in spaces. This doesn't always work, but it frequently does.
Amar Hadzihasanovic said:
You always get an interval as a cylinder on the terminal object, but it may not be obvious how to get a "good" cylinder from an interval.
Let me check for myself to see if this is true... a cylinder object for a terminal object would be a factorization . There's two projection morphisms , so composing indeed does give two morphisms from to its cylinder. This does define an interval object. So indeed there's a good way to, given a category with equivalences, define an interval object from it.
But my question was two-sided. Not only did I want to get an interval out of the notion of cylinder or map space object, I wanted to "go backwards" and get to a notion of category with weak equivalences from an interval object. As suggested this might work if the category is monoidal, but I don't want to assume any additional structure, and I want to define a construction that works for all conditions no matter what.
I think you're out of luck then.
Mike Shulman said:
I think you're out of luck then.
Why do you give up so easily? Math is supposed to be a fun exploration of possibilities, isn't it?
For any category (even without a terminal object!) one can generate . Let an "interval object in C" be an object in the image of with two morphisms from the terminal object of . Since preserves limits we never have to worry if the terminal object of doesn't align with that of . Now has all products and coproducts and an interval object, so we seem set to be able to define a model structure based on that.
Ok, after more reflection... Let be a category with finite products, a terminal object , and an internal hom (maybe it's the presheaf category for some other category, but let's brush that aside for now). Equip with an interval object of the form . Now define a "cylinder object of " to be and define a "map space object of " to be . Here's my big question: What is the step by step procedure to generate the corresponding infinity category for this setup?
John Onstead said:
Mike Shulman said:
I think you're out of luck then.
Why do you give up so easily? Math is supposed to be a fun exploration of possibilities, isn't it?
Mike tends to give up when he knows something won't work. That is not a bad thing. :upside_down:
I didn't read much of this, but Cisinski model structures are reasonably close to answering to the question of how to produce a homotopy theory just from an interval object.
Without any sort of monoidal structure on the category? That's what John O. wanted.
Cisinski doesn't require a pre-existing monoidal structure such that the cylinder on coincides with some if that's responsive. You do need to be able to write down so you can ask to factor it via the cylinder.
But the Cisinski theory only applies on presheaf categories, which maybe John will find much too restrictive.
John Baez said:
Mike tends to give up when he knows something won't work. That is not a bad thing.
I understand. I just happen to be a very stubborn person sometimes :)
Kevin Carlson said:
But the Cisinski theory only applies on presheaf categories, which maybe John will find much too restrictive.
Not exactly, I actually like to think that every category is secretly a presheaf category in disguise due to the Yoneda embedding.
Though I do have a question about something. I'm currently reading the definition of Cisinski model structure and it seems close to what I was actually trying to define above. The only difference is that for some reason the morphism needs to be a monomorphism? While the presheaf category condition is ok, this seems a little too restrictive... is there any way to remove this restriction and still get a coherent theory (and even if you can't get a model structure, maybe you could at least get a category with weak equivalences?)
Can you explain why this seems too restrictive?
The monomorphisms are the cofibrations in Cisinski model categories so they're always going to play a major role.
I just happen to be a very stubborn person sometimes :)
That's actually okay so I shouldn't tease you about it (except when I need to).
Kevin Carlson said:
Can you explain why this seems too restrictive?
Right. I had to read more on the subject, that was just my first pass take... Now from what I see a Cisinski model structure is that generated by a presheaf category equipped with some class of "anodyne extensions" as well as an "elementary homotopical datum" endofunctor. A special case of an elementary homotopical datum functor seems to be the cartesian product functor with an interval object, which is exactly what we want. I could be wrong, but the monomorphism condition, in this case, seems like it comes from the requirement that the interval object be "disjoint" in that the pullback of the two morphisms from the terminal object is the initial object. In hindsight that doesn't seem too restrictive, but I guess I was just wondering if that's as good as you can get when it comes to relaxing conditions on the interval object.
After going over things again I think that's probably as close as I'll get... Thanks for the help!
Kevin Carlson said:
I didn't read much of this, but Cisinski model structures are reasonably close to answering to the question of how to produce a homotopy theory just from an interval object.
I'm not sure what you're thinking about here; didn't you say in your next comment that Cisinski uses a cylinder, not just an interval?
John Baez said:
Mike tends to give up when he knows something won't work. That is not a bad thing.
Heh. In this case it's not exactly knowing that it won't work, but thinking it seems unlikely to give anything useful, and not having any motivating examples. I would ask, why do you want to obtain a homotopy theory from just an interval object without even cartesian products? Is there some category you're interested in that has an interval object but not cartesian products?
Oh, okay - that's different, and it's good to note the difference. I thought you meant it couldn't work.
Sorry, I think I got my memory or timings confused; I should take more time writing my replies. Looking back, I realize at the moment when I said "I guess you're out of luck then" I did think that it couldn't work. The presheaf-category approach hadn't occurred to me. Now I will admit that that might give something, but I'm doubtful that it'll be useful, as in my last comment.
Mike Shulman said:
I'm not sure what you're thinking about here; didn't you say in your next comment that Cisinski uses a cylinder, not just an interval?
Yes the Cisinski model structure is based on a "elementary homotopical datum" which itself contains the specification of a "cylinder object endofunctor" that sends an object to a cylinder object in a functorial way. But the point being made was that, given any interval object in the presheaf topos, the corresponding endofunctor satisfies all the properties of being a cylinder object endofunctor and thus is a valid component of a elementary homotopical datum.
Mike Shulman said:
I would ask, why do you want to obtain a homotopy theory from just an interval object without even cartesian products? Is there some category you're interested in that has an interval object but not cartesian products?
I'm not a mathematician who has to solve problems, so I don't see doing mathematical things in terms of a useful/not useful dichotomy. Instead I'm interested in these concepts as a matter of principle- I want to understand the relationships between different concepts at the deepest/most general level such relationships exist. So I suppose I'm after more of a philosophical rather than practical matter- at least in this case. Anyways I hope this gives some insight into my thought process!
Guhh, I am not explaining myself at all well. I'm sorry, my brain always turns to mush during the last 2 weeks of the semster after Thanksgiving. Probably you should just ignore everything I've said today.
If I can try one more time: based on my experience of a fair number of examples and theorems, I am doubtful that "categories with an interval object but without finite products or another monoidal structure" is a conceptually useful level of generality. I don't think I've ever encountered a category like that for which the induced "homotopy theory" was interesting. The closest thing I can think of is the [[simplex category]] which has an "interval object" , but in that case the homotopy theory we care about is really that of the presheaf category itself, not that of the domain category induced by the cylinder functor on the presheaf category.
But it's not usually a good use of time to argue about whether something else is a good use of time, and you're absolutely right that math is supposed to be fun, so if you're having fun, more power to you! And maybe I'll come around after a while.
I was trying to help, but maybe that was the kind of help we all can do without.
Mike Shulman said:
I'm sorry, my brain always turns to mush during the last 2 weeks of the semster after Thanksgiving
I can relate!
Mike Shulman said:
I was trying to help, but maybe that was the kind of help we all can do without.
No problem, I very much appreciate all your help!
Still on the subject of interval objects, it seems there's some relation between the notion of an interval object and that of a "fundamental infinity category" as detailed on this nlab page. However, this page mentions this subject isn't fully developed at the bottom, so I am unsure if I should accept what is written on there.
In any case, here are some questions I had about this: first, what is a fundamental infinity category- the highest level of "fundamental" is a fundamental infinity groupoid, right? Second, the page uses things unfamiliar to my previous study of model categories. It refers to an interval object somehow generating an operad (how does it do this?) and somehow this operad is an internal infinity category- whatever that is? I guess if I had to ask one question about this, it would be: how, precisely, does this setup tie into (and perhaps determine) a model structure on the category at hand such that this notion of "internal infinity category" can be more easily discerned by looking at the resulting infinity category for that model structure?
Ok, maybe that subject in general hasn't developed enough for these answers to be readily available...
Switching gears slightly, I want to better understand modeling of higher categories in general. Starting with the base case, the nerve of any notion of -groupoid lands in in its subcategory on Kan complexes. In the infinity category setting with the standard model structure on , this nerve gives rise to an equivalence of categories between Kan complexes and infinity groupoids. Therefore, enriching in the part of on Kan complexes is basically the same as enriching in infinity categories, which is why these "simplicially enriched categories" are good representations of infinity categories. Moving one level up, the nerve of any notion of infinity category (that is, -category) lands in the subcategory of on "quasicategories", and an analogous relationship ensues between infinity categories and quasicategories to that of infinity groupoids and Kan complexes. Interestingly, one of the notions of infinity category this works for is the case of simplicially enriched categories, which allows us to view simplicially enriched categories as quasicategories that are "semi-strict".
I have a few resulting questions from this. First, what if, instead of enriching in on Kan complexes, one weakly enriches in the full infinity category generated by the standard model structure on whose objects are, essentially, all Kan complexes? Does this "weak enrichment" (analogous to how bicategories are "weakly enriched" in ) make the weakly enriched simplicially enriched category into a full quasicategory? My second question is: can the process above be repeated? That is, can you enrich in on quasicategories to make an -category, and in which subcategory of does the nerve of these quasicategory-enriched categories land (and so on)? My last question is: given a quasicategory-enriched category/-category, what structure can I put on a 1-category (or maybe this time a 2-category) that will make that a good model? I guess I'm looking for a completion of the analogy "categories with weak equivalence are to Kan complex enriched categories/-categories, as categories with ??? are to quasicategory enriched categories/-categories". Thanks!
Those are good questions!
Yes, you can get a notion of -category by weakly enriching in Kan complexes. Depending on how you formulate a notion of "weak enrichment" this might be an [[A-infinity category]] or a [[Segal category]]. These notions are different from both quasicategories and simplicially enriched categories, but are also models for -categories.
And yes, you can enrich in quasicategories, either strictly or weakly, to get models of -categories. The simplicial nerve of an -category, including an -category, is a [[weak complicial set]]. However, things are not as well-behaved when you go higher than quasicategories, because while Kan complexes and quasicategories are full subcategories of simplicial sets, weak complicial sets are not: part of the data of a WCS is the collection of higher cells that are "equivalences", and these need not be preserved by arbitrary simplicial maps. This is why the [[model structure for weak complicial sets]] is on the category of [[stratified simplicial sets]] instead of simplicial sets.
Finally, I don't know offhand of anything analogous to a category with weak equivalences that can be used to present -categories. I have a vague sense that I might have seen something like this before, like maybe categories with two different classes of "weak equivalences"? But I can't remember what it was called or where to find it, so maybe I'm making it up.
I guess one natural thing to try would be a 2-category with weak equivalences. But I don't remember whether it's known that those suffice to present all -categories.
Thanks! That's very interesting!
Mike Shulman said:
Depending on how you formulate a notion of "weak enrichment" this might be an [[A-infinity category]] or a [[Segal category]].
A-infinity categories were actually one of the ones mentioned in the "fundamental infinity category" article I was trying to understand earlier, so I read the page for it. It seems these are only models for stable infinity categories- is that true?
Also, I'm curious to know of any connections or relationships between Segal categories and quasi-categories. For instance, are the two notions different because the weak enrichment "overshoots" in some capacity- that is, does it make certain properties even weaker than in a quasicategory? Or does it weaken something other than the part of a simplicially enriched category that makes it "semi-strict" over the quasicategories?
Mike Shulman said:
Finally, I don't know offhand of anything analogous to a category with weak equivalences that can be used to present -categories.
I was interested in this because of the analogies between Cat and homotopy theory, via the "model structure on 1-Cat". For instance I was wondering if there was any way I could put some model structure on 1-Cat to eventually get 2-Cat out of it. With just the usual model structure on 1-Cat, the best you can do is get the category of categories, functors, and natural isomorphisms between functors, not all natural transformations.
John Onstead said:
A-infinity categories were actually one of the ones mentioned in the "fundamental infinity category" article I was trying to understand earlier, so I read the page for it. It seems these are only models for stable infinity categories- is that true?.
No, they are models of arbitrary -categories, as Mike said.
The [[A-infinity operad]] is the operad that controls the coherence laws for associativity, like the the pentagon identity for the associator, the law governing the 'pentagonator' as we move up to the next rung of the ladder, and so on. As we climb up this ladder we get all the [[Stasheff polytopes]], which were discovered when Stasheff asked the question: what structure does a topological space get if it's equipped with a homotopy equivalence to a topological monoid? It becomes an [[A-infinity-space]], which is an algebra of the A-infinity operad. Later this idea was used, often in the simplicial context, to develop the concept of A-infinity category.
If you're interested in the stable analogue of this story, you want the [[E-infinity operad]], E-infinity spaces, and so on.
It's true that of all literature on " categories" is on the algebras for the chain complex-enriched operad, which is a stable kind of thing, though. I don't remember, from when I checked on it a decade ago, anybody every having written anything like a paper constructing a model structure on categories in spaces Quillen equivalent to quasicategories.
I guess I've focused a lot on the % which treats the operad as a topological or simplicial operad. @Todd Trimble explained a really nice simplicial construction of the operad using the bar construction.
I see now that Bergner's survey of (infinity,1) categories has naught but a brief mention of -categories. Since you can strictify them to simplicially enriched categories, she talks instead about those!
It's a real shame she does so, though! I think it's one of the most natural ways to think about an -category with all the weakness explicitly there and yet semi-manageably parameterized.
Not her fault, though, but that of the collective.
For a while, at least, it was conjectured but not proven that -categories for a topological or simplicial operad were a model for -categories equivalent to the nonalgebraic ones. I think that was the case when Bergner wrote her survey. Is it now proven somewhere?
John Onstead said:
Also, I'm curious to know of any connections or relationships between Segal categories and quasi-categories. For instance, are the two notions different because the weak enrichment "overshoots" in some capacity- that is, does it make certain properties even weaker than in a quasicategory? Or does it weaken something other than the part of a simplicially enriched category that makes it "semi-strict" over the quasicategories?
The two notions are not different up to homotopy: both have model structures that are Quillen equivalent. As for why they "look" different, one way to describe it is that quasi-categories are a homotopification of the one set of morphisms definition of category while Segal categories are a homotopification of the many sets of morphisms definition of category.
Have you read the Cheng-Lauda illustrated guide book to higher categories? You may find it helpful to get intuition for the various ways of modeling higher categories and how they relate.
Mike Shulman said:
For a while, at least, it was conjectured but not proven that -categories for a topological or simplicial operad were a model for -categories equivalent to the nonalgebraic ones. I think that was the case when Bergner wrote her survey. Is it now proven somewhere?
No, as far as I know that's never been proven, which makes me slightly sad though not nearly sad enough to try to do it myself.
With just the usual model structure on 1-Cat, the best you can do is get the category of categories, functors, and natural isomorphisms between functors, not all natural transformations.
That's true if you forget everything about 1-Cat except that it's a 1-category with a model structure. But in fact that model structure is enriched over itself, i.e. it is a "model 2-category", and when regarded in that way it presents the 2-category of categories, functors, and all natural transformations.
Kevin Carlson said:
No, as far as I know that's never been proven, which makes me slightly sad though not nearly sad enough to try to do it myself.
Any graduate students looking for a thesis problem, take note!
This would be a good thing to put on the list of open problems in category theory that was discussed in another thread.
I was interested in this because of the analogies between Cat and homotopy theory, via the "model structure on 1-Cat". For instance I was wondering if there was any way I could put some model structure on 1-Cat to eventually get 2-Cat out of it.
I don't know of such a model structure. It's not inconceivable that it might exist, but I do see some obstacles to overcome. For instance, there's the [[Thomason model structure]] on 1-Cat whose homotopy theory is equivalent to -groupoids, where a category is defined to present the same -groupoid as its simplicial nerve. So since 2-categories also have simplicial nerves, you could try to declare that a 1-category presents the same 2-category as its simplicial nerve and get a model structure on 1-Cat that way. But the issue of simplicial maps not preserving thin 2-cells that arose for weak complicial sets is probably going to rear its head here too, and it's not clear to me whether nerves of 1-categories would be sufficiently general to present all 2-categories.
John Baez said:
The [[A-infinity operad]] is the operad that controls the coherence laws for associativity, like the the pentagon identity for the associator, the law governing the 'pentagonator' as we move up to the next rung of the ladder, and so on.
This is very different from the simplicial stuff that I've learned infinity categories through! I guess that justifies the distinction of a "geometric" vs "algebraic" definition of an infinity category. I'll have to learn more about this when I can!
Mike Shulman said:
Have you read the Cheng-Lauda illustrated guide book to higher categories? You may find it helpful to get intuition for the various ways of modeling higher categories and how they relate.
No, I'll have to check that out!
Mike Shulman said:
I don't know of such a model structure. It's not inconceivable that it might exist, but I do see some obstacles to overcome.
I asked this as part of a tangential line of inquiry that probably should deserve its own thread. It's on the matter of the following problem: given some random 1-category, what are some of the nontrivial ways it can be extended to a 2-category with the same objects and morphisms? Obviously the classical example of a solution to this problem is the extension from 1-Cat to 2-Cat. I was hoping model structures might be a way this can be done in general (since they can extend to the infinity case, just only with all higher morphisms being equivalences), but it seems it might be a dead-end! Though I do think this is something worth following up on in the future!
Really quick additional question. Given a 1-category , the category and its global model structure present the infinity category of infinity presheaves on . However, what if we wanted to itself be an infinity category- how would we present its infinity category of infinity presheaves?
For this question, let be a category with weak equivalences that models the infinity category . Is there a way, in terms of , to write down a model category that presents the infinity category of infinity presheaves on ? For instance, some sort of modification we could do to the global model structure on ? Thanks!
It should work to left Bousfield localize a global model structure on at the Yoneda images of the weak equivalences in .
However, another approach that's sometimes better-behaved is to choose a simplicially enriched category that presents and just consider a global model structure on the category of simplicially enriched presheaves.
Mike Shulman said:
It should work to left Bousfield localize a global model structure on at the Yoneda images of the weak equivalences in .
Thanks, I was suspecting something like that...
Mike Shulman said:
However, another approach that's sometimes better-behaved is to choose a simplicially enriched category that presents and just consider a global model structure on the category of simplicially enriched presheaves.
That makes sense. But this could just be me... I prefer doing model category theory with 1-categories only. It seems almost redundant to put a model structure on a simplicially enriched category since a simplicially enriched category, all by itself, is already a model for an infinity category- you don't need to complicate things further by putting yet another model structure on top of it (when the whole point of a model category is to simplify the craziness of infinity categories)! But I guess I can see how in some cases it might be a useful shortcut to doing things with 1-categories only, as in this example.
For my next question, I was looking into loop space objects and noticed they were defined in an infinity category by the powering between the circle object in infinity groupoids and some object in the infinity category. The generalized elements of this space act like loops in the object. This reminded me of another thing: the "path space object" of a category with weak equivalences whose generalized elements are supposed to be interpreted as paths in an object. This made me confused: what about the object you get when you power the interval object in infinity groupoids and some object in the infinity category?
It can't be always true that these are the same, right? The path space object as defined by factorization of the diagonal in a category with weak equivalences becomes equivalent to its object in the infinity category setting (since the factorization must be into a weak equivalence between it and the object). If these two were to be the same, then the powering between the interval and an object in an infinity category would always have to end up equivalent to the object itself. So my question is: what is going on here? How do these two notions precisely relate to one another and which one of the two do we consider the "actual" space of paths?
John Onstead said:
It seems almost redundant to put a model structure on a simplicially enriched category since a simplicially enriched category, all by itself, is already a model for an infinity category- you don't need to complicate things further by putting yet another model structure on top of it (when the whole point of a model category is to simplify the craziness of infinity categories)! But I guess I can see how in some cases it might be a useful shortcut to doing things with 1-categories only, as in this example.
If you accept that model categories are just presentations of (certain) higher categories, then it seems a very common thing to present new algebraic structures by feeding them other algebraic structures without necessarily having to "flatten" everything to presentations. When you take a semidirect product of groups, do you insist that the definition be reduced to generators and relations of the factors?
John Onstead said:
the powering between the interval and an object in an infinity category would always have to end up equivalent to the object itself
Yep, that's true. The interval is contractible, so it's homotopy equivalent to a point, so powering by it doesn't do anything up to homotopy.
The circle, by contrast, is not contractible, and so powering by it does something nontrivial, producing the loop space object.
Mike Shulman said:
Yep, that's true. The interval is contractible, so it's homotopy equivalent to a point, so powering by it doesn't do anything up to homotopy
I see, good to know!
While I'm at it already with "dotting my I's and crossing my t's", I want to make sure that the infinity category for some category with weak equivalences is really consisting of the things that we want it to consist of. First off, in any category with weak equivalences with coproducts, we can define a notion of cylinder object and with it left homotopy (dually for categories with products, map space objects, and right homotopy). Given a left homotopy defined by a cylinder object in a category with weak equivalences (IE, by the two sided commutative triangle diagram), is it really true that it corresponds to a higher morphism in the resulting infinity category? That is, are we truly "right" to interpret the 2-morphisms of an infinity category as the homotopies as defined by the cylinder objects in a category with weak equivalences that presents it? The construction of the simplicial localization is so convoluted I haven't even begun to understand it, so I have no way of checking for myself on this!
Secondly, once we define a notion of homotopy (via cylinder objects in a category with weak equivalences), we can move forward and define a notion of "homotopy equivalence". This would be by the standard definition of homotopy equivalence- if two morphisms and existed such that is homotopic to the identity of and is homotopic to the identity of . My question is: what is the precise relation between these "homotopy equivalences" and the actual weak equivalences in our category with weak equivalence structure? For instance, are we able to start with a selection of weak equivalences, generate cylinder objects from it, generate homotopies from that, generate homotopy equivalences from that, and then use those to go all the way back and recover the weak equivalences we started with? Thanks!
It's true that if you have a cylinder object in a category with weak equivalences, and a homotopy with respect to it, you get a 2-morphism in the -category it presents. Suppose is a cylinder for , with and a weak equivalence such that . Then given a homotopy such that and , once we map into the -category, becomes an equivalence with inverse-up-to-homotopy , and so we have an induced 2-morphism
It then follows that any "homotopy equivalence" defined in terms of cylinders will become an equivalence in this -category.
However, for an arbitrary category with weak equivalences and coproducts:
The various extra structures that people assume in addition to just a class of weak equivalences, up to and including a Quillen model structure, are intended to partially or wholly solve those problems.
Mike Shulman said:
- An object may have more than one cylinder, and the notion of "homotopy" might be different for different cylinders.
- A homotopy using any cylinder will give rise to a 2-morphisms, but not every 2-morphism might arise from a homotopy for any cylinder.
A few days ago you mentioned that any homotopy generated with respect to one cylinder object is always equivalent to a homotopy generated by any other cylinder object, so that it doesn't end up "mattering" which one you choose. Is the same the case for the second point? That is, even if there are 2-morphisms that do not arise directly or strictly from a cylinder, that all 2-morphisms are equivalent to one that is? I think that ought to be true, in which case it "doesn't matter" if there are more 2-morphisms than the ones the cylinders generate since any such morphisms is equivalent to one that is.
John Onstead said:
A few days ago you mentioned that any homotopy generated with respect to one cylinder object is always equivalent to a homotopy generated by any other cylinder object
If I said that about an arbitrary category with weak equivalences, it was wrong. That's true in a model category, but not in an arbitrary category with weak equivalences.
Mike Shulman said:
If I said that about an arbitrary category with weak equivalences, it was wrong. That's true in a model category, but not in an arbitrary category with weak equivalences.
How not? The model structure does not determine the infinity category, the weak equivalence structure does.
Not every category with weak equivalences admits any model structure.
Although, hmm, actually, there's a subtlety even in the case of model categories: in order to get the "all cylinders are equal" property, you need to require that the inclusion map is a cofibration. Otherwise, itself counts as a cylinder object, and the only homotopies for that cylinder are identities.
This is certainly more confusing than I thought it'd be!
How about this. What if we had a category with weak equivalences, generated cylinder objects, used those to define left homotopies, used them to define homotopy equivalences, and then took all those as a "class of weak equivalences". In this way, every category with weak equivalences, through the homotopy equivalences you can define with it, induces yet another category with weak equivalences structure. I think this one should be better behaved. Of course, now it is true by definition that all homotopy equivalences are the weak equivalences. Is there any other "nice" things about this construction?
This setup, if it's possible, reminds me a little of the distinction between the Strom model and classical model structure on topological spaces. I can almost imagine that if you started with the classical model and its weak equivalences in Top and chose the usual cylinder object, you could rederive the Strom model structure like how I described above. And as you mentioned above, the Strom model structure is precisely the one you get by making all homotopies into the higher morphisms, and all homotopy equivalences into the actual infinity equivalences. Anyways just a thought I had!
If you just start with an arbitrary category with weak equivalences, and an arbitrary choice of cylinder objects, I don't think the "left homotopies" you get will satisfy any good properties like being composable or whiskerable. In particular, therefore, the "homotopy equivalences" won't satisfy the 2-out-of-3 property to be a collection of weak equivalences. I also don't immediately see why the cylinder objects for the original weak equivalences would also be cylinder objects for the homotopy equivalences, so it's not clear that the "homotopy equivalences" defined from the second structure would be the same as those defined from the first structure... even if you ignore the fact that you had to choose particular cylinder objects and the construction depended on those.
I think the lesson is that an arbitrary category with weak equivalences really is no good for doing "internal" homotopy theory with, using e.g. cylinders, homotopies, path objects. etc. If you want to do that, you really need more structure like a [[cofibration category]] or a Quillen model structure. The only sensible way a category with weak equivalence presents an -category is by the simplicial localization.
Mike Shulman said:
If you just start with an arbitrary category with weak equivalences, and an arbitrary choice of cylinder objects, I don't think the "left homotopies" you get will satisfy any good properties like being composable or whiskerable. In particular, therefore, the "homotopy equivalences" won't satisfy the 2-out-of-3 property to be a collection of weak equivalences.
I don't think that's true. A homotopy is a symmetric/invertible entity, which we know it must be given that they form the higher morphisms of an infinity category (all of which must be invertible). So two homotopies are trivially composable by the "transitive property": if has some homotopy to , and has some homotopy to , then must have a homotopy to . Such a homotopy is the composition.
In any case, I guess the main "gist" of all this is that the notion of "homotopy" defined using the machinery of a weak equivalence category is not the same as the notion of "homotopy" in the actual corresponding infinity category. Which leaves me wondering what the point of defining a "homotopy" within a weak equivalence category even is (if there even is a point!)
John Onstead said:
In any case, I guess the main "gist" of all this is that the notion of "homotopy" defined using the machinery of a weak equivalence category is not the same as the notion of "homotopy" in the actual corresponding infinity category.
Right, that's the point I'm making. Which is why what I said above is true about the "homotopies" in a weak equivalence category.
John Onstead said:
what the point of defining a "homotopy" within a weak equivalence category even is (if there even is a point!)
What I'm saying is that I don't think there is a point if you only have a category with weak equivalences. If your category has the additional structure of a model category, or something weaker like a cofibration category, then you can define a notion of "homotopy" inside of it that does correspond to those in the resulting -category.
I recently stumbled upon the mouthful that was the nlab article "fundamental infinity groupoid in a locally infinity-connected infinity topos". Earlier, we covered that the notion of "fundamental groupoid" generalized to model categories was as a hom-groupoid in the corresponding infinity category. But it doesn't initially seem clear how the conception of a fundamental groupoid as a hom out of a model category, and the notion of fundamental groupoid presented in this nlab article, are related. So my question is: what is the precise relation between these ideas, and can one potentially be converted into the other? Thanks!
So, in the nLab definition one considers locally -connected -toposes where you have an adjoint triple between and , right?
Here is the global sections functor, so it's precisely a “hom--groupoid functor”, that is, it is where is terminal.
Now, that definition tells you that is a generalised fundamental -groupoid of an object of . But when is the -topos of topological spaces, then all functors in the triple are equivalences, so in fact it makes no difference whether you take or , and both compute the fundamental -groupoid of a topological space.
At the same time, since -groupoids (or topological spaces) are the “free homotopy cocompletion of the point”, in a certain sense the hom-groupoid out of a point, that is, the “global sections” functor, is the only one that it makes sense to ask, that is, you don't get anything out of the generalised parametrised ones...
So I think both are generalisations by extrapolation of this somewhat “degenerate” situation. I do not know much about the use cases of cohesive -toposes, but it would seem that usually the cohesive structure captures some extra “geometric” information which is not seen by the global sections functors, so presumably the -groupoids computed by are strictly more informative than those computed by , and seen as more deserving of the name “fundamental -groupoid”.
On the other hand, it seems reasonable that in other situation where there is no extra geometric structure, which seems to be typical of -categories presented by model categories which are “generalised homotopy theories”, it is instead more useful to generalise from “the -groupoid of global elements” to “the -groupoids of -shaped elements” for varying , which was the suggestion from earlier in this discussion.
TL;DR “the fundamental -groupoid of a topological space” is a special instance of both definitions due to the fact that in the -topos of -groupoids, but otherwise the two generalise this special case in somewhat incomparable directions.
Right, the intuitive meaning of "fundamental -groupoid" is that it makes geometric information into homotopical information: continuous paths become 2-morphisms and so on. is not a such a functor, because it forgets the geometric information.
In the case of classical topological spaces, the "making geometric information into homotopical information" has morally already happened in the process of putting a model structure on topological spaces that considers continuous paths as the homotopies. So the which is higher-categorically the identity functor is computing the fundamental groupoid when you implicitly consider it as "composed" with the map from the 1-category Top to the -category presented by its model structure.
Amar Hadzihasanovic said:
TL;DR “the fundamental -groupoid of a topological space” is a special instance of both definitions due to the fact that in the -topos of -groupoids, but otherwise the two generalise this special case in somewhat incomparable directions.
Thanks for the in-depth answer! So it seems that the nlab page's definition works for locally connected toposes when there is some "geometric information" involved, while in other infinity toposes without this "geometric structure" the hom groupoid definition makes more sense.
Mike Shulman said:
Right, the intuitive meaning of "fundamental -groupoid" is that it makes geometric information into homotopical information: continuous paths become 2-morphisms and so on. is not a such a functor, because it forgets the geometric information.
I'm a little confused about what this "geometric information" actually is that somehow the homs in an infinity category can't capture but instead forget. By the Yoneda Lemma, the homs out of a category should contain all the information about the objects in the category, right? The only difference I could find between these locally connected toposes and a normal infinity topos is that they have extra structure of these functors, but this extra structure is on the category as a whole and not the objects in the category. Maybe my intuition is leading me astray here, please let me know where my reasoning is going wrong!
I would never call the hom-groupoid a "fundamental -groupoid". To me "fundamental" requires a process of making geometric information into homotopical information.
The collection of all homs into an object remember all the information about it, but is specifically the hom out of the terminal object, which forgets a lot in general.
Think of the category of topological spaces. is the set of continuous maps , which is just the underlying set of .
In that case equips a set with the discrete topology, and (which is defined only on locally-connected spaces) sends a space to its set of connected components, which you can call the "fundamental set" of a space -- it makes geometric information (continuous paths) into homotopical information (equalities).
I like that term "fundamental set" - a nice case of taking a famous concept and going down to the simpler (more fundamental!) concept on which it rests, which was never exciting enough to get a cool-sounding name. This is the sort of term James Dolan would use.
I'm actually a bit surprised if he never did.
I guess then one could also say the "fundamental proposition" or "fundamental truth value" of a space is whether it's nonempty.
I'm still a little confused. A morphism from the terminal object to an object is a point in that object and the functor it represents is the underlying set functor, sure. But that's in the 1-category setting. In the infinity category setting, we have higher morphisms. For instance, if you have a homotopy (which is a higher morphism in the infinity category setting) between two functions into the same object from the terminal object, this represents a "path" in that object. Then the hom groupoid for the point sends you to the path groupoid. If you then decategorify and identify objects along equivalences, this is the same by quotienting out by homotopies. In the end, you are left with the set of all path connected components of a topological space. Maybe I'm being misled because this is in Top where the two kinds of "fundamental groupoid" converge, but it seems like the hom out of a point should still be a useful thing to consider and that it does capture geometric information about an object. In any case, this isn't to mention that you can get even richer information by using another object than the point to consider homs out of.
The categories that nLab article is about have two separate "directions" of structure, one geometric and one higher-categorical. So the objects are, for instance, *topological -groupoids": -groupoids that have additional compatible topologies on the sets of objects, 1-morphisms, etc. The functor forgets these topologies and remembers only the given -groupoid structure, while the adds new higher morphisms made from the paths (the two words have to be carefully distinguished in this context) that are continuous relative to the topology.
Mike Shulman said:
The categories that nLab article is about have two separate "directions" of structure, one geometric and one higher-categorical. So the objects are, for instance, *topological -groupoids": -groupoids that have additional compatible topologies on the sets of objects, 1-morphisms, etc. The functor forgets these topologies and remembers only the given -groupoid structure, while the adds new higher morphisms made from the paths (the two words have to be carefully distinguished in this context) that are continuous relative to the topology.
Thanks, it's good to have another example of an infinity topos other than Top (infinity groupoids) to consider things from the perspective of. In this case it's a lot like stuff, structure, property with the global sections functor as the "forgetful functor" that forgets the topological structure on the topological groupoid. But what exactly do you mean by "add new higher morphisms"? Where are these new morphisms being added to and how? (or maybe this is a metaphor I am taking too literally?)
They're added to the underlying -groupoid . For example, if is an ordinary topological space, regarded as a topological -groupoid with only identity higher morphisms, then is its usual fundamental -groupoid -- the identity higher morphisms have been "added to" by new morphisms induced by the continuous paths in .
By the way, I would recommend not using "Top" to refer to . Once you're in the world of -categories, the objects of that -category are not really "topological spaces" in any sense any more.
Oh ok I think I understand... in the same way that a morphism from the point in Top isn't sensitive to the topological structure of a topological space, a morphism from the point- and higher morphisms between them- aren't sensitive to the topological structure of the topological groupoid. The functor is sensitive to this extra structure. To be fair, this is all a little dizzying to me... infinity groupoids already act like topological spaces (CW complexes, at least) so it's hard for me to process what it means to add topological structure on top of that! Maybe this is why I like the concept of a model category so much since working with 1-categories is much less confusing than with infinity categories!
Moving on, now that I've been thoroughly confused by "homotopy", I want to get even more confused by cohomology. I guess I have to ask: what "IS" cohomology in general? For instance, I've seen cohomology discussed as measuring the obstruction from local to global. But does cohomology always deal with local to global, or is that just a very very special case and in general it's just some general pattern of assigning an algebraic invariant to an object?
John Onstead said:
To be fair, this is all a little dizzying to me... infinity groupoids already act like topological spaces (CW complexes, at least) so it's hard for me to process what it means to add topological structure on top of that!
First, -groupoids don't act that much like topological spaces or CW complexes: you can't talk about convergent sequences of points in them, etc. What -groupoids really act like - or indeed 'are' - is [[homotopy types]]. So [[homotopy type]] is the right word to use, rather than 'topological space', when you want to talk about how an -groupoid captures a bit of information about a topological space.
Second, yes, James Dolan always used to be puzzled by things like [[topological groupoids]], i.e. groupoids internal to , which have the 'doubling of structure' that's bugging you here. I recommend pondering topological groupoids because 1) people do a lot of stuff with them, soon reaching [[stacks]], and 2) they may be the simplest example where this 'doubling of structure' shows up.
Well, perhaps even simpler are 'double groupoids', i.e. groupoids internal to the category of groupoids! Here the doubling is very literal.
Another good example is [[bicomplexes]], i.e. chain complexes in the category of chain complexes. A chain complex of abelian groups is secretly a strict stable -groupoid.
Another great example is [[bisimplicial sets]], i.e. simplicial objects in the category of simplicial sets.
John Baez said:
What -groupoids really act like - or indeed 'are' - is [[homotopy types]]. So [[homotopy type]] is the right word to use, rather than 'topological space', when you want to talk about how an -groupoid captures a bit of information about a topological space.
I guess I can think of objects of these categories of infinity groupoids like that- it makes the connection with homotopy type theory clearer as well!
When it comes to the "double structure" I think I'll just have to somehow reconcile that there's two notions of "homotopy" to consider depending on the situation. There's the native intrinsic homotopy from the structure of an infinity category (whose higher morphisms can be thought of as higher homotopies), and then there's "geometric homotopy" which works specifically for locally connected infinity toposes (and not for other infinity categories, though I'm sure you can consider infinity categories where the objects are groupoids equipped with extra structure but are not locally connected toposes, so no idea how homotopy is supposed to work in those cases!)
Wait a minute! New idea I just suddenly got! So the point we reached above is that the objects in a locally connected infinity topos are like a "doubled up" structure, and that's why the natural infinity category homotopy structure isn't sensitive to it, since it's only sensitive to the homotopy structure of the underlying groupoids. But what if that just means the homotopy construction also needs to be doubled- maybe then the "geometric homotopy" is the same as a usual homotopy, just an iterated variant.
Here's what I mean. First, consider some site of topological spaces used to make topological groupoids- this generates a model structure on the category of simplicial presheaves on the site. Doing the infinity category construction for this is then just generating homotopy information for the "underlying" groupoids. This gives the infinity category of topological infinity groupoids, equipped with a geometric homotopy functor . A subcategory of this infinity category on its truncated objects contains many topological spaces, and the action of on them is the fundamental group construction for them. But wait- we could get this another way, namely as the "normal" homotopy for topological spaces. That is, we could extract this subcategory of "normal" topological spaces, add into it the homotopies from , and then generate an infinity category again which then gives the usual homotopy for this selection of topological spaces.
So here's my "big idea". What if we took this infinity category of topological infinity groupoids and truncated it into the 1-category of topological infinity groupoids. Then, we added in a notion of "homotopy" to this 1-category, alongside a notion of homotopy equivalence, and made this a model category. This notion of homotopy obviously should coincide with the topological notion inside the subcategory containing topological spaces- for instance, maybe there's some monoidal product on the category of topological infinity groupoids and we can use this to generate cylinder objects and then the homotopies from that like in Top. Then, we use this new model structure and generate another infinity category from that. I'm wondering if then the geometric homotopy in the original infinity category will coincide with "normal" homotopy in this "revised" infinity category of topological infinity groupoids!
I got lost somewhere in there. Maybe describe it again with some symbols and notation?
By the way, non-locally-connected (-toposes) also have a "geometric" notion of homotopy, it's just that the "geometric homotopy type" of an object therein is too complicated to be encapsulated by a single ordinary -groupoid . Instead it's a "pro-homotopy type". See [[shape theory]] and [[shape of an (infinity,1)-topos]].
Mike Shulman said:
I got lost somewhere in there. Maybe describe it again with some symbols and notation?
Sure, no problem!
Start with the infinity category of topological infinity groupoids: . There exists a functor , such that for a "normal" topological space in gives its usual fundamental group .
Now the stage is set. First, truncate into the 1-category of topological infinity groupoids as the corresponding infinity category for the model structure we just built. Now, use the monoidal structure that presumably exists on this category, along with the usual topological interval, to construct homotopies across this entire category. In the subcategory of on topological spaces, this obviously converges with the usual notion of homotopy. Lastly, take the homotopy equivalences generated in this manner to be the weak equivalences for a Strom-like model structure on .
One can now generate another infinity category (for the "infinity homotopy category of topological infinity groupoids"). Now, given a topological space, will be the fundamental groupoid of , just like in the infinity category corresponding to the Strom model structure on Top (since that model structure is precisely a sub-model category of the one we created above for all topological infinity groupoids). My question is: what then is the nature of for a more general topological infinity groupoid?
I presume that by "truncate" you mean quotient the hom-groupoids by equivalences to make them sets?
There won't be an actual model structure on that category, since it doesn't have 1-categorical limits and colimits. But you may be able to give it some kind of homotopy theory. However, I don't think it will end up coinciding with anything known.
If it helps, here's what I'm trying to do in other words. Above, you mentioned that the functor is "adding higher morphisms" beyond those in . What I'm trying to do is find a way to globally add in these same higher morphisms to the entire infinity category of topological infinity groupoids, turning it into a new infinity category where it is now true . That is, the higher morphisms I'm "adding in" are precisely the ones that is "adding in". The method I gave was one I was working on that I thought might accomplish something like that.
Seems to me that your new -category would be just then.
Each would be identified with .
Mike Shulman said:
Seems to me that your new -category would be just then.
Yes, if we started with and localized at the homotopy equivalences. But if we started with the category , which is much bigger than just (some part of) , and localized at its homotopy equivalences instead, then we'd get an infinity category with as a full sub-infinity category but more interesting stuff beyond that. This was what I was trying to illustrate above by trying to define a model structure on topological infinity groupoids. (Upon reflection I might have been able to do this more directly with simplicial model categories. In this case, just find the simplicially enriched category corresponding to the infinity category of topological infinity groupoids and add the model structure onto that simplicially enriched category directly- no need to go back down to the 1-categorical world first!)
Although I did find this section about "homotopy localization" of topological infinity groupoids being equivalent to infinity groupoids- is this what you mean? If so, I'm not sure it's the same construction I'm describing- the morphisms they localize with respect to there are those of the form while I want to localize with respect to the homotopy equivalences, which sounds a lot "stronger" a localization that will therefore "collapse" things a lot less. But I could be wrong about this!
Edit: Oops! I guess is homotopy equivalent to after all (at least in topological spaces), my bad... So indeed the localization of topological infinity groupoids at homotopy equivalences is likely just infinity groupoids as you suggested.
Ok, I went back to work on this for another hour to two, but I came up empty again. I want to move on to other topics as I still have a lot left to cover in this topic, but I at least want some resolution to this. So I will ask one final question about it- it will for certain have a definite explicit answer, so that is what I am looking for.
Here's the setup. Take to be the infinity category of topological infinity groupoids, to be the infinity category of infinity groupoids ("homotopy types"), and let be a mystery infinity category with a mystery functor . The category will have an object meant to represent the point.
Here's my question: For which functor and infinity category does the triangle formed by the functors , , and commute? That is, for which is the following true in the category : ? Thanks very much!
Well, you could take and .
Mike Shulman said:
Well, you could take and .
Is this meant to be tongue in cheek? You know I want a nontrivial answer!
In any case, I did wonder what if we took the observation made above (that the homotopy category of topological groupoids is equivalent to the category of homotopy types) seriously and had be . Unfortunately, the standard functor is just the "forgetful functor" that, for instance, sends a topological space (viewed as a discrete topological groupoid) to its underlying set (viewed as a discrete groupoid) and more generally sends a topological groupoid to its underlying groupoid. But I've been thinking there should also be a "homotopy functor" that would send a topological space (viewed as a discrete topological groupoid) to its underlying homotopy type. This functor would also send topological groupoids to their underlying topological space's homotopy type. Though I don't know if this functor would serve the purpose (IE, make that triangle commute) or if it even exists. What's your say?
Your "homotopy functor" is precisely . That's the point I'm making.
Mike Shulman said:
That's the point I'm making.
Ok, point taken.
So, I have a few "rapid fire" questions to wrap up this topic of learning categorical homotopy theory (and of identifying a notion of "category of spaces" in general) for now. I saved these for the end since I think they should all have quick answers (though let me know if I'm wrong!) I hope you don't mind if I put them all below:
A topos can be viewed from two perspectives: as containing generalized spaces ("big" topos) or as a generalized space ("little" topos). It would therefore make sense if (infinity) toposes were themselves to form an (infinity) topos- this topos of toposes would be a "big" topos containing generalized spaces as objects, which would therefore be toposes viewed from the "little topos" perspective. So is this true- is there any universe where toposes form a topos? And in any case, which 1-category is a good model category for the infinity category of infinity toposes?
In the context of stable infinity categories, one can view the "stabilization" of an infinity category as the "category of spectrum objects" in that infinity category. If the category is a locally presentable infinity category like an infinity topos, then there's even an adjunction between the infinity category and its category of spectrum objects. On the other hand, under Isbell duality between space and quantity/geometry and algebra, one of the functors in the adjunction is called "Spec". This is meant to invoke a generalization of the "spectrum of a ring". Do these two notions of "spectrum" and their respective adjunctions have anything to do with each other, or is this just an unfortunate overuse of terminology?
I tried asking this question above, so here goes again. What is the basic "gist" of "cohomology" supposed to be in general? Does it always have to do with local-global correspondences (such as measuring the obstruction in a space of extending local behavior of functions defined on that space to global behavior) or is that just a very special case of cohomology? (Maybe some quick slogan for some intuition will help!)
How does delooping work for groupoids? So the loop space and delooping functors define an adjunction between the infinity category of pointed objects and the infinity category of monoid objects in some suitable infinity category. This makes sense for the infinity category of categories- the monoid objects in this category are the monoidal categories, and indeed the delooping functor sends the monoidal category to the one object category one level higher. However, I'm wondering how this works in the infinity category of groupoids. Sure, groups are monoidal objects in this category (and their delooping is the obvious example of a one object groupoid), but so are monoids themselves (they are monoidal objects over the discrete groupoids- in other words, over sets). So where does the corresponding delooping functor send a monoid to? Does it somehow "groupify" the monoid before sending that resulting group to its one object groupoid?
These are nice questions. Some thoughts:
1) I'd expect the thing of all topoi to be much better described as a 2-category of some sort than as a mere 1-category, e.g. a topos. Similarly I'd expect the thing of all -topoi to be an -category of some sort.
Is the 2-category of all topoi a 2-topos? Should it be? I don't know. The 2-category of all categories is a 2-topos.
2) A warning: in homotopy theory, "spectra" are more general than stable -groupoids, since spectra can have nontrivial homotopy groups for negative, while stable -groupids only have nontrivial for . Stable -groupoids correspond to a special sort of spectra called [[connective spectra]].
More to your actual point: I think the meaning of "spectrum" in homotopy theory is so distant from its meaning in commutative algebra that it's bad to think they're somehow the same thing. A spectrum in commutative algebra is a generalization of the spectrum of a self-adjoint operator which is a generalization of the spectrum of light produced by, say, a hydrogen atom. I believe a spectrum in homotopy theory got its name simply because it was a list of topological spaces with maps from the suspension of to . A list is like a "spectrum" if you've got a good enough imagination.
In higher algebraic geometry we can "take the spectrum of an -ring spectrum", but there we are using the two concepts in the same sentence, not really blending them.
3) I think the gist of cohomology is studying an object by studying the thing of maps out of it, say , where is a "stable" object (i.e., commutative in the relevant up-to-homotopy sense), so that is an abelian group (or something similar).
Alternatively, we could say the point of cohomology is to simplify an object by "stabilizing" it - making it commutative in the relevant up-to-homotopy sense. But following the Yoneda philosophy, we can understand the stabilized version of by looking at its maps to stable objects , as above.
John Onstead said:
there should also be a "homotopy functor" that would send a topological space (viewed as a discrete topological groupoid) to its underlying homotopy type. This functor would also send topological groupoids to their underlying topological space's homotopy type.
Whoops, I missed your second sentence. The functor satisfies the first sentence, but not the second one. If by "underlying topological space" you mean the topological space of objects, then the second sentence isn't a sensible thing to ask for, since it would violate the principle of equivalence. For instance, if the topological space were discrete, this would be like taking the set of objects of an ordinary -groupoid, which is not a well-defined -functor.
To your other questions:
As for 3), I agree with @Mike Shulman. For example, once we get into "nonabelian cohomology" we are stretching the word beyond the limits of my attempted highly general concept of cohomology: now we are studying an object by mapping it to a non-stable object, and the main reason we call it "cohomology" is that it can still have some of the same flavor. It's still fun, for sure, but I think it's something new.
As for question 4), which I hadn't noticed: nowadays some people say the "delooping" of a monoid is the corresponding 1-object category . Then you can even take the nerve of this category and get a space! But the connection of loops in this space to elements of is vastly clearer when is a group.
Thanks very much John Baez and Mike Shulman for your help and answers! Very much appreciated!
John Baez said:
but there we are using the two concepts in the same sentence, not really blending them.
I'm sure that can be quite confusing!
Mike Shulman said:
then the second sentence isn't a sensible thing to ask for, since it would violate the principle of equivalence
To be honest, I wasn't exactly sure what the functor I was imagining would do :)
Mike Shulman said:
- is a "category of spaces", but not every category of spaces is a big topos.
But thanks to this whole thread we finally know what a "category of spaces" actually is in full generality! That is, a model category of some form- these include all big topos through the Bousfield localization on simplicial presheaves (on a site), as well as pretty much every other example I can think of. So I guess that's mission accomplished!
John Baez said:
For example, once we get into "nonabelian cohomology" we are stretching the word beyond the limits of my attempted highly general concept of cohomology: now we are studying an object by mapping it to a non-stable object, and the main reason we call it "cohomology" is that it can still have some of the same flavor. It's still fun, for sure, but I think it's something new.
Very interesting! I somewhat expected this flexibility of the word "cohomology", that is, a term to describe many similar phenomena even if it's not always possible to draw direct connections.
John Onstead said:
thanks to this whole thread we finally know what a "category of spaces" actually is in full generality! That is, a model category of some form
Well, if that's your notion of "category of spaces" then since any category admits a trivial model structure, every category is a "category of spaces".
It's funny because I'm inclined to think of any given category as a sort of space. I'll sometimes think of a morphism as directions from to .
But now is [[point]] a space?
If you have a board game with one piece on one square, and you're allowed to defer your move. Then yeah I think so, it's the space of the one square the piece is on.
But if our point is supposed to model a pebble in a set, or a point on a plane - then no I don't think it's really a space.
But maybe I'm conflating , , and ?
This reminds me of when I got in trouble for not being specific enough. I said "I like geometry" and the professor thought "algebraic geometry" and when I "clarified" further they got annoyed that I really meant "differential geometry".
But really I had no idea what I was referring to XD. I meant something like "spatial thinking", but vaguely described stuff I thought was cool from calculus.
Alex Kreitzberg said:
This reminds me of when I got in trouble for not being specific enough. I said "I like geometry" and the professor thought "algebraic geometry" and when I "clarified" further they got annoyed that I really meant "differential geometry".
But really I had no idea what I was referring to XD. I meant something like "spatial thinking", but vaguely described stuff I thought was cool from calculus.
There are two things which come to mind when I think of differential geometry. There's the usual approach which is about differential and smooth manifolds. Then there's synthetic differential geometry, which takes place inside of a [[smooth topos]] and to me is fairly similar to algebraic geometry.
I'm positive they didn't have synthetic differential geometry in mind. Or at least that they inferred I was interested in the usual approach.
I should add they did start to understand me, and then were quick to point out "geometry" in as large as a sense I meant, wasn't a real field of study. Which was jarring for me at the time. It was a good lesson for a relatively young mathematical thinker.
Even if there isn't a real field of study of "geometry", there seems to be plenty of anecdotal evidence that a "geometric" approach or viewpoint or culture exists in mathematics, opposed to an "algebraic" one, with varying demographics across fields, and a visible split among algebraic geometers...
I've heards lots of variations on the story of two algebraic geometers given a polynomial, one starts doing computations in a quotient ring and the other starts drawing curves and surfaces
They are working in opposite categories, with their arrows going in opposite directions like two trains passing in the night. One thinks they've got a quotient ring and other thinks they've got an subscheme .
Mike Shulman said:
Well, if that's your notion of "category of spaces" then since any category admits a trivial model structure, every category is a "category of spaces".
Yes, and I think that's a reflection of what I'm starting to learn in this thread. That asking "what categories are categories of spaces" isn't the right question, since the notion of "space" is way too nebulous and there's so many diverging directions of generality one could take. Instead, the right question is "how can we think of the objects of any category as spaces?" Certainly, model structures are one such answer, since obviously you can pick out many classes of weak equivalences in any category imaginable. (But obviously, I therefore now want to establish and ensure that any possible way of thinking about the objects of some category as spaces comes with a canonical associated model or weak equivalence structure, thus making these structures "the most fundamental" way of viewing a category as one of spaces)
Though if you have a more concrete definition of "category of space" that includes both the category of toposes as well as all gros toposes, then feel free to share it! You mentioned above that the category of toposes is "a" category of spaces, no? Would this mean you could tell the category of toposes is a "category of spaces" without being told what the objects of the category are supposed to represent (without being told they are toposes)? And if so, how?
Mainly I just meant it's a category whose objects we think of as "spaces".
John Onstead said:
Yes, and I think that's a reflection of what I'm starting to learn in this thread. That asking "what categories are categories of spaces" isn't the right question, since the notion of "space" is way too nebulous and there's so many diverging directions of generality one could take. ... (But obviously, I therefore now want to establish and ensure that any possible way of thinking about the objects of some category as spaces comes with a canonical associated model or weak equivalence structure, thus making these structures "the most fundamental" way of viewing a category as one of spaces)
It sounds like maybe the latter quoted sentence means you haven't fully learned the lesson yet? (-:O
John Baez said:
For example, once we get into "nonabelian cohomology" we are stretching the word beyond the limits of my attempted highly general concept of cohomology: now we are studying an object by mapping it to a non-stable object, and the main reason we call it "cohomology" is that it can still have some of the same flavor. It's still fun, for sure, but I think it's something new.
For an account that looks to portray continuity in the passage to nonabelian cohomology, see the nLab page nonabelian cohomology.
I find this story compelling, but it will always be a matter of convention whether one continues with the term 'cohomology'. Is it enough that central constructions carry over and continue to be important, such as nonabelian Poincaré duality?
I recall Colin McLarty telling me that Poincaré used unrelated terms for what we call 'group' and 'abelian group'. Ah, here it is in Voir-Dire in the Case of Mathematical Progress
Actually, I lied and have a few more questions (that are all closely related):
David Corfield said:
I recall Colin McLarty telling me that Poincaré used unrelated terms for what we call 'group' and 'abelian group'. Ah, here it is in Voir-Dire in the Case of Mathematical Progress [....]
"Fascieau". Did this have any effect on the later use of "fascieau" in French mathematics to mean "sheaf"?
Mike Shulman said:
- Also higher homotopies.
That's really interesting, I wonder why this is! It seems almost an anomaly since in general when we have two "layers" of model structure on a category, one "weak" and the other "stronger", it's not true that there's such a fully faithful embedding.
Mike Shulman said:
- It's "almost" the case that a map in the Strom model structure is a weak homotopy equivalence iff the induced map is an equivalence of -groupoids for all . Intuitively, maps out of detect . But this isn't quite true because of basepoints.
But don't the groupoids contain the basepoint information? After all, given some basepoint x, the fundamental group at that basepoint is the automorphism group of the object corresponding to x in the fundamental groupoid.
John Onstead said:
That's really interesting, I wonder why this is! It seems almost an anomaly since in general when we have two "layers" of model structure on a category, one "weak" and the other "stronger", it's not true that there's such a fully faithful embedding.
What examples do you have in mind? This happens quite frequently in the case of "localization" and "co-localization", where we start with one model structure and make a new one by adding more weak equivalences, but in a controlled way so that the model structure shrinks the class of fibrant or cofibrant objects and on the new fibrant-cofibrant objects the homotopy theory coincides with the old one. In -category world this corresponds to considering a reflective or coreflective subcategory.
John Onstead said:
Mike Shulman said:
- It's "almost" the case that a map X→Y in the Strom model structure is a weak homotopy equivalence iff the induced map hom(Sn,Y)→hom(Sn,X) is an equivalence of ∞-groupoids for all n. Intuitively, maps out of Sn detect πn. But this isn't quite true because of basepoints.
But don't the groupoids contain the basepoint information? After all, given some basepoint x, the fundamental group at that basepoint is the automorphism group of the object corresponding to x in the fundamental groupoid.
Sorry, I misspoke. If we're willing to talk about equivalences of hom--groupoids, then we can simply say that a map is a weak homotopy equivalence iff the induced map (I also got the variance wrong, must have been too early in the morning) is an equivalence of -groupoids, since is the fundamental -groupoid and contains the basepoints and homotopy groups, as you say.
What I was remembering is that this is no longer true if instead of hom--groupoids you look at hom sets in the homotopy 1-category, and enlarging the collection of "test spaces" from to all the spheres doesn't fix it, although you might think it should.
Wow, that's interesting. Do you know a counterexample offhand?
I don't have a specific counterexample as in a particular map that induces bijections but is not a w.h.e. (The notation is traditionally used among algebraic topologists for the set of unbased homotopy classes of maps, i.e. hom-sets in the homotopy 1-category.) But I have an example of the sort of thing that can go wrong when you consider only unbased homotopy classes of maps.
Consider the wedge (the "wedge", or coproduct in pointed spaces, the disjoint union with the basepoints identified). You can map an into it by squishing the hemisphere containing the basepoint down to an interval, wrapping that interval around the arbitrarily many times, and then mapping the other hemisphere (which is now a sphere in its own right) to the . In , this gives you a bunch of different maps depending not only on the degree of the map onto but also the number of times you wrap around , since one end of the interval is fixed at the basepoint and the other end can't be "got off" of (unless the degree is zero). But in , the basepoint isn't fixed, so you can just retract that end around the and all that's left is the degree of the map onto .
This suggests that perhaps some map like the inclusion would be a counterexample, but I'm not confident enough of that to assert it.
I'm not quite sure about maps that merely witness that the don't detect weak equivalences in the unbased homotopy category. I suspect Mike's counterexample will work. But I do know a counterexample to a stronger claim: the finite CW-complexes don't detect weak equivalences in the unbased homotopy category! Of course, Yoneda says that a counterexample to this claim must involve infinite complexes.
In fact the counterexample is really just in groupoids. Consider , where is the (discrete) group of permutations of with finite support, i.e. cofinitely many fixed points.
There is an endomorphism of that essentially shifts the action of a permutation to the right by one unit: you send a permutation to with except that
I claim that this endomorphism, call it , is not conjugate to the identity of Once you think through the words this is obvious, since if were the identity for some then you could un-conjugate to show that is the identity too.
Now in the unbased homotopy category, morphisms are exactly conjugacy classes of homomorphisms , and so the above shows that is not equivalent to the identity on in that category.
However, one can check that any finite CW complex thinks is equivalent to the identity, roughly because there's enough room in to conjugate any finite list of elements with their images under . This should be unsurprising if you remember that conjugacy classes in finite permutation groups are given by cycle shapes, and doesn't change cycle shapes. And maps into from a finite connected CW complex are determined by the induced map on so the question reduces to group theory.
So that's an example of a map in the unbased homotopy category that looks like an identity to every finite CW complex, certainly including the spheres, but is not an identity!
This example has been known for a long time but Dan Christensen and I came back to it a few years ago and generalized it to show that no small class of CW complexes detects identities in the unbased homotopy category. Nobody had given a correct proof of this fact until then, surprisingly enough.
Nice!
Mike Shulman said:
What I was remembering is that this is no longer true if instead of hom--groupoids you look at hom sets in the homotopy 1-category, and enlarging the collection of "test spaces" from to all the spheres doesn't fix it, although you might think it should.
Ok that makes more sense! I've been thinking about how to find an analogue of weak equivalences given some category with designated homotopy equivalences. Maybe one way of doing this is to find the infinity category for that system of homotopy equivalences and then use a hom-based approach to define the weak equivalences, similarly to the case when going between homotopy and weak equivalences in Top. But before I get to that I want to understand more about general "homtopy equivalences"...
To that extent, I came across this question from math exchange. In this question, the poster defines a notion of "homtopy" between maps between simplicial sets. However, they remark that this notion of homotopy "generally fails to be either symmetric or transitive." My first question is: is this what was meant above by the following:
Mike Shulman said:
If you just start with an arbitrary category with weak equivalences, and an arbitrary choice of cylinder objects, I don't think the "left homotopies" you get will satisfy any good properties like being composable or whiskerable. In particular, therefore, the "homotopy equivalences" won't satisfy the 2-out-of-3 property to be a collection of weak equivalences.
If so, then my second question is: how does the poster solve this problem? It seems they try to define an equivalence relation based off this merely reflexive relation, but I don't see what this relation is doing, nor how it retains the important information about the homotopies. They then go on to use this new relation to define a notion of "homotopy equivalence", which then makes me wonder what the higher morphisms in the resulting infinity category are (if they are not at least directly the homotopies). Any explanation of this solution is appreciated! (I'm interested in it especially because, though I don't understand it, it looks generalizable beyond simplicial sets into any category with a choice of cylinder objects, so it should always work to solve the above problem!)
It's worth noting that the spheres or finite complexes also do not detect weak equivalences in the based homotopy category (based spaces and based homotopy classes of maps). Requiring based homotopies avoids these problems on the basepoint component, but you can also have other components. Mapping out of ordinary based spheres doesn't "see" the other components at all; and if you add in some spheres with disjoint basepoints, you're back to the same problems that we had in the unbased homotopy category.
I think it does work in the category of connected based spaces, and in the category of spectra.
Yes, the fact that the spheres detect isomorphisms (which is sometimes called being weakly generating, and sometimes called being strongly generating...) in the connected based homotopy category is a highfalutin phrasing of Whitehead's theorem.
I would have said it's more like the definition of "weak homotopy equivalence".
Oh, wait, you said "isomorphisms".
I guess by "homotopy category" you mean the weak-homotopy category?
Ugh, yeah, or I was thinking of the connected based homotopy category of CW complexes. Need to be careful in stating these foundational things.
Certainly everything also breaks in the connected based strong homotopy category but in much more terrifying ways.
John Onstead said:
It seems they try to define an equivalence relation based off this merely reflexive relation, but I don't see what this relation is doing, nor how it retains the important information about the homotopies.
It's the standard notion of an equivalence relation generated by a relation.
Mike Shulman said:
It's the standard notion of an equivalence relation generated by a relation.
Ohhh ok. So it's like the "equivalence-relation-ification" of a relation. Kind of like if you had a category of all sets with relations, and then the subcategory of sets with equivalence relations, it's like the free functor adjoint to the inclusion of the latter into the former (whether or not that adjoint actually does exist).
Yep, exactly.
Ok. I want to get familiar with this topic, so I want to try an example (other than simplicial sets, since that was already done). I think a category I have a somewhat good handle on is just Cat, the category of categories. Now in the standard (canonical) model structure on Cat, the interval object is the "walking isomorphism". The reason is there's an equivalence of categories (a weak equivalence in this model structure) from the walking isomorphism to the point. This makes sense since homotopies are meant to be invertible.
But now let's take the interval object in Cat to be the standard category interval, that is, the "walking morphism" rather than walking isomorphism. Now, if we consider "homotopies" between objects (maps into a category from the terminal category), we see they correspond to general morphisms between the objects (so a morphism is then a "path" between objects). The corresponding relation is reflexive, and unlike in simplicial sets it's also transitive due to composition of morphisms. However, it's not a symmetric relation unless our category is a groupoid, since morphisms are not in general invertible.
So let's consider what "completing" the equivalence relation on this relation does. At first I thought it was related to the core groupoid, but that's just the maximal sub-groupoid, and two objects can share a morphism without being in the same connected part of that groupoid. Instead, I think it's more like the "groupoidification" of a category, which would be sort of the opposite of the core- it's the minimal groupoid the category would embed into.
And so that's my hypothesis: that given two categories and , the "best" 2-category of functors and "homotopies" between them according to using the "walking morphism" as the interval object would be the "groupoidification" of the functor category . I hope I did everything correctly so far!
Also, before we continue with the example or more examples, I want to review a few things first. 1). First, what is the relation between "groupoidification" of a category: that is, the minimal groupoid with that category "contained" within it, and the localization of that category at every morphism?
Second, I want to avoid confusing "homotopy as a relation" with "homotopy as a multirelation", since it's the latter that is important when determining the groupoid of homotopies between two objects. Let's say you have a reflexive and transitive homotopy relation- this corresponds to a preorder, which can be viewed as a category. That means that a reflexive and transitive homotopy multirelation is just a category! (where the morphisms are homotopies- maybe this is what directed homotopy theory is about?). 2). So the "equivalence relation" completion of the reflexive and transitive relation should then just be the "groupoidification" of the preorder viewed as a category. Is this right?
Lastly, what if the relation is merely reflexive, as in the more general case? This relation can be viewed as a directed graph, so the corresponding multirelation should be some form of directed multigraph (since transitivity = composition, so it can't be a category). But then what is the "equivalence relation completion" of this setup? 3). Is it related to the "free category on a graph", or is it another kind of construction?
John Onstead said:
Also, before we continue with the example or more examples, I want to review a few things first. 1). First, what is the relation between "groupoidification" of a category: that is, the minimal groupoid with that category "contained" within it, and the localization of that category at every morphism?
They're the same, at least, I would interpret your slightly loose description of the groupoidification as either the application of the left adjoint to the inclusion from groupoids to a category, or as the localization at all morphisms, and those are the same.
John Onstead said:
Let's say you have a reflexive and transitive homotopy relation- this corresponds to a preorder, which can be viewed as a category. That means that a reflexive and transitive homotopy multirelation is just a category!
I assume by a "multirelation" you mean a [[quiver]]. In this case, I would say morally yes, although technically speaking just making a quiver "reflexive and transitive" doesn't yet imply it's a category until you add the unit and associativity axioms. But they should "clearly" be there, so yeah.
So the "equivalence relation" completion of the reflexive and transitive relation should then just be the "groupoidification" of the preorder viewed as a category. Is this right?
Not exactly: the groupoidification of a preorder viewed as a category isn't necessarily a [[thin category]] any more, so it isn't an equivalence relation exactly. (In fact, if memory serves, every groupoid can be obtained, up to equivalence, as the groupoidification of some preorder!) The relation is in the other direction: the equivalence relation generated by a reflexive-transitive relation is the "connected components" relation of its groupoidification qua category.
More generally, the equivalence relation completion of any relation can be obtained by regarding the relation as a quiver, generating the free category on it as you suggest, taking the groupoidification of that, and then taking the equivalence relation of connected components.
Going back to your question about intervals, from any interval in a cartesian category you get a quiver of maps between any two objects, and you can then generate a free groupoid from it. That doesn't use any reflexivity or transitivity, since in general an interval of this sort won't admit identities or composition. If there is a map , then you have a reflexive quiver and you can use that information to generate the free groupoid on a reflexive quiver. And if there's a map , where is the pushout of the two inclusions of into , then you get a composition operation on your quiver. It might not be a category, but it will be if these maps on the interval satisfy the "duals" of the category axioms, i.e. if the interval is an internal [[cocategory]].
A more homotopy-theoretic thing to do is think of the interval as a 1-simplex and ask if there is also a 2-simplex object , with three inclusions as the boundary edges. If so, then you get not a composition operation on your "homotopies" but a composition relation whenever the three homotopies are the boundary of a triangle that extends to a map . You can then generate a free groupoid from your quiver modulo these composition relations, i.e. forcing them all to become actual compositions .
Then if you want to go on to higher homotopies, you can ask for all -simplices , which is to say a [[cosimplicial object]] of your category. The products are called a "cosimplicial frame" on , and induce an enrichment of the category over simplicial sets. Then you can regard each hom-simplicial-set as presenting a "freely generated" -groupoid, i.e. as an object of the Quillen model structure on simplicial sets that can be fibrantly replaced by a Kan complex, and you're back in classical homotopy theory.
Kevin Carlson said:
They're the same, at least, I would interpret your slightly loose description of the groupoidification as either the application of the left adjoint to the inclusion from groupoids to a category, or as the localization at all morphisms, and those are the same.
Thanks! It's interesting you mention this operation as left adjoint to the inclusion of groupoids into categories, since I also know that core groupoid construction is the right adjoint to this. When I was mentioning how the "groupoidification" is sort of like the opposite of finding the core groupoid I was speaking loosely, but it's interesting that they are "dual" to one another in a more formal sense as well!
Mike Shulman said:
I assume by a "multirelation" you mean a [[quiver]]. In this case, I would say morally yes,
Is it odd that I think about this in terms of generalized multicategories? In this setting, given a (virtual) double category of V-matrices, a monad in it is sort of like a relation that's transitive and reflexive- for instance, a monad in the double category of relations is a preorder. To get from normal relations to multirelations, just replace truth values with sets and so the category of relations with set-matrices. A monad in that category is just a set-enriched category- a "normal" category!
Mike Shulman said:
Going back to your question about intervals, from any interval in a cartesian category you get a quiver of maps between any two objects, and you can then generate a free groupoid from it.
Ok, that's mainly what I was wondering about- that is, how generalizable the "trick" I found in that math stack exchange post was. I've looked into this a bit more and I think this works even in a generic monoidal category, so long as you define an interval object to be where is the monoidal identity. Of course in a cartesian category this is the terminal object. I'm guessing this also works when you start with a category of weak equivalences and define homotopies based off the cylinder objects in such a structure. So it's widely applicable!
Mike Shulman said:
Then you can regard each hom-simplicial-set as presenting a "freely generated" -groupoid, i.e. as an object of the Quillen model structure on simplicial sets that can be fibrantly replaced by a Kan complex, and you're back in classical homotopy theory.
Let's say you used this method to generate an infinity category corresponding to some interval, call that infinity category . Now, go back to the interval object, generate the cylinders and from that the homotopies, then complete the quivers of homotopies to groupoids. Now, use this to define "homotopy equivalence" in the standard way, find the class of homotopy equivalences, and take that to be a class of weak equivalences. Designate the resulting infinity category for this category with weak equivalences as . Here's my first question: what's the difference between and ?
Mike Shulman said:
It might not be a category, but it will be if these maps on the interval satisfy the "duals" of the category axioms, i.e. if the interval is an internal [[cocategory]].
Second question: Given this, what if we only cared about "completing" the homotopy relation up to being reflexive and transitive (such as with the interval object in Cat)? In essence, we are only doing the free category operation on the homotopy quiver, not the subsequent "groupoidification". Then instead of hom groupoids, we would get hom categories, and instead of hom infinity groupoids, we'd get- well, I'm not sure, would they be hom -categories, or something even higher? I believe this is called "directed homotopy theory". It makes me wonder if there's any standard way to get the overall infinity category where the higher morphisms are directed homotopies, maybe from a similar simplicial trick to what was discussed above?
In any case, this is almost reminding me of that "fundamental infinity category induced by an interval object" article I tried to understand a week or so ago. Maybe if this is related I can eventually work my way up to understanding it!
John Onstead said:
In this setting, given a (virtual) double category of V-matrices, a monad in it is sort of like a relation that's transitive and reflexive- for instance, a monad in the double category of relations is a preorder. To get from normal relations to multirelations, just replace truth values with sets and so the category of relations with set-matrices
Yep!
John Onstead said:
what's the difference between M(I) and N(I)?
They're the same if your original category was the fibrant-cofibrant objects in a simplicial model category, with the simplicial structure induced by your cosimplicial object. In full generality, it all depends on what the cosimplicial object is. For instance, you could pick , in which case your "homotopy equivalences" are just isomorphisms so is your original category, whereas and higher could still be nontrivial making nontrivial.
John Onstead said:
Then instead of hom groupoids, we would get hom categories, and instead of hom infinity groupoids, we'd get- well, I'm not sure, would they be hom (∞,1)-categories, or something even higher?
It probably depends on how you do it. If you're only looking at the quiver, then the free category is the same as the free -category. If you include higher simplices or other shapes, then you can get any sort of presheaf, and choose one that will generate whatever sort of -category you're interested in. For instance, if you use a cosimplicial object but ask for it to interact well with the Joyal model structure on simplicial sets rather than the Quillen model structure (e.g. to be part of a model category enriched over that model category), then you'll get a meaningful notion of hom--categories.
Mike Shulman said:
They're the same if your original category was the fibrant-cofibrant objects in a simplicial model category, with the simplicial structure induced by your cosimplicial object. In full generality, it all depends on what the cosimplicial object is. For instance, you could pick , in which case your "homotopy equivalences" are just isomorphisms so is your original category, whereas and higher could still be nontrivial making nontrivial.
Hmmm... so your approach where you use a cosimplicial object generated from an interval is an alternative approach to using the homotopy equivalences generated from an interval. Though I think I now see the divergence: every interval object induces a single, unique set of homotopy equivalences, and thus there is a unique resulting infinity category structure for some choice of interval object. But in the cosimplicial object case, there can be a huge number of cosimplicial objects for which is the interval, so through this an interval object can induce a whole bunch of different infinity category structures.
Though this makes me wonder about the connections between the two. Given some cosimplicial object for which is the interval, one can generate an infinity category as you described, but this also means that there's some corresponding choice of "weak equivalences" on the original category that, after simplicial localization, will produce the same infinity category. What is that class of weak equivalences?
Mike Shulman said:
It probably depends on how you do it. If you're only looking at the quiver, then the free category is the same as the free -category. If you include higher simplices or other shapes, then you can get any sort of presheaf, and choose one that will generate whatever sort of -category you're interested in.
This really makes me wonder about what the "correct" notion of model structure should be for directed homotopy theory. In a category with weak equivalences, the weak equivalences are meant to act as if they are the homotopy equivalences for some notion of homotopy which already satisfies reflexivity, transitivity, and symmetry. So you can only use this to generate an infinity category where all higher morphisms are invertible. So if a "category with weak equivalence" structure won't do it, what structure will?
An [[enriched model category]] over the Joyal model structure on simplicial sets is one of the most convenient ways to present an -category.
John Onstead said:
this also means that there's some corresponding choice of "weak equivalences" on the original category that, after simplicial localization, will produce the same infinity category.
Why does it mean that?
It's possible that if you take the smallest class of weak equivalences containing all the maps induced by the simplicial operators, it would have that property.
But in general, as I've said before, as soon as you start doing "internal" homotopy theory with intervals, simplical frames, homotopies, and so on, you really want to assume more than just an interval object or a class of weak equivalences, something more like a model structure.
Mike Shulman said:
An [[enriched model category]] over the Joyal model structure on simplicial sets is one of the most convenient ways to present an -category.
I guess that makes sense, since it's just the analogue "one level up" of how a simplicial model category presents an -category. But I think I want to see an example of how this works. For instance, let's say we have the category of directed topological spaces. Could you walk me through precisely what to do with this category to eventually get the corresponding -category that contains the directed homotopies as its 2-morphisms? For instance, which Joyal-enriched model category do we do whatever the version of simplicial localization to?
Mike Shulman said:
But in general, as I've said before, as soon as you start doing "internal" homotopy theory with intervals, simplical frames, homotopies, and so on, you really want to assume more than just an interval object or a class of weak equivalences, something more like a model structure.
Right, I'm just trying to verify that there's still ultimately only one notion of "categorical homotopy" regardless of which precise approach you take to it. I can do this so long as I can verify that any approach to doing an "internal homotopy" corresponds to some category with weak equivalences. And also, I understand the usefulness of model categories as they are extremely useful- when they exist! Unfortunately, to even define a model category, you need to set one of the most stringent possible requirements on the category, that it needs to be bicomplete. Almost no categories are bicomplete. So in my view, while a full model structure is indispensable and extremely useful when it can be defined, it's usefulness is very relative- when dealing with the vast majority of categories that are not bicomplete, categories with mere weak equivalences are actually technically much more useful! At least, that's just my thoughts on the matter!
I don't know whether any notion of directed topological spaces present an -category or whether they are a Joyal-enriched model structure. It's a middling-to-okay intuition that higher categories are like a sort of "directed topology", but I don't think anyone has had a lot of luck making that literally precise.
John Onstead said:
I'm just trying to verify that there's still ultimately only one notion of "categorical homotopy" regardless of which precise approach you take to it. I can do this so long as I can verify that any approach to doing an "internal homotopy" corresponds to some category with weak equivalences.
I don't think that's true. For instance, a simplicially enriched category has a notion of "homotopy" and presents an -category, but it may not do it by way of localizing any 1-category with weak equivalences.
John Onstead said:
Unfortunately, to even define a model category, you need to set one of the most stringent possible requirements on the category, that it needs to be bicomplete. Almost no categories are bicomplete. So in my view, while a full model structure is indispensable and extremely useful when it can be defined, it's usefulness is very relative- when dealing with the vast majority of categories that are not bicomplete, categories with mere weak equivalences are actually technically much more useful!
I think that's a mistaken point of view, for many reasons. Firstly, there are the same number of bicomplete categories as there are of non-bicomplete categories, namely a proper class of them. (Ignoring size considerations.) And while the definition of model structure is often stated using bicompleteness, actually it really only requires finite bicompleteness to make sense, and that was in fact Quillen's original definition. Finally, the majority of categories that arise in practice where we want to do internal homotopy theory are bicomplete, and most of them admit either model structures or something slightly weaker like a "weak model structure". Those that don't even admit that, usually admit at least a "one-sided" version of a model structure like a fibration category or a cofibration category.
There's a reason why practicing homotopy theorists use model structures all the time.
Mike Shulman said:
I don't think that's true. For instance, a simplicially enriched category has a notion of "homotopy" and presents an -category, but it may not do it by way of localizing any 1-category with weak equivalences.
What do you mean? This is the whole point of a category with weak equivalences! Categories with weak equivalences, through simplicial localization, give rise to the simplicially enriched categories. More generally categories with weak equivalences give rise to any model of infinity categories, given the right notion of "localization".
Mike Shulman said:
Firstly, there are the same number of bicomplete categories as there are of non-bicomplete categories, namely a proper class of them. (Ignoring size considerations.)
Right, that's true. I guess when I said there's "less bicomplete categories than non-bicomplete categories" I was thinking along the lines of something like "there's less prime numbers than integers". Intuitively that seems true but formally there's the "same number" of primes as integers (countable infinity).
Mike Shulman said:
I don't know whether any notion of directed topological spaces present an -category or whether they are a Joyal-enriched model structure. It's a middling-to-okay intuition that higher categories are like a sort of "directed topology", but I don't think anyone has had a lot of luck making that literally precise.
I wish this field was more developed since it seems weird, at least to me, to stop at categories and undirected homotopies and say "good enough". It feels a bit like if mathematicians refused to ever consider the notion of a "directed graph", instead believing that undirected graphs were the "only way to go".
John Onstead said:
Categories with weak equivalences, through simplicial localization, give rise to the simplicially enriched categories. More generally categories with weak equivalences give rise to any model of infinity categories, given the right notion of "localization".
Correct, but if I give you a simplicially enriched category, it would be silly to produce its corresponding -category by first producing a category with weak equivalences (which will in general be a very complicated and abstract construction with many more objects than the category you started from) and then localizing it, when you can instead just work with the simplicial hom-spaces. I would call this a different "approach to doing internal homotopy theory" than to work with categories with weak equivalences, even though ultimately they are all capable of presenting the same objects.
John Onstead said:
I wish this field was more developed since it seems weird, at least to me, to stop at categories and undirected homotopies and say "good enough". It feels a bit like if mathematicians refused to ever consider the notion of a "directed graph", instead believing that undirected graphs were the "only way to go".
The problem is that "directedness" is a pretty vague thing. Higher categories are "directed" in one precise sense, but it's not at all clear whether there's a notion of "direction" one can impose on a topological space that means the same thing.
It's arguably an accident that ordinary topological spaces can be used as a model for -groupoids, not something we should expect to continue into higher dimensions.
(For instance, I don't think it's true in constructive mathematics.)
I see. This is all certainly a lot to think about!
I want to take a break from model structures and homotopy theory to return to a discussion above about categories of sheaves. In a previous thread, I learned about two notions of establishing some notion of "space": a relation of some sort between parts of a space, and as the "gluing together" of parts of a space. Things like (V,T) algebras abstract the former and sheaves abstract the latter. At the end of that discussion, I asked if the "gluing way" of looking at things could be encapsulated by the "relation way", but the answer was that this was not possible in general, since you can't specify how parts glue together with just a relation. But on further reflection, maybe I had it the wrong way, and I should instead have asked if the "relation way" could be encapsulated by the "gluing way"!
Indeed, there's many examples of something like this. The most obvious example is the "topological topos", which aims to include as many kinds of space as possible such as all subsequential spaces, though in a misleading twist it doesn't include all topological spaces. The notion of the quasi-topological space would be a sheaf construct that does include all topological spaces, but the starting site is so "big" that it's not very useful to consider. On the other hand, primarily due to the small size of its site, the example of the topological topos gives a perfect opportunity to understand the relationship between the "relation way" and "gluing way" of looking at spaces, since its objects can be understood through both.
First, you can view the objects in the topological topos (at least, the subsequential spaces) as being defined via a relation: a binary relation between the set of sequences and the set of points that satisfy some properties. Importantly, as the nlab says, a general object can be thought of as a subsequential space "such that a given sequence can converge to a given point in 'more than one way'." On the other hand, you can view them in terms of gluing- they are the "gluing together" of objects on the defining site, which consist of the point and the "one point compactification" of the natural numbers. No doubt the fact both definitions involve the natural numbers (sequences are defined in terms of natural numbers, and the gluing uses a space of natural numbers) provides a connection between the two. But my question is: what is the specifics of the connection? More concretely: How can I go between viewing the objects of this topos from the relation sense to the gluing sense and back?
I think that might be a good exercise for you to work out yourself.
I have a different question about the topological topos. The implicit restriction to topological spaces whose topology is determined by saying which sequences converge to which points is sad - at least to a former analyst such as myself. In functional analysis we routinely deal with spaces where this isn't true! So, we use nets, a generalization of sequences where the indexing set is replaced by an arbitrary directed set.
Has anyone tried to build something like the topological topos where we replace the role of by an arbitrary directed set? Not a fixed directed set, that's probably easy, but that doesn't solve the problem! We need to let it vary.
(In analysis we put no bound on the cardinality of the directed set, and I can imagine that causing problems somehow, but in practice, for the spaces we actually care about, it's probably enough to work with nets whose cardinality is bounded. I haven't actually checked, but you should be able to get away using the maximum cardinality of the collection of open subsets in the spaces you're interested in.)
In an interesting twist, I was actually thinking of this and was going to ask a similar question to it once I understood the simpler example with the topological topos! But if you can replace the natural numbers with a fixed directed set, then maybe the solution is just to add in all the directed sets (though you would have to choose whether or not to include morphisms between the directed sets instead of just considering the ones from the point)
Mike Shulman said:
I think that might be a good exercise for you to work out yourself.
I suppose, but I wouldn't even know where to start! For instance, the notion of "convergence" makes sense from the relation standpoint- in fact, the relation is literally describing convergence! But I don't see exactly what the interpretation of "convergence" is in the sheaf picture. In addition, a sequence is just a map from the natural numbers to some set, and this is easily encapsulated in the relation picture where one of the sets is the set of sequences for some set. But where does the notion of "sequence" come from in the sheaf picture? I understand the set of natural numbers is an object in the site, but where's the map from it to some other set when there is no "other set" in the site?
@John Onstead wrote:
But if you can replace the natural numbers with a fixed directed set, then maybe the solution is just to add in all the directed sets (though you would have to choose whether or not to include morphisms between the directed sets instead of just considering the ones from the point)
In the usual theory of nets we do need to consider morphisms between directed sets: for example if you have one convergent net of real numbers and another one , you want to be able to say that their sum converges, but this requires creating a new directed set with morphisms to and .
In order to talk about a sequence, you're talking about a sequence in some space, which means you're not in the site any more but in the topos. So a sequence in an object of the topos is a morphism , and a "convergent sequence" is a morphism .
John Baez said:
Has anyone tried to build something like the topological topos where we replace the role of N by an arbitrary directed set?
I don't think anyone has managed to make exactly this work, but the theory of condensed/pyknotic sets is very similar and serves a lot of the same purposes. I learned at https://mathoverflow.net/q/441610/49 that these topoi satisfy many of the same good properties as Johnstone's topological topos. Informally, the idea is to generalize even further: instead of just considering the 1-point compactifications of nets in the site, you include more general compacta.
Mike Shulman said:
In order to talk about a sequence, you're talking about a sequence in some space, which means you're not in the site any more but in the topos. So a sequence in an object of the topos is a morphism , and a "convergent sequence" is a morphism .
Oh ok. So the "one point compactification" of the natural numbers can be interpreted as adding an extra point, a "point of infinity". And a sequence converges to a point if the extra point is mapped into . Though I don't see exactly how a sequence can converge to a point "in more than one way" in the topological topos since there's only one map from that will do both what the underlying sequence did and will map this point at infinity to . So I guess I'm still confused about that.
As for the "patching", I think I can see how that works a bit better now. You can imagine an object in the topological topos as being a whole bunch of convergent sequences patched together. Each copy of that you patch acts like a sequence that will be part of the resulting space. So you are literally building up a space from sequences!
As for the next thing I want to understand, it's the axioms. The relation defining a subsequential space has three axioms: centrality, isotonicity, and the "star property" (which works like the one for pseudotopological spaces). Meanwhile, on the "gluing" side, a sheaf is a presheaf satisfying the gluing condition. How do these two work together? For instance, what kind of "space" do you get when you just look at presheaves on this site, and what happens when the sheaf condition is then imposed?
John Onstead said:
I don't see exactly how a sequence can converge to a point "in more than one way" in the topological topos since there's only one map from that will do both what the underlying sequence did and will map this point at infinity to .
Nope, there can be lots! That's the point.
John Onstead said:
How do these two work together?
If memory serves, Johnstone goes through this in a fair amount of detail in his paper.
At some point in this conversation @Chris Grossack (they/them) should be summoned.
I did some more work on this and I came across an interesting connection. I first wanted to simplify the problem further by considering a "topos of preorders". This would work just like Johnstone's topos, but instead of being based on relations between a set of sequences and the set of points they can converge to, it's based on relations between a set and itself- essentially asking which point another point in the set "converges to". To construct this, I started with Johnstone's site and replaced the natural numbers with a two element space (IE, a "one point compactification" of the point itself, if that makes any sense). That way, in the resulting topos, every morphism from this representable into a space will correspond to a point and another point that the first one "converges to".
Then I thought that if this topos was anything like Johnstone's topos, then there'd be more than one way a point can "converge" to another point. I then came to a sudden realization that this just sounds like quivers ("multigraphs"). And I actually think that's what I was describing all along! If a site consisted of just a point and a 2 element space, then it consists of two objects and two morphisms from one to the other- which happens to exactly be the "walking graph"! The two point space, as a representable, can then be interpreted to be a generic edge in a graph. If this is all correct, then Johnstone's topos is really just an "extension" of the idea behind the quiver topos.
Nice observation!
I've been summoned! It seems like people are talking about the topological topos? I can say some words about that, if there's questions ^_^. I'll follow this thread more closely over the next few days. It's been moving a bit fast for me to read everything, but it looks like it's been really interesting
Mike Shulman said:
Nice observation!
I'm glad I'm understanding this a bit better! It's also interesting how this (inadvertently) ties into the earlier discussion we were having on "multirelations". In a sense, then, the objects of the topological topos are like a "multirelation" version of subsequential spaces. Maybe there's even a fun way to think about this in terms of generalized multicategories, like trying to define objects of the topological topos in terms of a monad on Set-Matrices that sends a set to its set of sequences.
Chris Grossack (they/them) said:
I've been summoned! It seems like people are talking about the topological topos? I can say some words about that, if there's questions ^_^. I'll follow this thread more closely over the next few days. It's been moving a bit fast for me to read everything, but it looks like it's been really interesting
No problem! For a quick summary I'm trying to use the topological topos as an example to understand the connection between defining spaces (and "locality") through binary relations (such as the relation between the set of sequences of a set and the set itself) and through "gluing" (such as "gluing together" convergent sequences to generate objects of the topological topos).
There were some questions floating around, at least in the background....
What I found interesting in the nlab article on the category of quivers is that, if a quiver is thought of as a multirelation, then a separated presheaf (with respect to the double negation topology on the walking quiver) is a normal relation. I guess my next thing to wonder is if this carries over in general. For instance, consider some equivalent of the double negation topology on the site defining the topological topos. Would a separated presheaf with respect to this topology be a true relation (so there's at most one way a sequence can converge to a point)?
If so, consider this. A "bisheaf" is a sheaf with respect to one topology and a separated presheaf with respect to another- consider a bisheaf that is a sheaf with respect to the typical topology on the site and a separated presheaf with respect to the "double negation". As with any bisheaf, the category of such is a quasitopos known as a "Grothendieck quasitopos". Coincidentally, the category of subsequential spaces is also a quasitopos. Is it possible these are one in the same?
Here's something else I'm wondering about- is the double negation topology on the walking quiver the only topology on the walking quiver? After all there's only two morphisms, so it makes sense that it is. But if so, the only sheaves on the walking quiver are "total relations". So there wouldn't be a way to define preorders directly in terms of sheaves on some site... (unlike with subsequential spaces, whose relation satisfies similar same axioms as a preorder, just with a different set involved in the relation)
What is the resulting topos if one replaces in the site of the topological topos with the Cantor space ?
If I remember correctly, it isn't the topos of light condensed sets because the covers are different.
@John Onstead, check out Johnstone's original article on the topological topos; it answers many of those questions.
@Madeleine Birchfield -- I think that replacing by gives the Escardó-Xu topos defined in this article. I also remember that it isn't the topos of light condensed sets, but I only thought about that briefly and it was a while ago
John Onstead said:
I guess my next thing to wonder is if this carries over in general. For instance, consider some equivalent of the double negation topology on the site defining the topological topos. Would a separated presheaf with respect to this topology be a true relation (so there's at most one way a sequence can converge to a point)?
I guess I'm not sure exactly what you mean by a "true relation", since I missed a fair amount of discussion. But I can tell you that Mike is right to say this is in Johnstone's original paper. Indeed thm 3.6 shows that objects of the topological topos which are sheaves for the double negation topology are exactly the indiscrete spaces (i.e., every sequence converges to every point). The objects that are merely separated for the double negation topology are exactly the kuratowski limit spaces (sometimes called subsequential spaces). These have the property it sounds like you want where each (sequence, limit point) pair has a unique "witness" to its convergence.
Yes, that's exactly what I was looking for! So this is a general pattern then, how interesting!
Ah, I see Chris gave you the answer. I actually intentionally didn't quote the answer or a specific citation. Since this answer is quite easy to find in the paper (it's mentioned at the top of page 2), I was hoping to encourage you to start doing a little bit more reading on your own before asking a question. It's not that I mind answering questions (if I did, I would just stop doing it); rather, I think you would personally benefit from improving your self-learning ability.
Mike Shulman said:
Ah, I see Chris gave you the answer. I actually intentionally didn't quote the answer or a specific citation. Since this answer is quite easy to find in the paper (it's mentioned at the top of page 2), I was hoping to encourage you to start doing a little bit more reading on your own before asking a question. It's not that I mind answering questions (if I did, I would just stop doing it); rather, I think you would personally benefit from improving your self-learning ability.
Right, the problem was I can't seem to find access to it (IE, on google scholar), and the "pdf" link for it on nlab doesn't seem to be working anymore.
Though there is this page on the ncat cafe that might include some details from the paper, so I think I'll give it a read next!
Ah, fwiw you can find (illegal, unfortunately) copies of almost any paper you want on scihub. This is how I read almost every paper I've ever read, and in particular Johnstone's paper is on there. I'll leave it to you to figure out how to use it, but it's pretty self explanatory once you have a DOI in hand
With all this covered, I actually want to create a true "topological topos". A topological space can be defined in terms of convergence as a binary relation between a set of ultrafilters and the set itself. While Top is not a quasitopos and so topological spaces cannot be expressed directly as bisheaves, if one relaxes the axioms of the binary relation between the ultrafilters and the set, it will give the notion of "pseudotopological spaces", which do happen to form a quasitopos. So let's try and find the topos that contains the pseudotopological spaces (and thus the topological spaces).
If we try to do this straightforwardly following from the example of the topological topos, we need to replace the natural numbers with the "ultrafilter space". Then, we can do a one point addition (or compactification) of this space, and use that as the basis of our site. In the resulting presheaf category, every morphism from the corresponding representable to a pseudotopological space will select an ultrafilter and a point that ultrafilter converges to, just like with sequences in the topological topos. We can then think of both topological and pseudotopological spaces as sets of ultrafilters that have been "glued together".
That's all well and good, but when I looked onto nlab on the page "quasitopos", it mentions explicitly that pseudotopological spaces are not Grothendieck quasitopos while subsequential spaces are. This means, apparently, that they cannot be expressed as bisheaves, such as those that are sheaves with respect to the standard topology on my site and separated with respect to the double negation topology on this site. But this doesn't make any sense- I just replicated exactly what you can do for subsequential spaces over for pseudotopological spaces. Can someone explain what is going on?
John Onstead said:
Right, the problem was I can't seem to find access to it (IE, on google scholar), and the "pdf" link for it on nlab doesn't seem to be working anymore.
Ah, you didn't say that! Just ask. Plenty of people have copies of papers or access to them and are willing to share.
John Onstead said:
Can someone explain what is going on?
There's no such thing as an "ultrafilter space".
If you switch from filters to nets, then get basically back to the same question that John Baez was up above. And as he mentioned, while nets are "representable" in a way that filters aren't, in general there are a proper class of them, so you can't get literally all (pseudo)topological spaces that way in an ordinary topos.
Mike Shulman said:
If you switch from filters to nets, then get basically back to the same question that John Baez was up above. And as he mentioned, while nets are "representable" in a way that filters aren't, in general there are a proper class of them, so you can't get literally all (pseudo)topological spaces that way in an ordinary topos.
This does make me wonder a bit about the relation between two of Grothendieck's "things": his toposes and his universes. Let's say you have a large site (such as the site of nets); the presheaf category will not be a topos since, in a sense, the category is "too big" to be a topos. But what if you were then to enlarge your category of sets a universe or so up, to get from to ? Now all of a sudden, the category somehow becomes a topos, right? But what I don't understand is how the property of being a topos somehow depends on the universe you are using to look at the category, rather than anything intrinsic to the category. After all, the definition of a topos seemingly makes no reference to size: it has finite limits (which are finite no matter which universe you are working in), is cartesian closed, and has a subobject classifier. Which of these somehow depends on the size of the universe you are working in?
It's not "the universe you're using to look at the category" that changes: you are actually talking about a different category: the category of Set-valued presheavs versus the category of SET-valued presheaves.
Mike Shulman said:
It's not "the universe you're using to look at the category" that changes: you are actually talking about a different category: the category of Set-valued presheavs versus the category of SET-valued presheaves.
Ohhhh! Ah, that's so obvious in hindsight, I should've known! So if is a large category, then might not be a topos, while can be.
Ok, with that settled, then how do Grothendieck quasitoposes work? Let's say you have two site structures on a large category and you have some notion of "bisheaf" for these sites. This forms a subcategory of on the bisheaves. How does this category appear when we go back down to looking at ? In other words, does the Grothendieck-ness of a quasitopos depend on the size scale? I think it does since the notion of "generating set" depends on your notion of "set", which itself depends on the size. So if that's the case, then that brings up a second question: given some non-Grothendieck quasitopos, is there any way to tell if it is a Grothendieck quasitopos, just with respect to some larger universe?
You could just apply the standard Giraud theorem for quasitoposes at the larger universe.
But if the quasitopos you're thinking of is something like pseudotopological spaces, this is not going to happen, because a Grothendieck (quasi)topos in some universe is complete and cocomplete for that universe, whereas ordinary Set-large categories like groups, rings, and (pseudo)topological spaces are only Set-complete, not SET-complete.
Mike Shulman said:
But if the quasitopos you're thinking of is something like pseudotopological spaces, this is not going to happen, because a Grothendieck (quasi)topos in some universe is complete and cocomplete for that universe, whereas ordinary Set-large categories like groups, rings, and (pseudo)topological spaces are only Set-complete, not SET-complete.
Ok, so then what would be the exact step by step instructions for deriving the category of pseudotopological spaces using the sheaf method? So far, I have the following steps: first, define the category for the "site of nets", which is a large site, and also define a site structure on this same large category for some "double negation" topology. Now, you have the category . Then, take the subcategory of on bisheaves that are sheaves with respect to the net topology and separated with respect to the double negation topology. Define this category to be the Grothendieck quasitopos . Am I done? Or is there one more "size" related restriction I have to do, and in fact the usual is a subcategory, rather than equivalent, to ? And if it is the latter, then what kind of subcategory is with respect to ?
As I said essentially to John Baez above, I don't think this is something that's been successfully done before, so you are treading new ground! Your sketch sounds reasonable. It's not immediately obvious that it will be possible to define a "net topology" with the right properties, but you should try! It's also not immediately obvious to me that the separation topology you want will be the double-negation one, although there's a reasonable chance. In the last step, you'll want to restrict to the bisheaves that are valued in Set rather than SET; a priori this doesn't preserve anything useful, but in this case we know that PsTop is a quasitopos, and presumably its structure will turn out to be induced by PSTOP.
Mike Shulman said:
In the last step, you'll want to restrict to the bisheaves that are valued in Set rather than SET; a priori this doesn't preserve anything useful, but in this case we know that PsTop is a quasitopos, and presumably its structure will turn out to be induced by PSTOP.
That makes sense! I'm wondering if the objects in that are valued in have any special properties. For instance, if they are in some sense "small" compared to all the other objects. I know there's a notion of "small object" but I think that may refer to something separately.
Mike Shulman said:
As I said essentially to John Baez above, I don't think this is something that's been successfully done before, so you are treading new ground!
I do think this is something reasonable to consider, since it always helps to have multiple perspectives on something. While the (T, V) algebra and relation perspective is powerful, thinking about things in terms of sheaves instead allows one to put notions of "space" and notions of "manifold" that you can build from spaces on the same footing. Perhaps eventually that could be used to derive a complete concrete definition of "space" that includes both topological-esque and manifold-esque spaces (and also therefore a unifying notion of "locality", which is what I eventually want to discover). In any case, this analysis also serves as a way of generalizing the correspondence between graphs and relations beyond the usual setting.
John Onstead said:
I'm wondering if the objects in PSTOP that are valued in Set have any special properties. For instance, if they are in some sense "small" compared to all the other objects. I know there's a notion of "small object" but I think that may refer to something separately.
Maybe a good exercise to work out?
Mike Shulman said:
Maybe a good exercise to work out?
Ok, well I looked into it a bit more and I think that notion of "small object" was actually the way to go. An object is a "-small/compact object" for some cardinal if it preserves a certain kind of colimit (a filtered colimit) of size . This is also related to ind-completion: the category of finite sets can be ind-completed into , in which the original finite sets can be viewed as the compact objects with respect to this completion.
My guess is that an ind-completion acts as the categorical generalization of a universe enlargement. So if I ind-complete , then I'd get the category where the original sets are "compact objects" with respect to this ind completion. Thus going back to the original example, the objects in which are pseudotopological spaces would be the "compact objects" in this category.
I hope that at least makes some sense!
John Onstead said:
if I ind-complete Set, then I'd get the category SET where the original sets are "compact objects" with respect to this ind completion.
That's right if you adjust the cardinal. Set is the -ind-completion of the category of sets of cardinality (i.e. finite sets), and the original finite sets are the -small objects in Set. Similarly, SET is the -ind-completion of the category Set of cardinality , where is the inaccessible marking the "size of the universe", and the original objects of Set are the -small objects in SET.
That by itself doesn't immediately imply that the Set-valued objects of PSTOP are the -small ones: smallness doesn't always transfer pointwise to sheaves, since colimits of sheaves are not just pointwise.
Mike Shulman said:
That by itself doesn't immediately imply that the Set-valued objects of PSTOP are the -small ones: smallness doesn't always transfer pointwise to sheaves, since colimits of sheaves are not just pointwise.
Well, in this blog article, written by a mathematician you might be familiar with, there's a treatment in which a locally presentable category can be "universe enlarged" to one . The objects of are then the "-presentable objects" of . I'm not sure what this means, but here's one guess. Every locally presentable category has a -small set of objects you can use to generate all other objects in with using -small colimits. Maybe an object of being -presentable means you can generate it with a -small colimit of the objects in the generating set- this means that objects of that are "outside" of those of would require larger colimits to generate (again, I have no idea if this is a good explanation of what "-presentability" is or if it's even remotely correct, the nlab article confuses things further by throwing in another cardinal to make me even more confused!)
Later, the blog mentions that the universe completion of the locally presentable category of sheaves valued in on a site is the category of valued sheaves on that site. Putting this together, the sheaves valued in are the -presentable objects in the category of valued sheaves.
Admittedly, I don't quite see the connection to ind completions since the blog article does not directly or explicitly mention them. However, an alternative name for "-compact object" on nlab is given as "-presentable object" so maybe that's got something to do with it.
A locally presentable category is precisely one that occurs as the -ind-completion of a small category with -small colimits. So in particular, categories of sheaves have that property -- when the site is small! The point here is that the site is not small.
Mike Shulman said:
The point here is that the site is not small.
No, but similar logic applies. A large category embeds into via the Yoneda embedding. This means one can generate any -valued presheaves via large colimits of representables (IE, the objects of ). The -valued presheaves should be precisely the ones where you don't need colimits that are "too large" to construct using the representables. I hope that makes sense! In any case, this is certainly something I will keep thinking about in the background...
John Onstead said:
The Set-valued presheaves should be precisely the ones where you don't need colimits that are "too large" to construct using the representables.
Tempting, but nope. First, if is not locally small, then the representables won't even be Set-valued. The category of nets should be locally small, so this isn't a problem, and then all small colimits of representables will be Set-valued; but there's no reason for the converse to hold.
We can give an informal argument for this using cardinality. If the cardinality of is , as it will be for the category of nets, then there will also be only -many small colimits of representables (up to isomorphism). But there are many functors , and unless is very special I expect there will also be that many isomorphism classes of them.
Hmmm... well while I work more on that idea, I want to share another thing I've also been working on (my apologies if this is a lot of text!): extending the graph-relation connection to more obscure forms of relations- for instance, metric spaces. A metric space can be thought of as a relation, not indexed by the set of truth values, but rather indexed by the set of real numbers. More generally, as we learned in the discussion on (T,V)-algebras, we can index a relation by any quantale V. My question is: how can we fit this in with our discussion of relations vs gluings?
I've thought about this for a bit and have a few basic ideas. My first is that instead of having a site with a point and one other object, we have a site where the objects other than the point are indexed by some element of the quantale. For example, if we let the quantale be the real numbers, then every site object (and thus representable in the topos) will be a "ball" of some real number size around whatever it is we are studying convergence of (a point, sequence, net, etc.)
As an example, let's consider "subsequential approach spaces" which are exactly the same as subsequential spaces, except instead of a binary relation between sequences and points, it's a real number indexed relation- that is, there's a distance function assigning a real number "distance" between a sequence and a point. Our site may consist of the point and a whole bunch of copies of something like the one point compactification of the natural numbers, one for each real number.
Now, let's go to the sheaf topos and find the representable corresponding to the real number 3. A morphism from this representable will do the same thing as in the topological topos- pick a sequence and a point that the sequence converges to. But here, a sequence from the representable for 3 only "converges" to a point if the "distance" between the sequence and the point is less than or equal to 3. In other words, we can think of each morphism from a representable as a sequence "surrounded" by a ball of some radius that only exhibits a convergence to some point if that point is in the radius of the ball.
The site itself should actually look like the poset of real numbers, in decreasing order. In the resulting topos, this ordering of the representables is important. For instance, if the "distance" between a sequence and a point is 3 in some space, then there will be a corresponding morphism from the 3 representable to that space. But there should then be a corresponding morphism from any representable greater than 3 as well since we want a sequence to converge to a point so long as the "ball" around the sequence encloses the point. The decreasing ordering gives us that through composition- for instance, there's a morphism from 4 to 3, and so composing with the one from 3 to the space gives us the corresponding map from 4 to the space we want. A contravariant functor on this site representing some space sends an object for some real number to the set of pairs of sequences and points such that the distance function for the space assigns a value less than or equal to that real number. This creates a chain of functions of sets now in increasing order (due to contravariance) that sends pairs of sequences and points to themselves within the more expanded ball/scope.
Anyways, I have no idea if this idea is good or not. Let me know what you would do if asked to extend the discussion we've been having to general quantales!
I would probably say "Sounds interesting, but I don't have time to think about it deeply." (-: But your idea sounds potentially interesting, I'd encourage you to develop it precisely.
Mike Shulman said:
I would probably say "Sounds interesting, but I don't have time to think about it deeply." (-: But your idea sounds potentially interesting, I'd encourage you to develop it precisely.
Thanks, I certainly will!
That analysis was to solve the relation-graph correspondence for (T, V)-algebras halfway (the V part namely). But there's still the T part, which will ultimately determine the category and site we want to consider. To this extent, I want to find some sort of machinery that takes in a monad on Set and spits out some site such that the category of sheaves on that site will correspond in a nice way to relations from the monad to sets. For instance, given the identity monad on Set, this machinery will produce the walking quiver; given the sequence monad on Set, this machinery will produce the site for the topological topos; and given the net monad on Set, this machinery will produce the net site we've been discussing above.
However, I'm much less confident on how to find the solution to this problem than with the analysis given above. I know that monads do correspond to categories, finite product categories to be specific (and if you have a finitary monad, it corresponds to a special finite product category known as a Lawvere theory), but I'm not sure how or even if the information we need can be extracted from the finite product category corresponding to the monad. Do you know of any construction that would do this? If not, would you at least know if there's anything you can do to the identity monad on Set to get something like the walking quiver? Anything would help!
I would expect it'll only be possible for very special monads.
I know that monads do correspond to categories, finite product categories to be specific.
That's not really true, right? Or is there some theorem here I don't know? Maybe you're deliberately talking in a loose and hand-wavy way here.
(and if you have a finitary monad, it corresponds to a special finite product category known as a Lawvere theory)
That's right (for a finitary monad on Set).
It's true that every monad on Set can be represented by a certain sort of "infinitary Lawvere theory", right? Although that's not a finite product category but an infinitary product category.
Yes, I misspoke (or mis-typed?) I meant to say "monads correspond to product categories", not "finite product categories". My mistake!
Mike Shulman said:
I would expect it'll only be possible for very special monads.
Well, as you mentioned earlier, this is technically new ground. So maybe this process I'm looking for just hasn't been developed yet. That's a shame, but I'm confident it does exist in some form out there...
For instance, I was reading about ultrafilters in this section where apparently there's a way to make ultrafilters representable in a similar way to making sequences representable with the topological topos site. Though I'm not sure if there's any significant connection there, it's just something I noticed.
John Onstead said:
With all this covered, I actually want to create a true "topological topos". A topological space can be defined in terms of convergence as a binary relation between a set of ultrafilters and the set itself. While Top is not a quasitopos and so topological spaces cannot be expressed directly as bisheaves, if one relaxes the axioms of the binary relation between the ultrafilters and the set, it will give the notion of "pseudotopological spaces", which do happen to form a quasitopos. So let's try and find the topos that contains the pseudotopological spaces (and thus the topological spaces).
If we try to do this straightforwardly following from the example of the topological topos, we need to replace the natural numbers with the "ultrafilter space". Then, we can do a one point addition (or compactification) of this space, and use that as the basis of our site. In the resulting presheaf category, every morphism from the corresponding representable to a pseudotopological space will select an ultrafilter and a point that ultrafilter converges to, just like with sequences in the topological topos. We can then think of both topological and pseudotopological spaces as sets of ultrafilters that have been "glued together".
That's all well and good, but when I looked onto nlab on the page "quasitopos", it mentions explicitly that pseudotopological spaces are not Grothendieck quasitopos while subsequential spaces are. This means, apparently, that they cannot be expressed as bisheaves, such as those that are sheaves with respect to the standard topology on my site and separated with respect to the double negation topology on this site. But this doesn't make any sense- I just replicated exactly what you can do for subsequential spaces over for pseudotopological spaces. Can someone explain what is going on?
@John Onstead Here are some facts that might be of interest to you. (I haven't followed every detail of this discussion, so I apologise if these facts have already been mentioned somewhere.)
Thus, the quasi-topos hull of the category of topological spaces is given by the category of concrete sheaves w.r.t. the "smooth topology" on topological spaces. (N.B., in AG the smooth topology gives the same sheaf topos as the étale topology. A surjective map between sequential topological spaces is étale in this sense iff the lifts of the tails of the converging sequences are unique (possibly after restricting the tails a bit further). I don't know whether this gives the same Grothendieck topology as the "smooth" topology for topological spaces.)
Adrian Clough said:
- The category of pseudotopological spaces forms the quasitopos hull of the category of topological spaces
I hadn't heard of a quasitopos hull before, but it looks interesting! From what I can see so far it's something that's defined in terms of categories topological over Set (which is one of the kinds of "categories of spaces" I'm interested in!)
Adrian Clough said:
Thus, the quasi-topos hull of the category of topological spaces is given by the category of concrete sheaves w.r.t. the "smooth topology" on topological spaces.
That's quite interesting to note! I'll certainly be thinking about this more...
Though, I will mention that in the above discussion when trying to define spaces in terms of sheaves, I wanted to avoid using Top to do so, since- and this could just be me- it would feel "circular" to do so. While the "site of nets" discussed was a large category, its objects are not spaces so I guess I was still ok with using them to "build up" spaces.
In any case, I have a lot to think about over the holidays... I'm not sure how active I will be on here over the next two weeks (also since I don't know how active this server will be anyways). I might post here or there but I think I will end this present discussion- it's been very engaging and I sure learned a lot (to think, a few weeks ago I had no idea anything about categorical homotopy theory or what infinity categories had to do with it...)
But don't worry I'll be back in the new year with a whole bunch more hopefully engaging topics! One of my New Year resolutions is to finally solve what "locality" is in general in terms of category theory, and I think that will go smoothly now that I know a lot more about all the different kinds of "space" and ways "spaces" are handled in category theory. On the more concrete side of things, I also eventually want to return to the gauge theory discussion and see fiber bundles in action in describing all sorts of physical situations. In any case, enjoy the holidays and I will be sure to return to continue these interesting discussions soon!
Mike Shulman said:
It's true that every monad on Set can be represented by a certain sort of "infinitary Lawvere theory", right? Although that's not a finite product category but an infinitary product category.
I'm scared of "monads without rank", i.e. monads that correspond to algebraic structures that crucially involve arities that are arbitrarily large cardinals, like suplattices. The link seems to suggest that those are okay too, but I haven't put time into understanding it. (It says that some things haven't been proved yet, but those may be irrelevant to the conversation here. It also states a Conjecture followed by a bunch of lemmas that seem to be aiming at a proof.)
John Onstead said:
One of my New Year resolutions is to finally solve what "locality" is in general in terms of category theory...
You claimed you were not the sort of mathematician who works on problems, but I'm glad to see you are. If I don't see you before January 2nd, Happy New Year! I'll be here as usual all the time.
Another place in the nLab where the connection between algebraic theories and monads on (or on for a set of "sorts") is worked out is [[algebraic theory]], which treats the infinitary multisorted case.
Some of this material was written quite a while ago, and there are some nomenclatural kinks that should be worked out between this and similarly named articles, but the link I gave has theorems and proofs.
Okay, so that setup explicitly claims to handle monads of unbounded rank, so I will relax.
If ever you want to come back to this sometime to satisfy yourself that the claims hold, without the pain of having to read through the article [[algebraic theory]], then solving the following puzzle first will possibly be of help:
Define an infinitary Lawvere theory to be a locally small category equipped with a functor that is the identity on objects and which is product-preserving. (Similar to how ordinary finitary Lawvere theories are defined, except we replace the category of finite sets, or a skeleton thereof, by the category of all sets.)
Equivalently, is a left adjoint. If is its right adjoint, we get a monad . There is an induced comparison functor from the Kleisli category, and after some doing, one can show this comparison functor is an equivalence. Conversely, if you start with a monad on , then the free -algebra functor is the identity on objects and preserves coproducts, so that becomes an infinitary Lawvere theory. So "morally", infinitary Lawvere theories and monads on are essentially equivalent. One goes on to show that models of , i.e., product-preserving functors , are equivalent to -algebras.
By the way, back here:
Mike Shulman said:
John Baez said:
Has anyone tried to build something like the topological topos where we replace the role of N by an arbitrary directed set?
I don't think anyone has managed to make exactly this work, but the theory of condensed/pyknotic sets is very similar and serves a lot of the same purposes. I learned at https://mathoverflow.net/q/441610/49 that these topoi satisfy many of the same good properties as Johnstone's topological topos. Informally, the idea is to generalize even further: instead of just considering the 1-point compactifications of nets in the site, you include more general compacta.
I should also have linked to https://mathoverflow.net/q/441838/49.
Happy new year everyone!