You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
John Onstead (Nov 25 2024 at 12:12):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.
John Onstead (Nov 25 2024 at 12:13):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!
Morgan Rogers (he/him) (Nov 25 2024 at 16:45):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?
Morgan Rogers (he/him) (Nov 25 2024 at 16:47):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.
Morgan Rogers (he/him) (Nov 25 2024 at 17:13):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).
Julius Hamilton (Nov 25 2024 at 18:00):This one? :) https://philosophy.stackexchange.com/a/111165/72812
Morgan Rogers (he/him) (Nov 25 2024 at 18:25):That's the one, thanks!
John Onstead (Nov 25 2024 at 20:19):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.
David Egolf (Nov 25 2024 at 20:28):"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.
David Egolf (Nov 25 2024 at 20:29):I'm reminded of this picture, from "The Geometry of Schemes":
picture
David Egolf (Nov 25 2024 at 20:31):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”...
Morgan Rogers (he/him) (Nov 25 2024 at 20:42):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!
John Onstead (Nov 25 2024 at 21:45):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"
John Onstead (Nov 25 2024 at 21:50):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?
Peva Blanchard (Nov 25 2024 at 21:54):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
John Onstead (Nov 25 2024 at 23:48):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!
Peva Blanchard (Nov 26 2024 at 00:26):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?
John Onstead (Nov 26 2024 at 10:51):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!
John Onstead (Nov 26 2024 at 12:21):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!
John Baez (Nov 26 2024 at 15:43):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.
Mike Shulman (Nov 26 2024 at 18:43):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.
John Onstead (Nov 27 2024 at 00:06):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?
John Onstead (Nov 27 2024 at 02:40):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!
Mike Shulman (Nov 27 2024 at 02:47):Yep, that's right! For a while this was an open problem but fortunately it got solved.
John Onstead (Nov 27 2024 at 03:22):Mike Shulman said:
Yep, that's right! For a while this was an open problem but fortunately it got solved.
Glad it was!
John Onstead (Nov 27 2024 at 03:23):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!
Posina Venkata Rayudu (Nov 30 2024 at 07:08):@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.
John Onstead (Dec 02 2024 at 12:23):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!
Mike Shulman (Dec 02 2024 at 16:01):It's mainly a question of how someone chose to define "cohesive site". Maybe it should have been called a "cohesive site with points".
John Baez (Dec 02 2024 at 17:10):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 Onstead (Dec 02 2024 at 18:32):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!
Mike Shulman (Dec 03 2024 at 00:52):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:
John Onstead (Dec 03 2024 at 07:46):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".
John Onstead (Dec 03 2024 at 12:14):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".
John Onstead (Dec 03 2024 at 12:17):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!
Mike Shulman (Dec 03 2024 at 17:11):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.
Mike Shulman (Dec 03 2024 at 17:13):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.
Mike Shulman (Dec 03 2024 at 17:14):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.
Mike Shulman (Dec 03 2024 at 17:33):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 .
Mike Shulman (Dec 03 2024 at 17:37):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.
John Onstead (Dec 03 2024 at 18:19):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?
John Onstead (Dec 03 2024 at 18:22):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"?
Mike Shulman (Dec 03 2024 at 18:23):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.
Mike Shulman (Dec 03 2024 at 18:26):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.
Mike Shulman (Dec 03 2024 at 18:34):I fixed the nLab.
Mike Shulman (Dec 03 2024 at 18:37):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]])
Fernando Yamauti (Dec 03 2024 at 19:19):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.
John Onstead (Dec 03 2024 at 19:22):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"!
Mike Shulman (Dec 03 2024 at 19:24):Where "sufficient" means "sufficient to determine it up to homotopy equivalence", which is much weaker than up to homeomorphism.
John Onstead (Dec 03 2024 at 22:34):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?
Mike Shulman (Dec 03 2024 at 22:45):Given that we don't have any examples of non-Grothendieck cohesive toposes, I wouldn't spend a lot of time worrying about that. (-:
Mike Shulman (Dec 03 2024 at 22:52):(By which I mean, of course, unbounded cohesive geometric morphisms.)
Morgan Rogers (he/him) (Dec 04 2024 at 10:36):Indeed, finite directed graphs are an example of a cohesive elementary 1-topos over finite sets