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Stream: learning: questions

Topic: One Space to Rule Them All?

John Onstead (Nov 11 2024 at 12:32):

In my previous discussion we discussed many kinds of space and generalizations of space, such as how locales generalize (sober) spaces and toposes generalize those. We also discussed pretopological spaces for a bit. So I looked more into all these different kinds of "topological-esque" spaces and found many interesting kinds. But I now want to organize them all into a hierarchy, and so I will need to know which one to put at "the top"- in other words, find a notion of "topological-esque" space that includes all the others, if such a thing is even possible!

To start, we can generalize topological spaces to convergence spaces, which also includes pretopological and pseudotopological spaces. Beyond that, we can directly relax axioms to get to filter space and eventually to preconvergence space. Meanwhile, separately, the nlab page for convergence space gives us another direction of generalization of convergence space: to cluster spaces. Furthermore, just as one can see topological spaces as a particular kind of approach space, one can see a convergence space as a kind of convergence approach space. Next, we can more directly generalize topological spaces to syntopogenous spaces and quasi-topological spaces, but the connection to convergence spaces there is unclear. Lastly, one has a notion of apartness space given under this nlab article which doesn't appear to generalize topological spaces, but also doesn't appear to be generalized by them. There might also be other kinds of non-topological (or generalizations of topological) "topological-esque" space that I'm missing, so feel free to fill me in on more if you have them!

John Onstead (Nov 11 2024 at 12:34):

So here's my questions: for all the spaces I mentioned above (the ones in italics), how would you arrange them in a hierarchy (thus, which one is the "most general")? And if this poset of generality doesn't have a greatest element - that is, none of the spaces I mentioned subsume all the other spaces I mentioned- then which notion of space IS "the most general"? That is, such that the categories of all "topological-esque" spaces I mentioned above embed fully faithfully into this category of "most general spaces". In other words- what is the one "topological-esque" space to rule them all? Thanks!

Madeleine Birchfield (Nov 11 2024 at 15:15):

There are also cohesive sets and cohesive $\infty$-groupoids in some synthetic approaches to topology.

Mike Shulman (Nov 11 2024 at 17:57):

which notion of space IS "the most general"?

None.

John Baez (Nov 11 2024 at 18:00):

Presumably as you keep generalizing you lose more and more useful properties until you get something like "object in an $\infty$-category".

Mike Shulman (Nov 11 2024 at 18:02):

But at some point before that you'll lose the fact that the less general notions embed fully-faithfully in the more general ones, which John Onstead specifically wanted.

Madeleine Birchfield (Nov 11 2024 at 18:12):

Is there any direct relation between the continuous functions between locales/formal topologies and the continuous functions between filter/convergence spaces?

Madeleine Birchfield (Nov 11 2024 at 18:19):

There are also topological space like structures where topological spaces do not embed fully faithfully in them, such as $\sigma$-topological spaces. The continuous functions between $\sigma$-topological spaces are defined such that the inverse image is only a $\sigma$-frame homomorphism rather than a frame homomorphism; thus the forgetful functor from topological spaces to $\sigma$-topological spaces is not fully faithful.

Madeleine Birchfield (Nov 11 2024 at 18:22):

One can also consider more general kinds of topological spaces where the open sets only form a distributive lattice or even a semilattice rather than a frame or $\sigma$-frame.

John Onstead (Nov 11 2024 at 20:38):

Thanks. I guess I figured it would be difficult to define a "most general" notion of space. But I at least still want to unpack more about the relationships between the kinds of space mentioned.

It seems that the first few kinds of space I mentioned take "filters" to be fundamental. A preconvergence space is any set equipped with a relation, known as the convergence relation, between the set of filters and the set itself. If a filter is paired with a certain point in the set by this relation, then it is said to "converge" to that point, and that point is the "limit" of the filter. In essence, this is defining the notion of "limit" as "extra structure" on a set. Of course, pseudotopological, convergence, and filter spaces are special cases as one then imposes additional axioms on these binary relations. I also speculate that cluster spaces are a special case of preconvergence space since they also involve a binary relation with the filter set. But this time it's a "clustering" relation rather than a "convergence" one. Still, since any such relation qualifies as a preconvergence space, I would say this makes cluster spaces preconvergence spaces.

John Onstead (Nov 11 2024 at 20:38):

I hope that all sounds reasonable so far! But I do have some more questions about filters because I want to make sure I understand them well enough to figure out how these kinds of space are interrelated. I'm still not clear on exactly what a filter is supposed to be. For instance, how do you construct the set of all filters on a set (the very set that the binary relation exists between to get a preconvergence space)? Is it related in any way to the powerset of the set, and if so, how?

Madeleine Birchfield (Nov 11 2024 at 21:55):

I've never heard of the term "preconvergence space", but the most general concept of such spaces built out of filters is a set equipped with a relation between that set and the set of its filters.

John Onstead (Nov 11 2024 at 22:03):

Madeleine Birchfield said:

I've never heard of the term "preconvergence space", but the most general concept of such spaces built out of filters is a set equipped with a relation between that set and the set of its filters.

Right, that's the definition of a preconvergence space!

Madeleine Birchfield (Nov 11 2024 at 22:06):

John Onstead said:

But I do have some more questions about filters because I want to make sure I understand them well enough to figure out how these kinds of space are interrelated. I'm still not clear on exactly what a filter is supposed to be. For instance, how do you construct the set of all filters on a set (the very set that the binary relation exists between to get a preconvergence space)? Is it related in any way to the powerset of the set, and if so, how?

A filter, as used here, is a subset of a powerset which is closed under finite intersections and supersets; i.e. given a set $S$, a filter $F$ satisfies the following conditions:

1. The improper subset of $S$ is in the filter $F$
2. If subsets $A$ and $B$ of $S$ are in $F$, then its intersection $A \cap B$ is also in $F$
3. Given subsets $A$ and $B$ of $S$, if $A$ is in $F$ and $A \subseteq B$, then $B$ is also in $F$.

Then since filters of $S$ are subsets of the powerset of $S$, they are elements of the double powerset of $S$, and one can define the set of all filters as a particular subset of the double powerset of $S$ - whose elements are the ones satisfying the conditions above.

Madeleine Birchfield (Nov 11 2024 at 22:23):

Oh I see, the nLab has an article on [[preconvergence spaces]]. I suspect this might be another instance of nLab authors coming up with new names for mathematical structures because there aren't any names being used for this structure in the existing published literature.

John Onstead (Nov 11 2024 at 22:42):

Thanks for the explanation! It's interesting (and maybe a little overwhelming) that the set of filters isn't just related to the powerset but the double powerset!

John Onstead (Nov 12 2024 at 12:22):

Here's my next main question. In math class we learn about the definition of a limit of a function in terms of "epsilon-delta". Meanwhile, in a convergence space, the notion of limit is given not in terms of some "epsilon-delta" machinery, but in terms of the actual structure put on the set to make it into a convergence space. What I want to know is: how do these two notions of specifying a limit converge? For instance, is it possible to, given a function and a point in the domain, find a corresponding filter such that all limits of that filter (in terms of convergence spaces) coincides perfectly with all limits of that function at that point (in terms of epsilon-delta)?

Also, let's say that there is a way of doing this. The whole reason why limits of functions are useful in the first place is because they can be defined for discontinuous functions. Otherwise we'd just always take the value of a function at a point! But according to nlab, the functions in the category $\mathrm{Conv}$ - even in spite of convergence spaces being more general than topological spaces- are still "continuous", at least in some sense of the term. So when the only functions between convergence spaces are continuous, what exactly would it even mean to take a limit of a discontinuous function- either in terms of convergence spaces or in terms of epsilon delta, categorically speaking?

Matteo Capucci (he/him) (Nov 12 2024 at 17:52):

John Onstead said:

The whole reason why limits of functions are useful in the first place is because they can be defined for discontinuous functions.

I'm not sure about that. Usually you take limits of functions which have gaps in their domain of definitions, but if $f(x) \to L$ as $x \to x_0$ then you can extend the domain of $f$ with $x_0$, put $f(x_0) = L$, and the resulting function is continuous. So the continuity is there.

Matteo Capucci (he/him) (Nov 12 2024 at 17:53):

Even with blow-up, the issue is about the codomain of definition of $f$

Mike Shulman (Nov 12 2024 at 18:03):

Anyway, it's not true that the only functions between convergence spaces are continuous. The morphisms in the category of convergence spaces are the continuous ones, but we can certainly talk about discontinuous functions between (the underlying sets of) convergence spaces.

Mike Shulman (Nov 12 2024 at 18:04):

Abstractly, $\epsilon$-$\delta$ arises by a two-step process. First, any metric space induces a topology whose subbasis is the balls of finite radius. Second, any topology induces a notion of convergence in the usual way.

Mike Shulman (Nov 12 2024 at 18:04):

That's $\epsilon$-$\delta$ for the limits of sequences, nets, or filters.

Mike Shulman (Nov 12 2024 at 18:06):

For "limits of functions", abstractly in terms of convergence, when we say $\lim_{x\to a} f(x) = L$ whan we mean is that for the neighborhood filter $N_x$ of $x$, the pushforward filter $f(N_x)$ converges to $L$. This makes sense whether or not $f$ is continuous.

John Onstead (Nov 12 2024 at 19:18):

Mike Shulman said:

Abstractly, $\epsilon$-$\delta$ arises by a two-step process. First, any metric space induces a topology whose subbasis is the balls of finite radius. Second, any topology induces a notion of convergence in the usual way.

I see, so $\epsilon$-$\delta$ is a special case of limits (for topological spaces, which are themselves special cases for limits of convergence spaces) when dealing with metric spaces. So the minimal structure needed to do $\epsilon$-$\delta$ is a metric space structure.

Mike Shulman said:

For "limits of functions", abstractly in terms of convergence, when we say $\lim_{x\to a} f(x) = L$ whan we mean is that for the neighborhood filter $N_x$ of $x$, the pushforward filter $f(N_x)$ converges to $L$. This makes sense whether or not $f$ is continuous.

I see, so the filter generated by a point and function is the "pushforward filter". I'm still not sure exactly what this is however, since I still haven't found a clear explicit definition for it. Let's say that an ultrafilter contains some subset $S$, is the pushforward filter just given by evaluating $f$ for each $S$? That is, it's the collection of all subsets in $Y$ that $f$ sends the filter's subsets in $X$ to? Or is it more complicated than that?

Mike Shulman (Nov 12 2024 at 19:27):

The pushforward $f(F)$ of a filter $F$ under a function $f:A\to B$ is the filter on $B$ generated by the set $\{ f(S) \mid X\in S \}$. This set may not be a filter in $B$ itself.

Mike Shulman (Nov 12 2024 at 19:29):

John Onstead said:

So the minimal structure needed to do $\epsilon$-$\delta$ is a metric space structure.

Weeell... you can make do with a pseudometric, or a quasimetric, or a quasipseudometric, ...

And if you're willing to add in some quantification over metrics, you can make do with a (quasi) [[gauge space]].

Madeleine Birchfield (Nov 12 2024 at 20:09):

Hypothetically, the minimal structure on a set $S$ needed to define epsilon-delta stuff is simply a ternary relation $x \sim_\epsilon y$ indexed by elements $x \in S$ and $y \in S$ and positive real number $\epsilon \in \mathbb{R}_{+}$.

John Onstead (Nov 12 2024 at 20:54):

Mike Shulman said:

The pushforward $f(F)$ of a filter $F$ under a function $f:A\to B$ is the filter on $B$ generated by the set $\{ f(S) \mid X\in S \}$. This set may not be a filter in $B$ itself.

I think this makes sense... the resulting set would be a "filter basis" or "prefilter" which I believe becomes a filter under upwards closure (so in this case, "generation" is just taking the upwards closure operator), right?

Mike Shulman said:

And if you're willing to add in some quantification over metrics, you can make do with a (quasi) [[gauge space]].

That's interesting, I've never heard of a gauge space before. On my first thought I thought it had to do with gauge theory (so principal bundles, smooth manifolds, and all that) but it looks like something very different to that! From what I have read so far, I can see epsilon-delta working just fine for a typical gauge space since gauge spaces are topological spaces. I might object to this working for quasi-gauge spaces since the article almost made it sound like not every quasi-gauge space was a topological space.

Mike Shulman (Nov 12 2024 at 20:59):

John Onstead said:

the resulting set would be a "filter basis" or "prefilter" which I believe becomes a filter under upwards closure (so in this case, "generation" is just taking the upwards closure operator), right?

Yes.

I thought it had to do with gauge theory (so principal bundles, smooth manifolds, and all that) but it looks like something very different to that!

Yes, totally different meaning of "gauge".

From what I have read so far, I can see epsilon-delta working just fine for a typical gauge space since gauge spaces are topological spaces. I might object to this working for quasi-gauge spaces since the article almost made it sound like not every quasi-gauge space was a topological space.

No, every quasigauge space has an underlying topology, and moreover every topological space can be induced by a quasigauge.

John Onstead (Nov 12 2024 at 22:30):

Thanks!

Mike Shulman said:

Anyway, it's not true that the only functions between convergence spaces are continuous. The morphisms in the category of convergence spaces are the continuous ones, but we can certainly talk about discontinuous functions between (the underlying sets of) convergence spaces.

I think I might know how this might be done, but I want to make sure it makes sense. So $\mathrm{Set}$ embeds into $\mathrm{Top}$ as part of the adjunction with the forgetful functor. So any category that $\mathrm{Top}$ embeds into, such as $\mathrm{Conv}$, will also have this feature by composition of the embeddings. The image of this embedding is on all "discrete space", allowing us to view sets as spaces. Using this, is it possible to define a discontinuous function as simply a "continuous" map from a set (viewed as a discrete space) into some other space? If so, we can do this (and thus consider limits) entirely within the context of $\mathrm{Top}$ or $\mathrm{Conv}$ without having to toss out all our structure.

Mike Shulman (Nov 12 2024 at 22:33):

Well, normally when you would talk about a discontinuous function, the domain as well as the codomain is given with a topology, so you have to "forget" the topology. The simplest way is to just say it's a function $U(X) \to U(Y)$, where $U: \rm Top \to Set$ is the forgetful functor. You can also say it's a morphism $\flat(X) \to Y$, where $\flat(X)$ is $X$ retopologized discretely, i.e. $U(X)$ embedded discretely as you describe.

John Onstead (Nov 13 2024 at 00:02):

Ah, I guess you could say that $\flat(X) \to Y$ is a Kleisli morphism of some kind for the space $X$ with the monad being the one you get by composing the forgetful functor $U: \mathrm{Top} \to \mathrm{Set}$ with the free functor back.
Maybe I'm thinking backwards since normally a Kleisli morphism is into the monad applied to an object, but I think this still works! Maybe it's like a "co-Kliesli" morphism of some sort...

Mike Shulman (Nov 13 2024 at 00:05):

Yep, it's a (co-)Kleisli morphism for a comonad!

John Onstead (Nov 13 2024 at 00:07):

When I was revisiting the nlab page for "filter", I saw the following paragraph: "Filters of subsets form a category whose simplicial category provides a somewhat more formalisation of the intuition of “nearness” than the usual topological one; in particular, it contains the categories of topological and of uniform spaces, of simplicial sets, and of filters themselves, allowing to reformulate in terms of this category notions such as limit, equicontinuity, locally trivial, and geometric realization."
I'm interested in understanding this passage. I know a topological space, and many of its generalizations, can be described as the set of filters equipped with a binary relation of "convergence". But this approach seems to be different- it's taking filters themselves as fundamental, not a relation on filters. I'm therefore wondering how exactly this generalizes topological spaces, and connects with the notion of a topological space as a relation on filters rather than a simplicial object in the category of filters.

Mike Shulman (Nov 13 2024 at 00:14):

I believe that paragraph was added by an anonymous crank and should be removed.

Madeleine Birchfield (Nov 13 2024 at 00:33):

That anonymous user put the paragraph in there on 9 April 2021. It seems the same anonymous user created the article [[situs]] on the nLab nine days prior on 31 March 2021, where it is linked from the filter article via the term "simplicial category" in the paragraph. Notably, the anonymous user claims something similar to the paragraph in the nForum's discussion page for the situs article here.

John Onstead (Nov 13 2024 at 01:02):

Oh, my bad. I'll disregard what it says for now...

John Onstead (Nov 13 2024 at 01:09):

I also think I figured out at least part of why I was so confused with limits above. The only reason I was insisting that we find some morphism in $\mathrm{Top}$ or $\mathrm{Conv}$ for some discontinuous function was because I thought that "taking a limit" was something that happened in the category $\mathrm{Top}$. But upon further reflection, under the filter definition of a limit, limits aren't being taken in the category $\mathrm{Top}$, they are being taken "inside of" the objects of $\mathrm{Top}$. Which, I guess, is "happening" in $\mathrm{Set}$, at least at some level in some capacity.
To clarify, consider the category $\mathrm{Grp}$. The act of "taking the product" of two elements of a group isn't something that's done in the ambient category- it's something that's done inside a group inside $\mathrm{Grp}$. There isn't anything in the ambient category that allows you to define "taking a product of elements" (and if it does, maybe it's a coincidence and probably doesn't generalize well to other "sets with binary products"). Instead, the product is defined- and taken- in $\mathrm{Set}$. So somehow, not everything you can do related to a mathematical object "makes it into" the actual category of those mathematical objects.
Anyways, I hope I'm making sense. Am I on to something here or am I going math crazy?

David Michael Roberts (Nov 13 2024 at 01:19):

I strongly suspect that the page [[situs]] was added by Misha Gavrilovich. Certainly has his own ideas, and that page has broken formatting etc (and is not really a good fit for the nLab), but I wouldn't call him a crank. Compare https://mishap.sdf.org/6a6ywke/6a6ywke.pdf

Mike Shulman (Nov 13 2024 at 01:25):

Ok, well, I think at least that paragraph should be removed from the "Idea" section as it suggests this is a fundamental and well-established use of filters. It would be more appropriate in a "related pages" section.

John Onstead (Nov 13 2024 at 12:40):

So I've been reading more into defining spaces in terms of "distance", and came across approach spaces. I think they are interesting but there's a few points I'm confused about. First, every quasi-pseudometric $d$ ("Lawvere metric space") gives rise to an approach metric $D$ in accordance with $D(x,A) = \mathrm{inf}(d(x,a): a \in A)$. But also, every quasi-pseudometric has an underlying topological space, which itself can be given an approach metric. However, the approach metric for a topological space is always two-valued. Thus, the approach metric for a quasi-pseudometric space and its underlying topological set are somehow different from one another! My question is: is there any way to connect them (well, besides a potential manual way of doing it, maybe I'm looking for a way of doing the conversion without ever stepping foot in $\mathrm{Top}$ and so working entirely with approach metrics, if possible).

My second question is about the specific way that gauge spaces- and also importantly quasi gauge spaces- intersect with approach spaces. The nlab for "approach space" gives a conversion from a gauge- in fact a gauge base- to approach metric, but it differs from the one that wikipedia gives here. The one on Wikipedia even seems to be defining not a gauge but a quasigauge, but I could be wrong. Wikipedia also insists that the conversion from gauge to approach metric is "invertible"- does this imply an equivalence of categories somewhere? Thanks for helping me untangle these notions!

Mike Shulman (Nov 13 2024 at 15:57):

I don't know the answer to either one, although I do note that the definition of "gauge" on that wikipedia article has an extra condition relative to the "usual" one at [[gauge space]].

Mike Shulman (Nov 13 2024 at 15:58):

You might also be interested in the circle of work around so-called (T,V)-algebras, a.k.a. "quantalic" generalized multicategories, which include many of these different kinds of space.

Jean-Baptiste Vienney (Nov 13 2024 at 16:01):

There is also this book on the topic Mike mentioned: Monoidal topology: A categorical approach to Order, Metric and Topology (the authors of the linked paper are part of the authors of the book)

John Onstead (Nov 14 2024 at 03:16):

Hi! Thanks so much for the pdf link to the book. I've already gone through a large part of it and I'm blown away by the amount of different kinds of "space" are classified by the (relatively) simple (T,V)-algebra scheme (as well as its extensions to (T,V)-graphs and such).

John Onstead (Nov 14 2024 at 03:21):

I am a little disappointed the book never mentions "convergence space" by name directly. The closest it gets is to define an extension of the filter monad $F$ onto $\mathrm{Rel}$ (IE, by setting $V = 2$), but it only ever "goes up to" pseudotopological spaces. For instance on pg 220, it defines a category $(F,2)\mathrm{-RGrph}_{Ps}$, which is a full subcategory of $(F,2)\mathrm{-RGrph}$ on pseudotopological spaces, but it never tells you what $(F,2)\mathrm{-RGrph}$ itself is. I think there's good reason to believe this is exactly the category of convergence spaces, but I don't think I'm up to try to prove something like that, at least not yet!

John Onstead (Nov 14 2024 at 08:54):

Update to the above: I reviewed the book and it actually states outright that $(F,2)\mathrm{-RGrph}$ corresponds to the first two axioms ("centrality" and "isotone") of any convergence relation. So going by the nlab this would qualify as a "filter space", but many authors also call it a "convergence space". So I guess it depends on the terminology used! Then upgrading the right unital property to the unital property gets to the category $(F,2)\mathrm{-UGrph}$, which is just pretopological spaces. It thus makes me wonder if one can define a property "between" being right unital and unital to get the nlab's convergence space (whose third axiom of being "directional" is a much weakened version of the third axiom of pretopological spaces). One doesn't seem immediately apparent but it might just be a more complicated property.

Madeleine Birchfield (Nov 14 2024 at 12:19):

I think the nLab's convergence space article has a sentence saying that different authors use different definitions for what a "convergence space" is.

John Onstead (Nov 14 2024 at 12:38):

Here's a question I had about what I read yesterday. I've seen it mentioned that this philosophy of $(T,V)$-algebras is related to that of generalized multicategories. But I have some trouble seeing how this is. Here's why:

In the philosophy of $(T,V)$-algebras, the objects are just that- algebras (well, lax algebras). Indeed, given the category $\mathrm{Mat}(V)$ for relations from some quantale (or, as will be important later, general monoidal category) $V$, the monad $T$ on $\mathrm{Set}$ might extend to one $T'$ on $\mathrm{Mat}(V)$. A $(T,V)$-algebra is then just a lax algebra of $T'$ in $\mathrm{Mat}(V)$.

On the other hand, the philosophy of $(U,W)$-multicategories deals not with algebras of a monad, but monads themselves! Not only that, but these monads aren't in $W$. Instead, given a (virtual double) category $W$, a $(U,W)$-multicategory is a monad in (not on!) the category $\mathrm{HKleisli}(U)$. So we can't just substitute $W$ = $\mathrm{Mat}(V)$ and $U$ = $T'$ to get a convergence between the two philosophies, because such a generalized multicategory would be a monad in the category $\mathrm{HKleisli}(T')$ and not an algebra of $T'$ in $\mathrm{Mat}(V)$! So what is going on here?

Mike Shulman (Nov 14 2024 at 15:25):

Check out my paper with Geoff Cruttwell on generalized multicategories.

Nathanael Arkor (Nov 14 2024 at 16:05):

To be completely explicit, if you look at the expanded definition of a T-monoid (e.g. immediately after Definition 4.3), you will see that it has exactly the structure of a "lax loose T-algebra" in a suitable sense. So there really is no difference between T-monoids and lax T-algebras. (See also the remark at the end of section 2 in Day and Street's paper Lax monoids, pseudo-operads, and convolution.)

John Onstead (Nov 14 2024 at 20:35):

That seems quite odd! If it's really true, then wouldn't this imply that every single algebraic structure is a "generalized multicategory"? After all, if you define an "algebraic structure" to be a model of a Lawvere theory, then you know all such models are equivalent to algebras for some finitary monad on $\mathrm{Set}$. But if all algebraic structures are algebras of some monad, and all monad algebras are monoids in the horizontal Kleisli category for that monad (are generalized multicategories), then it follows that algebraic structures are generalized multicategories, no? I'm confused!

Mike Shulman (Nov 14 2024 at 20:47):

Well, there are some conditions on the monad so that it lifts to some double category and you can define a horizontal Kleisli construction for it. But modulo that, yes, every algebraic structure gives rise to a generalized multicategory, in the same way that every monoidal category gives rise to an ordinary multicategory. Of course, not every generalized multicategory arises in this way, only the "representable" ones.

Mike Shulman (Nov 14 2024 at 20:48):

What generalized multicategories are the same as is lax algebras for the induced monad on the horizontal bicategory. Which are more general than ordinary strict algebras for the original monad on the vertical category.

John Onstead (Nov 14 2024 at 22:06):

Mike Shulman said:

Well, there are some conditions on the monad so that it lifts to some double category and you can define a horizontal Kleisli construction for it.

Wouldn't every monad $T$ on $\mathrm{Set}$ lift automatically to one on $\mathrm{Sq(Set)}$? In which case, the horizontal Kleisli category is exactly the same as the vertical one (since the horizontal morphisms are the same as the vertical ones in that double category). Well, maybe except for the "lax" part of it, as you mentioned.

Mike Shulman (Nov 15 2024 at 01:53):

Yes, if you're willing to consider "degenerate" double categories like Sq(Set), you can do that. But note that Sq(Set) is not an equipment, and the theory of generalized multicategories works better with an equipment.

Mike Shulman (Nov 15 2024 at 01:54):

(And since Sq(Set) is locally discrete, lax algebras are the same as strict ones.)

John Onstead (Nov 15 2024 at 02:05):

Ah, good to know!
I did some more reading of the article in the context of $\mathrm{VMat}$ and I had some questions about it. First, the article only mentions $V$ up to being a monoidal category. But VDC theory allows you to enrich in objects beyond monoidal categories, up to and including VDCs themselves. In fact, a monoidal category is just a one object bicategory, which is just a special kind of double category, which is just a special kind of VDC. This leads me to wonder why the paper never talks about $\mathrm{VMat}$ for $V$ a general VDC, and if its category of monads $\mathrm{Mod(VMat)}$ will correspond precisely with VDC-enriched categories.

Secondly, that's not the only generalization of $\mathrm{VMat}$ I can think of. $\mathrm{VMat}$ is inherently based in $\mathrm{Set}$, as its objects are sets and "generalized relations" between them. But why can't you define a VDC $\mathrm{Mat}(C,V)$ for some arbitrary category $C$ and monoidal category $V$, such that $\mathrm{VMat}$ = $\mathrm{Mat(Set},V)$? Then maybe extending monads from $\mathrm{Set}$ onto $\mathrm{VMat}$ can be functorial, if there exists a functor $\mathrm{Mat}(-,V)$ that would send a monad on $\mathrm{Set}$ to some natural (perhaps even the canonical) lax extension on $\mathrm{VMat}$.

Mike Shulman (Nov 15 2024 at 02:58):

Yes, you can define $V \rm Mat$ for any VDC $V$, and its category of monads wil indeed be "categories enriched in $V$".

Mike Shulman (Nov 15 2024 at 03:00):

If you want to generalize from Set to some other category $C$, then you need a notion of "$C$-indexed family of objects of $V$", which is usually interpreted as meaning that you need $V$ to be a "$C$-indexed category", i.e. a pseudofunctor $C^{\rm op} \to \rm Cat$, or equivalently a Grothendieck fibration over $C$. In this context there is a notion of "enriched indexed category" that I wrote about in this paper, and which can indeed be obtained as monads in some VDC of matrices.

John Onstead (Nov 15 2024 at 12:49):

I've been trying to understand that article for a while now- I think I'm getting closer! Here's what I gather so far. It seems given some generic monoidal fibration $V: C^{op} \to \mathrm{MonCat}$, there exists a construct called the "frame construct" $\mathrm{Fr}(V)$ which can be used to find the virtual equipment of $V$-categories (by taking the Mod construction, as a special case of generalized multicategories). Given a monoidal category $M$, one gets a monoidal fibration $\mathrm{Fam}(M): \mathrm{Set}^{op} \to \mathrm{MonCat}$ that sends a set $X$ to the category of $X$-indexed families of $M$ objects. Apparently also, $\mathrm{Fr(Fam(}M)) = \mathrm{Mat}(M)$. So I guess my construction $\mathrm{Mat}(C,M)$ would involve somehow getting a monoidal fibration $\mathrm{Fam}(C,M): C^{op} \to \mathrm{MonCat}$ that sends an object $Y$ to the category of $Y$-indexed families of $M$ objects. But I'm not exactly sure if- or even how- this would work, since how can you index objects of one category with something other than a set?

In addition, I wanted to ask about the "chirality" of VDCs in general. Unlike lower categories, VDCs appear inherently chiral in that the appropriate notion of map between them are lax functors. But there's another notion of oplax functor that could also have been chosen. I guess my question is- how is this chirality resolved, and how do you define an oplax functor between VDCs? Is there a way to "convert" a general oplax functor into a lax functor? For instance, say I have an oplax functor between two double categories $F: C \to D$. Considering double categories as special cases of VDCs, what action can I take within the category/VDC $\mathrm{VDC}$ that will allow me to perfectly replicate the effects of this map $F$?

Mike Shulman (Nov 15 2024 at 15:40):

how can you index objects of one category with something other than a set?

By postulating it. You have to start with an indexed monoidal category; you can't expect to make one if you're just given a category C and a monoidal category M.

Although there are a number of general way to construct indexed monoidal categories from other input data, as described in the paper.

Put differently, an indexing of $V$ over $C$ is extra structure, which there might be many or no ways to construct given $V$ and $C$.

Mike Shulman (Nov 15 2024 at 15:43):

what action can I take within the category/VDC VDC that will allow me to perfectly replicate the effects of this map F?

I don't know of any. I mean, you can observe that your VDCs are representable and define an oplax map between representable VDCs, but that's really just going back to double categories.

John Onstead (Nov 15 2024 at 20:54):

Mike Shulman said:

Put differently, an indexing of $V$ over $C$ is extra structure, which there might be many or no ways to construct given $V$ and $C$.

I think I can see how this might work. If you start with the category $V^X$ for $X$ some set, and you consider some group structure on $X$ which we will call $G$, then you can turn the category $V^X$ into the category of $X$-indexed families of objects in $V$ with a group structure $G$ on top. We can then take $\mathrm{Fam(Grp,} V): \mathrm{Grp}^{op} \to \mathrm{MonCat}$ that sends $G$ to the described category. I haven't thought too much about what the resulting enriched index category would be, but it seems it would be some mix of internal and enriched category. It might have a group of objects and a group of hom-objects, but where the hom objects would be objects of $V$. If that's true, then I guess that would be more or less like I'd pictured!

John Onstead (Nov 15 2024 at 20:56):

What I want to think about now is how multicategories are dealt with from the monoidal fibrations perspective (since this wasn't, at least not that I believe, mentioned explicitly in the paper). So an internal category is one corresponding to the self-indexing monoidal fibration, and an enriched category is one corresponding to the family indexing monoidal fibration. What are the corresponding analogous monoidal fibrations that give rise to both internal and enriched multicategories? And how does this tie in with multicategories being $T$-multicategories where $T$ is the free monoid monad on $\mathrm{Set}$?

Mike Shulman (Nov 15 2024 at 21:20):

Multicategories are not an instance of enriched indexed categories.

John Onstead (Nov 15 2024 at 23:25):

I should've figured this would be the case!

John Onstead (Nov 15 2024 at 23:25):

I still want to see how the notions of monoidal fibrations and generalized multicategories relate. For instance, can they be combined- maybe to define a notion of "enriched indexed $T$-multicategory"? In the article "Unified Framework for Generalized Multicategories", it says that if you have a $T$-algebra in $\mathrm{Set}$, then a $V$-enriched virtual $T$-algebra is a corresponding $T$-multicategory in $V\mathrm{Mat}$. I think this is a case of representability, where vertical $T$-algebras in a virtual equipment might correspond with "representable" $T$-multicategories in that same equipment. Now as we know, $V\mathrm{Mat} \cong \mathrm{Frame(Fam(}V))$, which is the frame of a set-indexed monoidal fibration. Which leads me to wonder if we can extend $V\mathrm{Mat}$ in the above description of an enriched virtual $T$-algebra to an arbitrary $\mathrm{Frame(}M)$ where $M$ is an arbitrary (but still set-indexed) monoidal fibration. The problem then becomes trying to find out if it's possible to extend some monad $T$ on $\mathrm{Set}$ to $\mathrm{Frame(}M)$ so that the $T$-algebras are the same.

For instance, consider $T$ = free monoid monad. An algebra of this is a monoid, and the virtual equivalent is a multicategory. If we take $M$ to be the self-indexing of $\mathrm{Set}$, then this is a $\mathrm{Set}$-enriched multicategory. Of course, $\mathrm{Frame(}M)$ is then just $\mathrm{Span(Set)}$, and there's a simple extension of $T$ onto this, giving us "normal" multicategories. Next, we can choose $M$ to be $\mathrm{Fam(}V)$, then $\mathrm{Frame(}M)$ is just $V\mathrm{Mat}$. Extending $T$ onto this category is slightly more involved since it involves solving the lax extension problem, but it can still for the most part be done. This gives us $V$-enriched multicategories. So the question is: for a more general set-indexed monoidal fibration $M$, how easy is it to solve the extension problem to extend a monad $T$ on $\mathrm{Set}$ to the category $\mathrm{Frame(}M)$, and does this give an appropriate notion of $M$-enriched $T$-multicategory?

Mike Shulman (Nov 15 2024 at 23:38):

I would back up a further step and ask whether there are any other interesting Set-indexed monoidal categories. (If you said "indexed" you don't also have to say "fibration" -- an indexed category is equivalent to a fibration.) In fact any Set-indexed category that satisfies the natural condition of being "a stack for the canonical topology on Set" is automatically of the form Fam(V). There are Set-indexed monoidal categories that aren't stacks (e.g. any [[tripos]]), but they're arguably a bit esoteric.

Mike Shulman (Nov 15 2024 at 23:41):

On the other hand, even the question of whether a monad on Set extends to $V\rm Mat$ is nontrivial. In the case when $V$ is a poset, at least, the "monoidal topology" people have somewhat general constructions that I don't remember. And perhaps if $T$ is finitary, or induced by an operad, you can do something explicit.

Mike Shulman (Nov 15 2024 at 23:42):

You can also generalize the base category: given any category $C$ and a $C$-indexed monoidal category $V$, can we lift a monad $T$ on $C$ to $V\rm Mat$? I don't know a general answer to this question either, but there are certainly examples. E.g. there is an indexed monoidal category of abelian-group objects over any topos, and the free monoid monad on the topos should extend to it, yielding a notion of "internal additive multicategory".

John Onstead (Nov 16 2024 at 02:20):

This is all very interesting! It seems there's certainly still a lot to explore in this field and a lot of directions to take.

Mike Shulman said:

You can also generalize the base category

This also makes the problem 2-dimensional, since not only can you ask whether a monad on $C$ extends to one on $\mathrm{Mat}(C,V)$, you can also ask when a monad on $\mathrm{Set}$ extends to one on $C$. For instance, which class of categories admits a "free monoid" monad like the one on $\mathrm{Set}$ - this would be the class of categories for which multicategories can be internalized.

Then there's also the question of which categories have "internal fibrations" that can be functorially assigned a VDC of the form $\mathrm{Frame}(F)$. Of course, for the VDC $\mathrm{MonCat}$ an "internal fibration" is just a monoidal fibration, and the notion of $\mathrm{Frame}(F)$ is just the one discussed in the paper. But maybe this can be done for other VDCs than just monoidal categories.

John Onstead (Nov 16 2024 at 03:05):

Here's something I've been thinking of in terms of "extending monads". Every finitary monad on $\mathrm{Set}$ corresponds with a Lawvere theory. While monads aren't very extendable in general, theories very much are- all you have to do is just change the target category of your finite product preserving functor (so long as that category also has finite products). So while a "free monoid" monad only exists for some small number of categories, given a finite products category you can derive the category of monoids internal to it with ease via the Lawvere theory. Maybe something like this could be used in some way to figure out if a monad on $\mathrm{Set}$ can be extended. Or even better, perhaps generalized multicategory theory itself can be reformulated in a theory-centric way as opposed to a monad-centric way. For instance, just as one can restrict the category of algebras of a monad to the Kleisli category, perhaps we can construct a "category of horizontal models of a theory" in a VDC and then restrict that to get something analogous to $\mathrm{HKleisli(}T)$. Then you can simply take the Mod construction of that to get a generalized multicategory without having to worry about monads at all. But of course the main problem with this is that the monad-theory correspondence only works on $\mathrm{Set}$ in the unenriched case, but I'm still confident there's a way to extend this correspondence that might help.

Anyways that's all my thoughts for today... I'll probably get back to talking about spaces but I think I might need to rest for a bit first!

Madeleine Birchfield (Nov 17 2024 at 16:54):

John Onstead said:

Madeleine Birchfield said:

I've never heard of the term "preconvergence space", but the most general concept of such spaces built out of filters is a set equipped with a relation between that set and the set of its filters.

Right, that's the definition of a preconvergence space!

I did some digging around in the literature and ended up finding a few authors who did define a notion of "preconvergence space". However, they all defined a preconvergence space as to also satisfy an isotone condition; i.e. a different definition than the one found in the nLab article. See Wikipedia's article on convergence space and the following references:

• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Dolecki, Szymon (2009). Mynard, Frédéric; Pearl, Elliott (eds.). "An initiation into convergence theory" (PDF). Beyond Topology. Contemporary Mathematics Series A.M.S. 486: 115–162.
• Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function spaces and hyperspaces: local properties" (PDF). Houston J. Math. 40 (1): 285–318.

Madeleine Birchfield (Nov 17 2024 at 16:57):

Are there any authors who did define preconvergence spaces without the isotone condition in the existing literature, or do we just have the word of some anonymous user(s) on the nLab?

John Baez (Nov 17 2024 at 17:48):

Btw, if you find good references for concepts that are mentioned on the nLab, you can add them to nLab articles, without worrying too much about how to format them.

Madeleine Birchfield (Nov 17 2024 at 18:10):

John Baez said:

Btw, if you find good references for concepts that are mentioned on the nLab, you can add them to nLab articles, without worrying too much about how to format them.

I would add them, except the concept described on the nLab article is different from the preconvergence spaces defined in the references. So I'm wondering if I should instead rename this current nLab article to something else and start a new article on the preconvergence spaces actually found in the literature. Especially since John Onstead pointed out that the current article's topic also generalises cluster spaces which are meant to formalise clustering rather than convergence.

Madeleine Birchfield (Nov 17 2024 at 18:18):

I think a good name for the space currently described in the nLab article on [[preconvergence spaces]] would be "generalised filter spaces" since they generalise all the spaces defined using filters.

Madeleine Birchfield (Nov 17 2024 at 20:07):

In Non-symmetric Convenient Topology and its Relation to Convenient Topology the author Gerhard Preuss defines a "preconvergence space" to be this other structure:

• a preconvergence space is a preuniform convergence space which is ‘generated’ by its preconvergent filters

which doesn't have anything to do with the other definition by Dolecki and Mynard etc.

Madeleine Birchfield (Nov 17 2024 at 20:13):

See also,

• Lattice-valued preuniform convergence spaces by Jinming Fang
• Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions by Gerhard Preuss

John Baez (Nov 17 2024 at 20:53):

Madeleine Birchfield said:

John Baez said:

Btw, if you find good references for concepts that are mentioned on the nLab, you can add them to nLab articles, without worrying too much about how to format them.

I would add them, except the concept described on the nLab article is different from the preconvergence spaces defined in the references. So I'm wondering if I should instead rename this current nLab article to something else and start a new article on the preconvergence spaces actually found in the literature.

I believe the "official" best thing to do is to raise the issue on the nForum and see if anyone objects, and then change the page title if nobody objects after a while. I think it would be really good for you to join the nForum. But if you don't want to, we could just ask @Mike Shulman what he thinks - he's like the local representative of the nForum here.

In the meantime the easiest thing to do is add the references prefaced by a sentence saying "For a different concept of preconvergence space, see..." And maybe you could explain the difference. This would take 5 minutes and nobody could possibly be offended.

Madeleine Birchfield (Nov 17 2024 at 21:00):

John Baez said:

I believe the "official" best thing to do is to raise the issue on the nForum and see if anyone objects, and then change the page title if nobody objects after a while. I think it would be really good for you to join the nForum. But if you don't want to, we could just ask Mike Shulman what he thinks - he's like the local representative of the nForum here.

I've already raised those issues on the nForum a few hours ago here.

John Baez (Nov 17 2024 at 21:03):

Excellent!

John Baez (Nov 17 2024 at 21:04):

It's quite possible nobody knows enough about this to care very much, but it's also the weekend now, so if nobody replies after - what, a week? a few days? - it's good to just go ahead and take action before you lose interest. People can always change things back if they don't like what you did, but it's nice to see if anyone has opinions.

Mike Shulman (Nov 17 2024 at 21:55):

These days I spend more time here than I do at the nForum. But for whatever it's worth, I agree with your suggestion.

John Onstead (Nov 17 2024 at 23:28):

This is very interesting and important to cover, since it can be really confusing when there isn't a unified terminology for a particular mathematical structure! That's why I like the system of $(T,V)$-algebras and more generally $(T,V)$-graphs since you can uniquely specify a type of space just by specifying some (lax extension of a) monad $T$ and quantale $V$. In this case, what I call a "preconvergence space" is just a $(F, 2)$-graph for $F$ the filter monad and $2$ the poset of truth values- that is, an endofunctor algebra for the lax extension of $F$ to $\mathrm{Rel}$.

Mike Shulman (Nov 18 2024 at 06:07):

I'm not hugely surprised if there isn't uniform established terminology for obscure and centipedey notions like "preconvergence space", though.

John Onstead (Nov 18 2024 at 12:16):

I wanted to return to discussing spaces from the discussing of VDCs by asking a question I've had for some time now. So when you have a virtual equipment like those for generalized multicategories, you can define category theoretic machinery for the objects in that equipment, such as (weighted) limits and colimits. But topological spaces (and, as we've seen, many other kinds of space) can be seen as generalized multicategories. That leaves me wondering: what is the notion of "limit" and/or "colimit" internal to the virtual equipment of topological spaces? Does it correspond with any traditional/standard topological concepts? Is it useful in any way?

Madeleine Birchfield (Nov 18 2024 at 12:34):

John Baez said:

It's quite possible nobody knows enough about this to care very much, but it's also the weekend now, so if nobody replies after - what, a week? a few days? - it's good to just go ahead and take action before you lose interest. People can always change things back if they don't like what you did, but it's nice to see if anyone has opinions.

Well on the nForum, Urs Schreiber just gave me permission to go ahead and change the article so I'll do it right now.

John Baez (Nov 18 2024 at 17:00):

Great! Thanks for doing this.

Mike Shulman (Nov 18 2024 at 17:05):

John Onstead said:

What is the notion of "limit" and/or "colimit" internal to the virtual equipment of topological spaces? Does it correspond with any traditional/standard topological concepts? Is it useful in any way?

That seems like a good exercise for you to work out!

John Onstead (Nov 18 2024 at 20:15):

Mike Shulman said:

That seems like a good exercise for you to work out!

I'll give it a try when I have a little more time later!

John Onstead (Nov 18 2024 at 20:16):

In the meantime... I went back to the article "One Setting for All" you sent me about "$(V,T)$-proalgebras" that got us started talking about generalized multicategories and all that in the first place. In "Unified Framework for Generalized Multicategories", it actually brings up this article but then states "In general, given a VDC $X$, one can define a new VDC pro-$X$, and go on to describe pro-generalized multicategories. Further discussion of this, however, awaits a future paper". My question is: did this "future paper" ever come out?

Mike Shulman (Nov 18 2024 at 21:23):

That phrase wasn't intended to suggest that we had any plans to write that paper. As far as I know, no one has.

John Onstead (Nov 18 2024 at 21:40):

Mike Shulman said:

That phrase wasn't intended to suggest that we had any plans to write that paper. As far as I know, no one has.

Hmm... by any chance, have you ever considered revisiting the subject of generalized multicategories and writing up a "sequel" paper? Of course it wouldn't need to just be about pro-generalized multicategories. The "Unified Framework for Generalized Multicategories" came out all the way back in 2010. Since then, there have been a plethora of new concepts and developments that could be fit into, or help enhance, the framework. For instance, there's been a massive amount of further work in developing formal category theory, including in virtual equipments (such as what was covered in a recent double category seminar advertised on this server). There's also been other candidates for formal category theory settings, like augmented virtual double categories. We also have all the potential ways to expand the subject that we discussed right here above!

In my ideal world I'd of course very much like to do some of this myself. I have a lot of ideas for the subject of VDCs and generalized multicategories, and I'd like the opportunity to see for myself if they are actually useful or functional or not. Alas I'm no mathematician so I don't see that as something that can happen in the near future!

Mike Shulman (Nov 18 2024 at 22:08):

We did originally intend to write one more paper, about the functoriality of the HKl and KMod constructions. We wrote up a lot but never finished it. Maybe one day we'll finish that, but I don't currently foresee going back to generalized multicategories personally other than that.

John Onstead (Nov 18 2024 at 23:33):

Mike Shulman said:

Maybe one day we'll finish that, but I don't currently foresee going back to generalized multicategories personally other than that.

Ah that's too bad!

Kevin Carlson (Nov 18 2024 at 23:34):

John Onstead said:

Ah that's too bad!

Ars longa, vita brevis.

John Onstead (Nov 18 2024 at 23:41):

When it comes to functoriality, I'm wondering about the functoriality of the "pro" construction. It seems like it should be functorial since, as stated in the paper, there should be a "pro-VDC" for every VDC, and it also seems there's a natural way to lift a monad on a VDC to one on the pro-VDC.

But I also wonder... could the "pro" construction not just be one but a whole spectrum of different related constructions? For instance, the usual pro construction takes you from a category where morphisms are relations of a certain kind to a category where morphisms are directed sets of those relations. Might it be possible to come up with a "weaker" notion of "pro" construction that takes you to a category where morphisms are mere sets of the relations (rather than directed sets?) And on the opposite end, maybe there's a "pro" construction that takes you to a category where morphisms are filters of relations, in which case doing this kind of "pro" construction for $\mathbb{R}^+ \mathrm{Mat}$ would give a category such that a (quasi) gauge space is such a "pro"-algebra. If so, this could add a whole third dimension to the study of $(T,V)$-algebras (beyond varying $T$ as well as varying the "strength" of the algebra such as weakening the transitivity axiom), and also might allow us to include things like pregauge spaces or gauge base spaces into the picture!

Mike Shulman (Nov 19 2024 at 00:46):

The "one setting for all" paper mentions that monads in pro(R-Mat) are "prometric spaces". Those are like quasi-gauge spaces except that you don't demand that each distance function individually satisfy the triangle identity but rather that for each distance function there is another that "subdivides" it. They include quasi-gauge spaces as a special case.

John Onstead (Nov 19 2024 at 12:15):

Mike Shulman said:

The "one setting for all" paper mentions that monads in pro(R-Mat) are "prometric spaces". Those are like quasi-gauge spaces except that you don't demand that each distance function individually satisfy the triangle identity but rather that for each distance function there is another that "subdivides" it. They include quasi-gauge spaces as a special case

That makes sense... Above I was trying to find a way to explicitly characterize (quasi) gauge spaces themselves (and alone) as a $(T,V)$-algebra of some kind.

John Onstead (Nov 19 2024 at 12:17):

I was thinking more about the monad extension problem recently... Let's say you had a Lawvere theory $T$ and $\mathrm{Mod}(T, C)$ is the category of models of $T$ in $C$. Is $\mathrm{Mod}(T, -)$, for some fixed $T$, a functor? And if so, would applying it to $\mathrm{Set}$, and a monad on $\mathrm{Set}$, allow for an extension of that monad to any category of models of a Lawvere theory in $\mathrm{Set}$? I can't see how this would work since a vast array of categories take the form $\mathrm{Mod}(T, \mathrm{Set})$ (since we can even go beyond Lawvere theories) and yet don't seem to have equivalents of all $\mathrm{Set}$ monads on them, but at the same time I can't seem to see right away why it wouldn't work.

Mike Shulman (Nov 19 2024 at 17:56):

For a Lawvere theory T and a category C with finite products, Mod(T,C) is equivalently the category of finite-product-preserving functors from (the syntactic category of) T to C. Therefore, Mod(T,-) is a representable functor whose domain is the (2-)category of categories with finite products and finite-product-preserving functors. Thus, if a monad on Set preserves finite products, then you can apply this functor to it and obtain a monad on Mod(T,Set).

You can do the same thing for theories in other doctrines, and in each case the monad will have to preserve the relevant categorical structure in order to lift.

John Onstead (Nov 19 2024 at 20:34):

Mike Shulman said:

You can do the same thing for theories in other doctrines, and in each case the monad will have to preserve the relevant categorical structure in order to lift.

Ah, so that's the catch! Thanks, that makes sense! Especially since the more advanced the doctrine, the less functors will preserve that structure.
But this provides at least somewhat of a systematic answer to if a monad will lift to some category, specifically a category of models- just look at what structure the monad preserves.

John Onstead (Nov 19 2024 at 20:53):

When I was looking at the article for prometric space to learn more about them, I stumbled upon this table at the bottom which seems to summarize some of the $(T,V)$-algebra concepts. But it adds another dimension that we haven't explored yet: topogenies. A quasiproximity space is given as analogous to preorders and quasipseudometrics, which are $(T,V)$-algebras. In addition, a syntopogenous space is then given as analogous to pro-algebras like uniform spaces and prometric spaces. So I have a simple question: for which monad $T$ and quantale $V$ are quasiproximity spaces (and by extension syntopogenous spaces via the pro construction) the $(T,V)$-algebras of?

I tried looking into it myself, but I ran into a problem- a topogeny is a binary relation, but it seems to be one between a powerset and itself. But we need a binary relation between some monad applied to a set and the set itself.

Mike Shulman (Nov 19 2024 at 21:07):

I don't think I know of a (T,V)-style definition of proximity space.

Mike Shulman (Nov 19 2024 at 21:08):

It is possible to generalize generalized multicategories to "generalized [[polycategories]]", a.k.a. generalized [[PROPs]], where you have one monad for the domains of morphisms and a possibly-different monad for their codomains, related by a "horizontal distributive law". But I don't know whether you can make proximity spaces come out of that. (And I don't think anyone has yet written down a general theory of such things.)

John Onstead (Nov 19 2024 at 22:58):

Hmm... we'll here's some insights I have into the matter after thinking about it a bit. Let's say we want to find a set of form $T(X) \times X$ such that functions into $2$ are in bijection with functions from $P(X) \times P(X)$ into $2$. First, we express $P(X) \times P(X) \to 2$ as $2^X \times 2^X \to 2$, and then we can use "exponentiation rules" to get $2^{X + X} \to 2$ or $2^{2X} \to 2$. But this still isn't of the form we want, so let's make it into the form we want by doing the following: $\frac{2^{2X}}{X} \times X \to 2$. But any power of two is only divisible by two (or a lesser power of two), so unless the cardinality of $X$ is a power of two, we can't "divide" in this way (since the category of sets is the categorification of the natural numbers, not the rationals). This suggests it may be difficult if not impossible to find a $T(X) \times X$ for this purpose. Anyways I hope what I did makes sense and wasn't fully nonsensical...

John Onstead (Nov 19 2024 at 23:00):

But I have another solution to the problem! Consider the category $\mathrm{PRel}$ where objects are sets and the morphisms are powerset relations between the sets. Then a proximity space is easily a generalized multicategory for this category under the identity monad. (that is, if the category I described actually makes sense...)

John Onstead (Nov 20 2024 at 00:36):

John Onstead said:

(that is, if the category I described actually makes sense...)

I guess that'd be my next question... does this category actually exist, and if so how can it be constructed?

John Onstead (Nov 20 2024 at 00:37):

Mike Shulman said:

(And I don't think anyone has yet written down a general theory of such things.)

So there's no treatment of generalized polycategories analogous to "A Unified Framework for Generalized Multicategories"? If not, at least there should be something in the works, right?

Mike Shulman (Nov 20 2024 at 00:54):

I've suggested it as a project to more than one person, but so far no one has done it. I think it would make a nice PhD thesis.

John Baez (Nov 20 2024 at 01:41):

Now you've got a thesis topic, @John Onstead.

John Onstead (Nov 20 2024 at 02:04):

John Baez said:

Now you've got a thesis topic

I wish!

John Onstead (Nov 20 2024 at 12:34):

Here's my next question. In the nlab article "apartness space", it states that an apartness space is given by a binary relation between the powerset and the set itself. However, in the book Monoidal Topology, as well as other sources (even including the generalized multicategories article), this is referred to as a "closure space". Maybe they're the same thing? But if they are, then why does the nlab article claim that apartness spaces don't generalize topological spaces? Topological spaces are a case of closure space for sure, since one can describe topological spaces via a closure operator satisfying all the Kuratowski closure axioms. So what's going on here?

Mike Shulman (Nov 20 2024 at 15:21):

To start with, the relation is different: a closure space axiomatizes the relation "$x$ is in the closure of $A$" while an apartness space axiomatizes the relation "$x$ is not in the closure of $A$". In classical mathematics, the two are equivalent, but apartness spaces are usually used in constructive mathematics, and that's the context the page is working in when it says they're not equivalent.

John Baez (Nov 20 2024 at 15:37):

I haven't looked at the page, but I'd hope it would state Mike's first sentence somewhere near the start: it's such a simple and useful thing to say.

Mike Shulman (Nov 20 2024 at 15:54):

Well the "Idea" section says

An apartness space is a set equipped with an “apartness relation” that distinguishes between pairs of points or sets. They are particularly interesting in constructive mathematics; in classical mathematics they tend to be equivalent to better-known definitions expressed in terms of “closeness” rather than “apartness”. [emphasis added]

It can't be more precise than that at that point because the page as a whole is about several different kinds of "apartness space".

Mike Shulman (Nov 20 2024 at 15:55):

The section on "point-set apartness spaces", which this question is about, gives the axioms, then contraposes them and says that those axioms

are precisely the axioms of a topology expressed in terms of a closure operator. In constructive mathematics, of course, the law of contraposition does not hold.

But the lower section "Relation to topological spaces" could probably say more explicitly that the adjunction it discusses becomes an equivalence in classical mathematics.

John Onstead (Nov 20 2024 at 20:10):

Ah that makes sense... constructive mathematics is very confusing sometimes for me!

John Baez (Nov 20 2024 at 20:33):

Many beginning students find topology confusing because "not open" doesn't mean "closed", but little do they know how far the rabbit-hole of negation goes.

Mike Shulman (Nov 20 2024 at 21:04):

Constructive mathematics is very confusing most of the time for most people!

John Onstead (Nov 20 2024 at 21:40):

I did more research on the category I proposed above, $\mathrm{PRel}$. A type of category similar to this, and that also connects with distributivity conditions as mentioned, is a "double" Kleisli category. Given a monad $T$ and comonad $C$, satisfying the distributivity conditions, this kind of category has as morphisms the morphisms $C(X) \to T(Y)$ in the original category. This is close to what I want, but unfortunately it requires not two monads but a monad and a comonad, while the powerset monad is only a monad.

I tried to do something a little different next. First, I considered the category $\mathrm{Rel}$ and the powerset monad extended onto it $P$. But instead of finding its algebras, I applied the Kleisli category construction to get the category $\mathrm{Kleisli}(P)$. The morphisms in $\mathrm{Kleisli}(P)$ are binary relations from a set to the powerset of another set. Here's my question: can you extend the powerset monad again to this Kleisli category? And if so, what would be an algebra for this monad- would it be of the form of a binary relation $P(X) \to P(X)$?

Mike Shulman (Nov 20 2024 at 21:56):

I don't know about that last question. But re: the first paragraph, this is why I think the proper setting for generalized polycategories is a pair of "vertical" monads (not a monad and a comonad) on a VDC, together with a horizontal distributive law between them. That should suffice to define a horizontal double-Kleisli VDC. The connection with the monad/comonad picture is that in an equipment, with both companions and conjoints, a vertical monad induces both a horizontal monad and a horizontal comonad, by taking companions or conjoints respectively.

John Onstead (Nov 20 2024 at 23:32):

Mike Shulman said:

But re: the first paragraph, this is why I think the proper setting for generalized polycategories is a pair of "vertical" monads (not a monad and a comonad) on a VDC, together with a horizontal distributive law between them. That should suffice to define a horizontal double-Kleisli VDC. The connection with the monad/comonad picture is that in an equipment, with both companions and conjoints, a vertical monad induces both a horizontal monad and a horizontal comonad, by taking companions or conjoints respectively.

I see!

John Onstead (Nov 21 2024 at 12:16):

I covered most of the kinds of space I wanted to cover (IE, the topological-esque spaces), so now I want to situate this discussion in terms of the larger concept of "locality". In summary, it seems that from the perspective of $(T,V)$-algebras, a "topological-esque space" is a set equipped with some set of "parts" of the space (whether it be subsets via the powerset monad, filters via the filter monad, ultrafilters via the ultrafilter monad, etc.) and relations involving these parts (that can be either binary or valued in some more general quantale) that "quantify" how "close" (or "apart") the parts of the set are to each other. As we've seen I think this covers most if not all examples of such spaces.

But being "topological-esque" isn't the only notion of "locality" in math. For instance, there are kinds of space with a "higher order" of locality- most notably manifolds and "manifold-esque spaces". So my question here is: how does the approach to locality given by the above summary connect to the bigger picture of locality in general, and more specifically to the example of manifolds? For instance, is there a way to express a manifold (doesn't matter which type, could be topological, differentiable, smooth, etc.) directly as a $(T,V)$-algebra, or if not directly, is there at least some connection that can be made?

Mike Shulman (Nov 21 2024 at 16:38):

I doubt it.

John Onstead (Nov 21 2024 at 21:38):

Mike Shulman said:

I doubt it.

Could you expand on that? At least, tell me what is fundamentally different between the "locality" encapsulated by topological-esque spaces and "locality" as encapsulated by manifolds. If these are two separate constructions, why would we say that both have a notion of "locality"- whatever that means? They have to have at least something in common for this observation to make sense...

Mike Shulman (Nov 22 2024 at 00:03):

I think a lot of different answers could be given to that question. The one that occurs to me right now is that "topological-esque" spaces have a way to measure "how far apart" things are (where the "things" could be points, subsets, filters, etc. and the "measurement" could be with real numbers, filters, truth values, etc.) whereas manifold-esque spaces are built by gluing together "local models" (which could be parts of $\mathbb{R}^n$, parts of $\mathbb{C}^n$, spectra of rings, etc.).

John Onstead (Nov 22 2024 at 02:08):

Mike Shulman said:

The one that occurs to me right now is that "topological-esque" spaces have a way to measure "how far apart" things are (where the "things" could be points, subsets, filters, etc. and the "measurement" could be with real numbers, filters, truth values, etc.) whereas manifold-esque spaces are built by gluing together "local models"

That makes sense! So I guess if there was a common theme, it would be relating "parts" together, whether it be by measuring how "far apart" they are, or by gluing them together.

John Onstead (Nov 22 2024 at 02:12):

Though that does make me wonder if gluing itself can be expressed as some sort of binary relation. For instance, let's say you have $\mathrm{Open}(\mathbb{R}^n)$, the set of open subsets of Euclidean space. What if you then imposed a binary relation on this set that describes "overlap"- that is, two open subsets of Euclidean space are paired with each other under this binary relation if they are meant to "overlap" with one another, perhaps in the context of using these open subsets to build up some manifold. Then the "gluing" can be encapsulated by imposing some axioms on this binary relation. Have you encountered such an "overlap" relation before, and how close would such a thing be to describing an actual manifold?

Mike Shulman (Nov 22 2024 at 06:44):

It's not just a relation on the set of open subsets, since you have to specify exactly how two opens overlap. So really it's a relation between each pair of open subsets themselves. If you make this precise you get a notion of "many-object relation" or "many-object congruence", and manifolds can indeed be described in terms of such things. We discussed this here a couple of months ago in the thread about motivating sites. There's a slightly simpler version on the nLab at [[k-ary exact category]].

John Onstead (Nov 22 2024 at 12:18):

I see... that topic and paper was actually something I wanted to go into more detail on, since it relates back to another concept I'm trying to understand, that of sheaves. It might be something I'd like to discuss in the near future, though I'll be sure to check this out before then!

John Onstead (Nov 22 2024 at 12:19):

Speaking of relations from a categorical perspective. I think have one last question for this topic before moving on... is there a "categorical" way to think about $(T,V)$-algebras? Here's what I mean. We covered $(T,V)$-algebras in the context of sets and the category $\mathrm{Set}$, where we have functions of the form $T(X) \times X \to V$ for $V$ some quantale. But an interesting thing to note is that, for many $T$, these are also objects in $\mathrm{Poset}$, the category of posets. Obviously $X$ (viewed as a discrete category) and $V$ are posets. But when $T$ is the powerset, filter, ultrafilter, and a few other examples of monads, $T(X)$ is also a poset (for instance, for the powerset monad, we get the poset of subset inclusions for $X$). The main problem is with the additional structure on $T(X)$ and $V$, it may no longer be true that all functions of the form $T(X) \times X \to V$ in $\mathrm{Set}$ have corresponding (order preserving) functions of the same form in $\mathrm{Poset}$.

So here's my question. Is there any good ways to think about $(T,V)$-algebras from the perspective of posets instead of sets? Maybe there's some way to factor in profunctors or enrichment or something creative. And also, thanks for all the help so far, learning this topic has been very interesting!

Mike Shulman (Nov 22 2024 at 15:41):

Well, you can define generalized multicategories for any monad on any VDC. So it could be the VDC of posets and profunctors, if you happen to have a monad there.