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Stream: theory: category theory

Topic: Dealing with Large Sites

Emilio Minichiello (Aug 01 2024 at 19:36):

I've been thinking about some sites lately and I just want somebody to reassure me that I'm thinking about this in the right way.

For the category of finite dimensional smooth manifolds $\mathbf{Man}$, the Whitney embedding theorem allows you to embed all of your manifolds in a single $\mathbb{R}^\infty$ and thus $\mathbf{Man}$ is an essentially small category, and so we can use the corresponding small category to define a small site of finite dimensional smooth manifolds with open covers.

I want to consider say the category of topological spaces $\mathcal{C} = \mathbf{Top}$. Now this is a large category that is not essentially small, as it contains all discrete sets within it.

Now if $\mathcal{C}$ is not a small category, then the category of presheaves $\textbf{Pre}(\mathcal{C})$ is not necessarily locally small. Now (I believe) this implies that sheafification doesn't necessarily exist, because you might end up taking a large colimit in the construction, which doesn't necessarily exist. This is bad for me because I want to sheafify things.

There's a bit of discussion of this kind of stuff on the nlab. There's also sometimes the possibility that your site will satisfy some nice conditions like WISC, and then you don't have to worry about this size issue (in this case $\mathbf{Top}$ does satisfy WISC, but there are other sites that don't). There's also this nice, but kind of complicated discussion on the Stacks Project about this worry for the category of schemes. But I don't want to have to worry about any of these things. I just want to construct a small category of topological spaces and move on with my life.

So lets suppose we fix three Grothendieck universes $u, U, V$, such that $u \in U$ and $U \in V$. I'll call elements of $u$ really small, elements of $U$ small and elements of $V$ large. I'll develop all of my usual category theory in $U$ and $V$ until I want to introduce a site that's supposedly large. So now I construct the category $\mathcal{C}$ of topological spaces whose underlying sets belong to $u$. So when I represent a topological space as some really complicated encoding of sets of sets of sets, they still end up being elements of $u$. Then by construction, $\text{Obj}(\mathcal{C}) \subseteq u$ and therefore $\text{Obj}(\mathcal{C}) \in U$, and similarly for morphisms, so $\mathcal{C}$ should be a small category.

So I'm done right? Like, no topologist is gonna care that I can't construct a topological space whose cardinality is inaccessible. I think this is what Jardine had in mind when he writes about "cardinality bounds" in his notes.

Is there anything I'm missing here? Any sneaky set-theory that's gonna bite me later on?

Kevin Carlson (Aug 01 2024 at 19:45):

I don't understand how you could imagine a non-essentially small category of finite-dimensional complex manifolds.

Kevin Carlson (Aug 01 2024 at 19:46):

I mean, every complex manifold is a real manifold too, right, and so there can't be a large number of complex manifolds unless some real manifold admits a large number of distinct complex structures.

Emilio Minichiello (Aug 01 2024 at 19:52):

That's a good point. I can't imagine this being non-essentially small, but I also can't prove it right away. The complex manifold thing was just an example, I just wanted to use this method to deal with all examples I could think of.

John Baez (Aug 01 2024 at 19:52):

One crazy way to get a non-essentially-small category of finite-dimensional complex manfolds would be to allow manifolds that are not sigma-compact: i.e., not a countable union of compact sets. Then a disjoint union of $\kappa$ copies of $\mathbb{C}$ is a complex manifold for any cardinal $\kappa$, so there's a proper class of isomorphism classes of objects!

John Baez (Aug 01 2024 at 19:54):

But most complex geometers are happy to work only with sigma-compact complex manifolds, and I bet the resulting category is essentially small.

John Baez (Aug 01 2024 at 20:04):

To prove this, I'd try to show such a manifold has a countable chart, and use this to put an upper bound on the cardinality of the set of isomorphism classes of such manifolds, by bounding the cardinality of the set of transition functions. It looks like the cardinality of the continuum will do.

Emilio Minichiello (Aug 01 2024 at 20:08):

Alright I've edited the post, the complex manifold thing was probably a bad example.

Kevin Carlson (Aug 01 2024 at 20:13):

If you do "normal category theory" with $U$ meaning "small" and $V$ meaning "large", but then defining your "category of topological spaces" to be $u$-small for purposes of knowing you have a sheafification, then this category will for instance not be complete and cocomplete, since those words mean "have all small (co)limits." You could relatedly get into all kinds of creeping issues around adjoint functor and representability theorems, in particular around sheafification: you'll be hiding the nonexistence of certain sheafifications (except that Top will satisfy WISC) behind the fact that you've decided to take presheaves valued in what, as far as the topological spaces can tell, are large sets. The main point here is that there's a real mathematical issue here and you don't get to just completely forget about it without risking saying false things; it's not just set-theoretic throat-clearing. It would be safer to explicitly say you're taking as a site the category of all topological spaces with underlying sets of cardinality at most $\kappa$ for some cardinal $\kappa$ with whatever properties you actually want, such as perhaps regularity. Your proposal amounts to letting $\kappa$ be inaccessible, which is quite a lot to assume (are you really going to take transfinitely iterated powersets of your spaces anyway?) and then leaving this $\kappa$ out of the notation, which is likely to confuse.

Kevin Carlson (Aug 01 2024 at 20:14):

This is exactly the same as Jardine's idea, yes. I'm not immediately sure what he means by the claim that these bounds "don't matter homotopy-theoretically."

Kevin Carlson (Aug 01 2024 at 20:16):

Incidentally, I wouldn't make semantically significant edits to posts people have already responded to, especially not on a platform that doesn't make the edit history visible. The meaning of this thread is no longer a function of its content, which is no fun for anybody else who might come try to participate.

Emilio Minichiello (Aug 01 2024 at 20:24):

Kevin Carlson said:

This is exactly the same as Jardine's idea, yes. I'm not immediately sure what he means by the claim that these bounds "don't matter homotopy-theoretically."

Yeah, I don't either, which is what led me down this path lol. You are right now that $\mathbf{Top}$ is no longer small-(co)complete. However for me, as the site underlying a sheaf topos I don't tend to care too much about its underlying (co)limits beyond maybe finite limits. I just wanted to think of a quick way to render all the sites I want to work with as small.

I think if I was writing a paper about sheaves on some kind of site in differential geometry, keeping around the $\kappa s$ might be more confusing than just stating the Grothendieck universe convention once. Or as you said I could just say they are all have cardinality less than some infinite regular cardinal $\kappa$ and just not include it in the notation. But isn't there some possibility that you could construct a topological space that is interesting where you go past $\kappa?$ I just used Grothendieck universes because then there's like no way to go past it.

Emilio Minichiello (Aug 01 2024 at 20:24):

Kevin Carlson said:

Incidentally, I wouldn't make semantically significant edits to posts people have already responded to, especially not on a platform that doesn't make the edit history visible. The meaning of this thread is no longer a function of its content, which is no fun for anybody else who might come try to participate.

Oh whoops! Point taken.

Emilio Minichiello (Aug 01 2024 at 21:07):

Reading more about this, obviously taking just an infinite regular cardinal is not good enough because $\mathbb{N}$ is an example.

From here, if you assume its an infinite, regular, weak limit cardinal, i.e. weakly inaccessible, that's certainly good enough, and being an infinite, regular, strong limit cardinal is the same as being inaccessible / a Grothendieck universe.

I guess I just wanted to figure out a quick yet rigorous way to couch my discussions without resorting to "use a cardinality bound." I agree with you that this is mathematically important and I'd definitely want to be up front with the actual kind of cardinality bound I want when introducing such a site.

David Michael Roberts (Aug 02 2024 at 03:13):

@Emilio Minichiello I've read the above, and I'm unclear as to what you actually want to use the large site for. In practice, people working with large sites often only need to use small presheaves or small sheaves, in which case there's actually no problem. If one is hoping to use a full elementary topos structure, including powerset, say, then this is obviously a problem. But very often what is actually only needed is the pretopos structure. Any for me, coming from the world of manifolds, I'm used to having to deal with function spaces in a special way, rather than just using diffeological spaces to get cartesian closedness for free.

So, what is the intended application for your site of topological spaces?

Emilio Minichiello (Aug 02 2024 at 03:42):

Hi David. Well my real motivation is the following: I want to think of categories of sheaves over a bunch of various sites that pop up in differential geometry as generalized smooth spaces of different flavors, and compare them with morphisms between the underlying sites. There are quite a few of these sites as you very well know: smooth manifolds, complex manifolds, topological spaces, manifolds with vector fields, dirac manifolds (these last two taken from Villatoro's thesis).

I guess the question I should have asked is the following: with the above goal in mind, what's the most expeditious way to handle set-theoretic issues for such examples (and possibly more that pop up as I think of them). I figured that just always using a really small Grothendieck universe makes all of these sites small and I don't have to worry about anything, but perhaps there's a more elegant solution? I didn't know about small (pre)sheaves, I will have to take a look into that.

I do very much want to use the full topos structure of the categories of sheaves on these sites, as I want to manipulate these sheaves in very hands-on ways, and eventually also develop the same idea for higher sheaves.

Kevin Carlson (Aug 02 2024 at 03:47):

(All of those sites of manifolds are naturally going to be essentially small, just to keep being annoying.)

Emilio Minichiello (Aug 02 2024 at 03:50):

Ah, maybe it would be easier to come up with like a meta-argument that says that. Then I wouldn’t need to worry about any of this.

Emilio Minichiello (Aug 02 2024 at 03:52):

But for Top, and other geometric sites whose objects have less structure, a cardinality bound would still be necessary