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Stream: theory: category theory

Topic: "Flow" to presheaf categories

Alonso Perez-Lona (Nov 17 2022 at 03:02):

One of the nice properties of the presheaf category PSh(C) of a small category C is that it is (co)closed. I suppose, depending on how one regards this, PSh(C) is in a sense the canonical (co)complete category in which C embeds. However, the process has always looked a bit weird to me. In a sense, the category is constructed "at once", the presheaves simply appear. So I was wondering, is there a sensible notion of C "converging" or "flowing" towards PSh(C)? Ideally, I would like to observe some "smooth flow" of C to PSh(C) but I am not sure what should be the ambient category. Nevertheless, this seems to me to be a reasonable idea.

Mike Shulman (Nov 17 2022 at 04:31):

Since Psh(C) is the free cocompletion of C under small colimits, you could consider the sequence of free cocompletions of C under $\kappa$-small colimits for increasing cardinals $\kappa$ which starts at C and "converges" to Psh(C).

Alonso Perez-Lona (Nov 17 2022 at 04:37):

Would there be any natural notion of smoothness one could introduce here?

Mike Shulman (Nov 17 2022 at 04:43):

Well, when $\kappa$ is a limit cardinal, the free $\kappa$-cocompletion is the union/colimit of the free $\lambda$-cocompletions for $\lambda<\kappa$.

Mike Shulman (Nov 17 2022 at 04:43):

So the resulting functor $\rm Ord \to Cat$ is (co)continuous...

Alonso Perez-Lona (Nov 17 2022 at 04:48):

What I had in mind was something more like the usual geometric scenarios of moving/flowing into a singularity in a moduli space. But maybe this particular picture phrased like that isn't really meaningful here.

Morgan Rogers (he/him) (Nov 17 2022 at 06:50):

It's not common in ordinary category theory to be able to interpolate "smoothly" between things, because categories are rather discrete. Consider freely adding a bottom element to a poset, for example. You can't do this partially, and for simple posets it isn't going to be "close" to the rest of the elements.

On the other hand, there are variants of category theory which might be more amenable to smooth transitions. If you look at enriched categories of topological categories, or possibly the more obscure continuous categories (a generalization of continuous posets) you have a chance of capturing notions of smooth transitions between categories.

Xuanrui Qi (Nov 18 2022 at 07:54):

I guess you can't do general, random categories, but if your categories are geometric in some sense, then they should be points in some moduli space. Still, that doesn't guarantee smoothness. But hey there's the whole enterprise of hidden smoothness! (unfortunately, which I don't understand).
I think this (categories-as-points-in-moduli-spaces) is indeed an important perspective that AFAIK hasn't been studied well enough. But yes, you could only study "nice enough" categories this way.