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Stream: learning: questions

Topic: checking if my functor preserves filtered colimits

Ralph Sarkis (Feb 21 2022 at 10:07):

I would like to check if a functor preserves filtered colimits. Is there a concrete and simple way to do this? For finite colimits, I only need to check it preserves the initial object and the pushouts. Is there such a set of "irreducible" filtered colimits which generate all other? What if I restrict myself to $\omega_1$--filtered colimits? From my understanding, coequalizers of idempotents and identities should be part of that set, but I am not sure.

Ralph Sarkis (Feb 21 2022 at 10:12):

Informally, what should I look for in a functor to convince me it preserves (or not) filtered colimits? I know the equivalence between finitary and preserving filtered colimits (my case is not in Set, but it is in a variant of Met which has a similar equivalence), but I don't see how to intuitively see that a functor is determined by what it does on finite sets/space.

Zhen Lin Low (Feb 21 2022 at 10:35):

Well, you can restrict to filtered posets, a.k.a. directed sets. But that's still quite complicated.

Zhen Lin Low (Feb 21 2022 at 11:13):

I haven't thought about it carefully, but maybe you can reduce to chains (= diagrams indexed by a well-ordered set). What I know for sure is that if you have a $\kappa$-directed poset of cardinality $\le \kappa$ then you have a cofinal chain of length $\kappa$.

Zhen Lin Low (Feb 21 2022 at 11:15):

(The case where you have a $\kappa$-directed poset of cardinality $< \kappa$ is actually trivial: such a thing has a top element. The interesting case is where you have a $\kappa$-directed poset of cardinality exactly $\kappa$.)

Mike Shulman (Feb 21 2022 at 19:36):

There's some discussion of this towards the beginning of Adamek-Rosicky Locally presentable and accessible categories, I believe.

Reid Barton (Feb 21 2022 at 19:59):

This is sort of like how to check if your function is continuous--you use various facts you already know about continuous functions but mainly what you need to do is determined by the function in question.

Reid Barton (Feb 21 2022 at 20:00):

Assuming Met is some flavor of metric spaces and short maps, IIRC it is only locally $\omega_1$-presentable and so you shouldn't necessarily expect there to be a good theory of functors that preserve filtered colimits, but there should indeed be a good theory of functors preserving $\omega_1$-filtered colimits.

Ralph Sarkis (Feb 21 2022 at 20:17):

Reid Barton said:

This is sort of like how to check if your function is continuous--you use various facts you already know about continuous functions but mainly what you need to do is determined by the function in question.

Do you have a reference where this is done by hands (for accessibility not continuity :smile:)? I think every proof of accessibility I have seen are very high level (e.g. using the fact that adjoints on locally presentable categories are accessible).

Reid Barton (Feb 21 2022 at 20:31):

So an example that you can check by hand is that the functor F : Set -> Set, F(X) = X^n for n a natural number preserves filtered colimits. That's because if X = colim_i X_i is a colimit over a filtered diagram, then every n-tuple of elements of X already arises as an n-tuple in some X_i (using filteredness), and likewise any two elements of X_i^n with the same image in X^n already get identified in X_j^n for some i <= j.

Reid Barton (Feb 21 2022 at 20:32):

Or, using the general theory, this is just an instance of finite limits commuting with filtered colimits.

Reid Barton (Feb 21 2022 at 20:33):

All $\omega$-accessible functors F : Set -> Set are built up from these examples under colimits.