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Stream: theory: category theory

Topic: final functors into filtered categories

Jonas Frey (Feb 08 2023 at 02:37):

If $F : C \to D$ is a final functor into a filtered category, then is $C$ necessarily filtered? If not, does somebody have a counterexample? Thanks!

El Mehdi Cherradi (Feb 08 2023 at 11:58):

Unless I am mistaken, F should induce a final functor between the slice, and contractibility of D's slices will imply that of C's
edit : as stated, it is not clear that I meant "comma over a finite diagram" rather than (usual) slices

Tim Campion (Feb 08 2023 at 18:18):

This appears eg as Exercise 1.o.3 in Adamek and Rosicky, Locally Presentable and Accessible Categories

fosco (Feb 08 2023 at 21:30):

Maybe Jonas was trying to solve that exercise and looked for help? :-)

Jonas Frey (Feb 09 2023 at 00:10):

Thanks everybody! @El Mehdi Cherradi are you saying that for every finite diagram $G: I \to C$, the induced functor $Cocone(G)\to Cocone(FG)$ is final (provided that $F$ is final)? Then we could indeed deduce inhabitedness of $Cocone(G)$ from inhabitedness of $Cocone(FG)$, and conclude that $C$ is filtered whenever $D$ is. But I don't see how to show that the functor between categories of cocones is final, showing that seems to amount to doing what I was more or less already trying by hand.

Jonas Frey (Feb 09 2023 at 01:02):

fosco said:

Maybe Jonas was trying to solve that exercise and looked for help? :-)

I wasn't aware of this exercise, so it's definitely good to know that the statement seems to be true!

Jonas Frey (Feb 09 2023 at 01:41):

The statement seems to be wrong after all! The terminal category 1 is filtered, and the terminal projection $C\to 1$ is final whenever $C$ is connected (or weakly contractible for $\infty$-categories), which is weaker than filtered.

Thanks to Mathieu Anel for this counterexample!

Jonas Frey (Feb 09 2023 at 01:51):

(this doesn't contradict the AR exercise, which has the additional assumption that $F$ is fully faithful)

Notification Bot (Feb 09 2023 at 10:51):

This topic was moved here from #general: mathematics > final functors into filtered categories by Matteo Capucci (he/him).

Tim Campion (Feb 14 2023 at 17:28):

Jonas Frey said:

(this doesn't contradict the AR exercise, which has the additional assumption that $F$ is fully faithful)

Ah, sorry about that! And if $C \to 1$ is fully faithful, then $C$ is indiscrete and hence filtered, so everything checks out.

I suppose Mathieu's counterexample also shows that "fully faithful" can't be weakened to "faithful", since $C \to 1$ is faithful iff $C$ is a preorder, but preorders can have arbitrary homotopy type.

Tim Campion (Feb 14 2023 at 17:34):

I'm trying to think through now all the times one has a cofinal functor to a filtered category and wants to know the domain category is filtered. Mostly the fully faithful hypothesis is satisfied. One place this is not the case is when you have a filtered category and look for a cofinal functor from a directed poset. That functor won't be fully faithful, so it seems you really do have to check that the poset you construct is directed...

El Mehdi Cherradi (Feb 15 2023 at 07:29):

Tim Campion said:

I'm trying to think through now all the times one has a cofinal functor to a filtered category and wants to know the domain category is filtered. Mostly the fully faithful hypothesis is satisfied. One place this is not the case is when you have a filtered category and look for a cofinal functor from a directed poset. That functor won't be fully faithful, so it seems you really do have to check that the poset you construct is directed...

After thinking a bit about it, I would say that the fully faithful condition is necessary: my guess would be that the functors induced between the slices over a finite diagram $C \downarrow X \to D \downarrow FX$ ($X : I \to C$, where $I$ is finite) are final if $C$ is assumed to be filtered, with only $I = *$ this would imply that the relevant square (for $F$ to be fully faithful) is exact.

Note that this does not seem to contradict theorem 1.5 of the AR book which you hinted at (where the final functor constructed is moreover fully faithful).