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Stream: theory: category theory

Topic: shape of finitary functors

James Deikun (May 12 2022 at 11:27):

When a category $C$ has finite limits, then a functor $F : C \to D$ preserves these limits exactly when each comma category $(d/F)$ is cofiltered.

Is there a similar statement for finitary functors from categories with filtered colimits, relating the colimit-preservation of the functor to the shape of its comma categories $(d/F)$ or perhaps $(F/d)$?

James Deikun (May 12 2022 at 12:52):

I'd assume it would be some kind of saturation of the class of finite diagrams but is there a nice way to describe this saturation?

Tom Hirschowitz (May 12 2022 at 14:20):

Not directly an answer, but there is stuff of this kind in functors with arities.

James Deikun (May 16 2022 at 12:34):

I'd assume it would be some kind of saturation of the class of finite diagrams but is there a nice way to describe this saturation?

Well, it turns out the saturation is the class of [[L-finite categories]] ...

James Deikun (May 18 2022 at 12:58):

For a functor $F : C \to E$, the coYoneda extension $Ran_F^{\mathrm{op}} : [C,\bold{Set}]^{\mathrm{op}} \to [E,\bold{Set}]^{\mathrm{op}}$ is computed by the limit

$$(\mathrm{Ran}_F\,X)(c) \simeq \mathrm{lim}\left((c/F) \to C \stackrel{X}{\to} \bold{Set}\right)\,,$$

so this should preserve filtered colimits exactly when the comma categories $(c/F)$ are L-finite. Because the coYoneda embedding $ʎ : C \to [C,\bold{Set}]^{\mathrm{op}}$ preserves colimits whenever they exist, then for categories $C, E$ with all filtered colimits, these should be preserved by $F$ exactly when all comma categories $(c/F)$ are L-finite.

It's interesting that this also involves the "coslice-like" categories $(c/F)$ rather than the "slice-like" $(F/c)$ because of the changes in variance cancelling each other out...

James Deikun (May 18 2022 at 13:06):

(Actually I wouldn't be surprised if I messed up variance somewhere though.)

James Deikun (May 19 2022 at 14:55):

Let me go through this more carefully since nobody seems to have double checked me ... the Yoneda extension is a left adjoint and always preserves the (freely added) colimits in $[C^{\mathrm{op}},\bold{Set}]$, so the coYoneda extension of $F$ should preserve the (freely added) limits in $[C,\bold{Set}]^{\mathrm{op}}$. That means, though that as a map from $[C,\bold{Set}]$ to $[E,\bold{Set}]$ it should preserve colimits, i.e., it should be $\mathrm{Lan}_F^{\mathrm{op}}$, not something with $\mathrm{Ran}$.

James Deikun (May 19 2022 at 15:03):

Therefore the formula to compute it is instead

$$(\mathrm{Lan}_F\,X)(c) \simeq \mathrm{colim} \left((F/c) \to C \stackrel{X}{\to} \bold{Set}\right).$$

James Deikun (May 19 2022 at 15:07):

Filtered colimits in $[C,\bold{Set}]^{\mathrm{op}}$ are the same as cofiltered limits in $[C,\bold{Set}]$ so these should be preserved precisely when the categories $(F/c)$ are co-L-finite, or in other words when $(F/c)^{\mathrm{op}}$ are L-finite.

James Deikun (May 19 2022 at 15:08):

At this point I am pretty sure I didn't screw up the variance.

Morgan Rogers (he/him) (May 19 2022 at 16:59):

James Deikun said:

Filtered colimits in $[C,\bold{Set}]^{\mathrm{op}}$ are the same as cofiltered limits in $[C,\bold{Set}]$ so these should be preserved precisely when the categories $(F/c)$ are co-L-finite, or in other words when $(F/c)^{\mathrm{op}}$ are L-finite.

Ah, but the classes of limits and colimits which commute in Set are not self-dual!

Morgan Rogers (he/him) (May 19 2022 at 16:59):

It's not true that cofiltered limits are precisely the ones commuting with finite colimits, for example.

James Deikun (May 19 2022 at 17:29):

:oh_no: ... I knew that wasn't true in general, but for some reason thought it was true in this particular case ...

James Deikun (May 19 2022 at 17:30):

I wonder then what does commute with cofiltered limits ...

James Deikun (May 19 2022 at 17:38):

https://www.matem.unam.mx/~omar/notes/fingrpcomm.html ... this hardly seems like it would be a saturated class though?

Tom Hirschowitz (May 19 2022 at 19:23):

James Deikun said:

I wonder then what does commute with cofiltered limits ...

See [[commutativity of limits and colimits]]. Apparently it's taking orbits under the action of a finite group.

James Deikun (May 19 2022 at 20:20):

That's what's at the link above; however, I don't think it's a saturated class of limits; for example I don't think it's closed under precomposition by initial functors.

James Deikun (May 19 2022 at 20:21):

More than that, I feel like there must be something fundamentally wrong with my reasoning above if it leads to this weird class instead of L-finite categories.

Morgan Rogers (he/him) (May 19 2022 at 21:13):

One day I'll get around to finishing my translation of Foltz' paper and have enough intuition about commutation to clear up this kind of question

James Deikun (May 25 2022 at 18:26):

James Deikun said:

More than that, I feel like there must be something fundamentally wrong with my reasoning above if it leads to this weird class instead of L-finite categories.

After thinking about this more and checking specific examples, like functors that take limits of finite diagrams in $\bold{Set}$, I think the weird class is actually correct, although it's hard to be sure since I don't have a good complete characterization of that class. But basically I think functors that preserve $\bold{\Phi}$-colimits have slicelike comma categories that are co-$\bold{\Phi}$-filtered, not $\bold{\Phi}$-atomic as one might expect. So for finitary functors they are cofiltered-filtered (whatever that actually is), not L-finite.

James Deikun (May 25 2022 at 18:31):

(And it groups as (co-$\bold{\Phi}$)-filtered, not co-($\bold{\Phi}$-filtered) as in the limit-preserving case.)