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Stream: learning: questions

Topic: fibrations and opfibrations between presheaf categories

John Baez (Dec 27 2023 at 21:55):

Suppose I have a functor $j: C \to D$ between small categories. This induces a functor between the presheaf categories

$j^\ast : \mathsf{Set}^{D^{\text{op}}} \to \mathsf{Set}^{C^{\text{op}}}$

called restriction along $j$: on objects it sends $f : C^{\text{op}} \to \mathsf{Set}$ to $f \circ j$, and it does something similar on morphisms.

What I'd like is necessary and/or sufficient conditions for $j^\ast$ to be a fibration or opfibration!

John Baez (Dec 27 2023 at 21:56):

I discussed this with @Todd Trimble a long time ago, focused on the case where $j$ is the inclusion of a subcategory, and I think we came up with a sufficient condition for $j^\ast$ to be an opfibration.

John Baez (Dec 27 2023 at 21:57):

But the issue came up again in work using CatLab in AlgebraicJulia.... @Nathaniel Osgood raised it today.

James Deikun (Dec 27 2023 at 23:52):

A fibration $p : E \to B$ can be characterized in terms of the following square:

$$\begin{CD} \overrightarrow{E} @>{\overrightarrow{p}}>> \overrightarrow{B} \\ @V{cod}VV @VV{cod}V \\ E @>>{p}> B \end{CD}$$

If the comparison arrow that measures how far this square is from being a strict pullback has a right adjoint right inverse, $p$ is a cloven fibration.

James Deikun (Dec 28 2023 at 00:12):

In the case you're interested in, this is determined by another square:

$$\begin{CD} C @>{j}>> D \\ @V{C \times 0}VV @VV{D \times 0}V \\ C \times \rightarrow @>>{j \times \rightarrow}> D \times \rightarrow \end{CD}$$

If the comparison determining how far this is from a pushout square -- the [[pushout product]] of $j$ and $0$ -- induces, under the free cocompletion functor, an arrow with a right adjoint right inverse, then $j^*$ is a cloven fibration. One way for this to happen is if this arrow itself has (if I got the variances right) a right adjoint right inverse, but it can also happen because of limits existing in $\mathsf{Set}$, an example being the codomain fibration of $\mathsf{Set}$.

Mike Shulman (Dec 28 2023 at 00:24):

(I think you either want $j:D\to C$ or $j^* : \mathrm{Set}^{D^{\mathrm{op}}} \to \mathrm{Set}^{C^{\mathrm{op}}}$.)

James Deikun (Dec 28 2023 at 00:25):

Basically I think it the necessary and sufficient condition comes down to all right Kan extensions along $(j \, \square \, 0)^{\mathrm{op}}$ into $\mathsf{Set}$ existing, and for opfibrations I think it will be left Kan extensions along $(j \, \square \, 1)^{\mathrm{op}}$.

James Deikun (Dec 28 2023 at 00:29):

And it probably wouldn't take much more work to parlay this into a computation of the actual (op)Cartesian lifts.

James Deikun (Dec 28 2023 at 00:36):

(The conditions are probably more interesting when the target category is not $\mathsf{Set}$ ...)

James Deikun (Dec 28 2023 at 00:46):

Hm, actually come to think of it my condition gives a right adjoint but not a right inverse. For the Kan extensions to be inverses there may be a slightly more interesting condition. The failure of this condition would manifest as the putative Cartesian lift not being a lift at all.

James Deikun (Dec 28 2023 at 01:16):

I think the extra condition in question is that the Kan extensions in question should be genuine extensions, i.e. the restriction of the extension of a presheaf should give back the original presheaf.

Then the formula for the Cartesian lift is you take your arrow in $\mathsf{Set}^{C^{\mathrm{op}}}$ and your $D$-presheaf over the target and combine them into a presheaf in $\mathsf{Set}^{((C \times \rightarrow) \amalg_C D)^{\mathrm{op}}}$ and Kan extend it to a presheaf in $\mathsf{Set}^{(D \times \rightarrow)^{\mathrm{op}}}$ which you interpret as the lifted arrow. The lift, assuming it truly is one, is Cartesian because of the universal property of Kan extensions.

James Deikun (Dec 28 2023 at 08:53):

In particular, since pointwise extensions into $\mathsf{Set}$ always exist, being a fully faithful functor should be a sufficient condition in itself.

Mike Shulman (Dec 28 2023 at 09:06):

Another way to see that $j$ being fully faithful is sufficient is to use the fact mentioned here (Theorem 2) that any pullback-preserving functor with a fully faithful right adjoint is a fibration, and dually. Since the Kan extension functors along a fully faithful functor are fully faithful, and restriction preserves all limits and colimits, the result follows.

John Baez (Dec 28 2023 at 09:09):

Thanks! I fixed the typo. More importantly, $j$ being fully faithful seems like a pretty practical sufficient condition for the examples I'm running into!

Mike Shulman (Dec 28 2023 at 09:12):

Conversely, Theorem 1 at that link says that a fibration that preserves terminal objects must have a fully faithful right adjoint, and dually. Thus, if $j^*$ is a fibration, then $\mathrm{Ran}_j$ is fully faithful, and if $j^*$ is an opfibration, then $\mathrm{Lan}_j$ is fully faithful. I think $\mathrm{Lan}_j$ being fully faithful actually implies $j$ is fully faithful, since $\mathrm{Lan}_j$ restricts to $j$ on representables and the Yoneda embedding is fully faithful. Not sure about $\mathrm{Ran}_j$, but it seems like full-faithfulness of $\mathrm{Ran}_j$ shouldn't be much weaker than full-faithfulness of $j$.

Morgan Rogers (he/him) (Dec 28 2023 at 11:42):

There's a characterization in section A4 of Sketches of an Elephant of functors which induce inclusions between presheaf toposes, and iirc it is exactly full faithfulness