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Stream: theory: category theory

Topic: pushing forward a diagram at its “sinks”

Amar Hadzihasanovic (Oct 09 2022 at 08:40):

I am looking for a reference, or anything that would give me a more “conceptual” understanding of the following explicit construction, perhaps in terms of formal 2-category theory.

I'll start from a familiar example. Given a category $\mathcal{C}$, any morphism $f: x \to y$ in $\mathcal{C}$ induces a functor $f_*: \mathcal{C}/x \to \mathcal{C}/y$ between the slices over $x$ and $y$.
This seems to be one instance of the following general phenomenon.

Given a small category $J$, let $\mathrm{s}(J)$ be the set (or discrete category) of sinks of $J$, where an object $i$ of $J$ is a sink if the only outgoing morphism of $i$ is the identity.
(I am aware that this is not invariant under equivalence! But $J$ is supposed to be a “diagram shape”, so in most examples I have in mind it is the free category on a graph, or something quite strict like that.)

Given an $\mathrm{s}(J)$-indexed family of objects of $\mathcal{C}$, that is a functor $x: \mathrm{s}(J) \to \mathcal{C}$ we can consider the category $\mathcal{C}^{J/x}$ whose

• objects are extensions of $x$ to a diagram $F: J \to \mathcal{C}$, and
• morphisms are natural transformations of diagrams that leave $x$ fixed (i.e. their components at the sinks are all identities).

Then I think we have the following:
Claim. Given a natural transformation $f: x \Rightarrow y$ between functors $x, y: \mathrm{s}(J) \to \mathcal{C}$, we have a functor $f_*: \mathcal{C}^{J/x} \to \mathcal{C}^{J/y}$, given by “pushing a diagram $F$ forward along $f$ at the sinks”: that is,

• $f_*F$ is equal to $F$ at non-sinks and at incoming morphisms of non-sinks,
• if $i$ is a sink, we let $f_*F(i) := y(i)$ and, for all non-identity morphisms $h: j \to i$, we let $f_*F(h) := F(h);f(i)$ (well-defined because $F(i) = x(i)$).

The “slice category” example is the instance of this where $J$ is the walking arrow, whose only sink is its codomain.

Amar Hadzihasanovic (Oct 09 2022 at 08:49):

(I didn't say what $f_*$ does on morphisms, but in fact a morphism of $\mathcal{C}^{J/x}$ is also a diagram in $\mathcal{C}$ whose shape -- a partially collapsed cylinder on $J$ -- has the same sinks as $J$, and the action of $f_*$ is the pushforward along $f$ of this diagram)

Amar Hadzihasanovic (Oct 09 2022 at 08:59):

Btw, it should be clear that, like in the slice category example, this is functorial in the sense that it defines an indexed category $\mathcal{C}^{\mathrm{s}(J)} \to \mathbf{Cat}$

Mike Shulman (Oct 09 2022 at 17:16):

I think you can generalize this by replacing $s(J)$ by any category $A$ at all and $J$ by the [[collage]] $[j]$ of some profunctor $j : A \to B$. Because the inclusion $A \to [j]$ is a "co-opfibration" (it's one side of a [[codiscrete cofibration]]), when you exponentiate by it you get an opfibration $C^{[j]} \to C^A$, which gives you your pushforwards.

Amar Hadzihasanovic (Oct 09 2022 at 17:44):

Wonderful, exactly what I was hoping for! Thanks!