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Stream: deprecated: id my structure

Topic: relatively final functors

Mike Shulman (Jan 08 2023 at 00:39):

A functor $u:A\to B$ is called a [[final functor]] if colimits of shape $B$ are equivalent to colimits of shape $A$, by restriction along $u$. That is, for any $X:B\to C$, the induced map $\mathrm{colim}^A (X\circ u) \to \mathrm{colim}^B X$ is an isomorphism. This is equivalent to all comma categories $(b \downarrow u)$ being connected (or, in the $\infty$-case, contractible).

In the context of a [[derivator]] $D$, a functor $u:A\to B$ is called a $D$-equivalence if colimits in $D$ of constant functors of shape $B$ are equivalent to colimits of shape $A$, by restriction along $u$. That is, if $\pi : B\to 1$ is the unique functor to the terminal category, then for any $X \in D$ regarded as a functor $1\to D$, the induced map $\mathrm{colim}^A (X\circ \pi\circ u) \to \mathrm{colim}^B (X\circ \pi)$ is an isomorphism. This holds for all representable $D$ (i.e. all ordinary categories) iff $u$ induces an isomorphism on connected components, and for all $D$ (i.e. all $\infty$-categories) iff $u$ induces an equivalence of nerves.

Now suppose in addition to $u:A\to B$ we have an arbitrary functor $p:B\to I$. We can consider the intermediate property that for any $X:I\to C$ the induced map $\mathrm{colim}^A (X\circ p\circ u) \to \mathrm{colim}^B (X\circ p)$ is an isomorphism. If $I=B$ or $I=1$ this reduces to the two previous notions respectively. Has the more general notion been studied? Does it have a name? Does it have a concrete characterization in terms of $u$ (by analogy, I would guess something about $u$ inducing an equivalence of connected components or nerves of comma categories under objects of $I$)?

Tim Hosgood (Jan 10 2023 at 23:28):

if I haven't misunderstood, then this is sort of something that we considered in https://arxiv.org/abs/2204.01843 (with the caveat being that we study initial instead of final, and it's 2-categorical instead of $\infty$-categorical) — see Definition 9.10 and onwards

Evan Patterson (Jan 11 2023 at 00:43):

Thanks for noticing this, Tim. It does seem very similar to what we worked on. We propose a definition (Def 9.10) for when a functor $R: \mathsf{J} \to \mathsf{J}'$ is initial relative to a natural transformation $\rho$ of form:

image.png

When $\rho$ is an identity, so that we just have a commutative triangle, we get the setup that I think Mike is describing. Now, our definition is a concrete/combinatorial one, analogous to the concrete description of initial functors. We show that this definition has some good properties, including limit preservation when $\rho$ is a natural isomorphism, but we don't give an abstract characterization. So it's fair to say we don't have the full story.

Mike Shulman (Jan 12 2023 at 19:28):

Thanks for the pointer! I like the idea of generalizing to transformations rather than commuting triangles. Your definition doesn't quite match what I would have expected, though. For one thing, since I would want relative initiality to be equivalent to limit-preservation, it ought to be defined for transformations that induce a comparison map of limits, i.e. in the case when $\rho$ goes the other way. In this case, I would have guessed that the right condition for relative initiality is that for each $c\in C$, the induced functor $D/c \to D'/c$, which sends $f:Dj \to c$ to $D'(Rj) \xrightarrow{\rho} Dj \xrightarrow{f} c$, is a bijection on connected components. Is that related to your condition?