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Stream: theory: algebraic topology

Topic: extranatural transformations

Patrick Nicodemus (Dec 16 2021 at 08:15):

Just had an insight: If we draw an analogy between categories and spaces, continuous maps and functors, and homotopies and natural transformations, then we have an analogy between extranatural transformations between functors and a certain kind of homotopy which can be defined between maps with different domains.

If $f: A\to C$, $g : B\to C$ then call a homotopy from $f$ to $g$ a homotopy in the ordinary sense between $f\circ \pi_A , g\circ \pi_B : A\times B\to C$. This definition wouldn't normally strike me as too interesting but I came up with this definition while I was thinking about the universal property of the join of two spaces, $A\ast B$; I think that $A\ast B$ should be initial in the category of such pairs $(C, f : A\to C, g : B\to C, h : f\Rightarrow g)$. I already happen to consider the join a very important operation, so now I find this notion of homotopy to be interesting too, and the parallel with extranatural transformations occurred to me immediately.

Patrick Nicodemus (Dec 16 2021 at 08:24):

Perhaps the parallel isn't exactly perfect... perhaps there is a better translation. Thinking out loud, for two categories $\mathcal{A},\mathcal{B}$ one can define a category
$\mathcal{A}\ast\mathcal{B}$, which has as its underlying objects the disjoint union of $\operatorname{Obj}(\mathcal{A}) \amalg \operatorname{Obj}(\mathcal{B})$ and whose morphisms are given in the expected way for $\operatorname{Hom}(a,a')$ and $\operatorname{Hom}(b,b')$; the morphisms $\operatorname{Hom}(a,b)$ could plausibly be given by any profunctor from $\mathcal{A}$ to $\mathcal{B}$ taken as a parameter in the definition; taking the terminal profunctor gives a fairly naive notion of natural transformation of functors with straightforward coherence conditions.

Patrick Nicodemus (Dec 16 2021 at 08:29):

Namely, the usual notion of extranaturality in the degenerate case where the functors are only covariant instead of mixing co- and contravariance.

Patrick Nicodemus (Dec 16 2021 at 08:30):

So it's a pretty good analogy, modulo the fact that the join doesn't let you discuss extranatural transformations in full generality with the mix of co/contravariance

Mike Shulman (Dec 16 2021 at 14:34):

$A*B$ should be initial in the category of such pairs

Yes, that's the statement that the join is the homotopy pushout of $A\leftarrow A\times B \to B$.

Mike Shulman (Dec 16 2021 at 15:48):

In categories you can have an analogous thing to a homotopy pushout called a cocomma object, and indeed cocomma objects are precisely the collages of profunctors.