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Stream: theory: category theory

Topic: "expanded" composition

Christian Williams (Mar 08 2023 at 01:31):

Composition of profunctors is normally defined by a coend $\int^b R(a,b)\times S(b,c)$, a sum over $B$ with a quotient by its action $(r\cdot b, s)\equiv (r,b\cdot s)$. But what if we just defined $(R\circ S)(a,c)$ to be triples of morphisms $(r,b,s)$? This seems to be an equivalent notion of composite, without needing any quotienting. It seems like the "coYoneda-expanded" form of composition. What's going on?

Christian Williams (Mar 08 2023 at 01:38):

Thinking of profunctors as discrete two-sided fibrations $A\leftarrow R\to B\leftarrow \vec{B}\to B\leftarrow S\to C$, the composite span is a two-sided fibration consisting of these triples $(r,b,s)$.

Christian Williams (Mar 08 2023 at 01:39):

So why quotient at all? Just to collapse to a normal form of pairs $R,S$?

Christian Williams (Mar 08 2023 at 01:42):

Oh, maybe this is about representing a virtual double category.

Christian Williams (Mar 08 2023 at 01:45):

but the "expanded" composite span seems to be a composite in that sense as well.

Christian Williams (Mar 08 2023 at 01:48):

So I'm wondering, why not define $(R\circ S)(A,C)$ to be a sum over morphisms rather than objects?
$(R\circ S)(A,C) = \sum B:\vec{B}.\; R(A,B_0)\times S(B_1,C)$
It seems simpler, because no quotient is needed.

Mike Shulman (Mar 08 2023 at 02:06):

Well, to start with, that sort of composition wouldn't have identities...

Christian Williams (Mar 08 2023 at 02:07):

Ah, yes of course.

Christian Williams (Mar 08 2023 at 02:07):

Thanks. It felt too good to be true, but I couldn't see the problem.

Matteo Capucci (he/him) (Mar 08 2023 at 12:25):

the archetypal 'identity crisis' :grinning_face_with_smiling_eyes:

fosco (Mar 10 2023 at 08:06):

Matteo Capucci (he/him) said:

the archetypal 'identity crisis' :grinning_face_with_smiling_eyes:

Initial object in the category of identity crises