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Stream: learning: questions

Topic: Composing two-sided discrete fibrations

Amar Hadzihasanovic (Dec 19 2024 at 14:29):

How can one express, formally, the composition of two profunctors in terms of the corresponding two-sided discrete fibrations of categories?

Mike Shulman (Dec 19 2024 at 16:56):

The simplest way is to compose them as spans and then take the discrete reflection.

Amar Hadzihasanovic (Dec 19 2024 at 17:19):

I guess I deserve this answer for not actually asking the real question I'm interested in :)
Given two functors $F: C \to D$ and $G: D \to E$, I can take the comma categories $\mathrm{id}_D/F$ and $\mathrm{id}_E/G$ with the associated two-sided fibrations, which are a kind of "non-discrete" version of the two-sided discrete fibrations classifying representable profunctors induced by $F$ and $G$.

I would like to understand the relation between the composite span of these two, with tip $\mathrm{id}_D/F \times_D \mathrm{id}_E/G$, and the comma $\mathrm{id}_E/GF$. Explicitly, the latter seems to be a “coend-y” quotient of the first with respect to a two-sided action of morphisms of $D$, so the relation between the two is very similar to the discrete case; but of course, since the latter is not discrete, it cannot be obtained as a discrete reflection.

I asked about the “discrete” case because it was a quicker question to ask, but actually I would like to know if there is an answer that also applies to this non-discrete variant. :)

Mike Shulman (Dec 19 2024 at 17:44):

I'm pretty sure comma categories are discrete two-sided fibrations. The fiber of $\mathrm{id}_D/F$ over $c\in C$ and $d\in D$ is $D(d,Fc)$, which is a discrete set.

Amar Hadzihasanovic (Dec 19 2024 at 17:52):

Oh, I got confused by the fact that the individual legs of a two-sided discrete fibration are not themselves discrete.

Amar Hadzihasanovic (Dec 19 2024 at 17:53):

(I was working under the erroneous assumption that the "discrete" span has as tip the "connected components of slices" category that shows up in the comprehensive factorisation of a functor.)

Amar Hadzihasanovic (Dec 19 2024 at 17:56):

Do you know a reference about the discrete reflection for two-sided spans, whether the discrete two-sided fibrations are the right class of a wfs, etc? The nLab was not enlightening.

Mike Shulman (Dec 19 2024 at 19:37):

• Hermida, From coherent structures to universal properties
• Street, Fibrations in bicategories (primarily concerned with the dual situation, codiscrete cofibrations)
• Carboni-Johnson-Street-Verity, Modulated bicategories (both this case and the dual, but I think also focused more on the dual)

Mike Shulman (Dec 19 2024 at 19:37):

Those are all about general 2-categories.

Mike Shulman (Dec 19 2024 at 19:39):

Discrete fibrations and discrete opfibrations are the right classes of the [[comprehensive factorization system]] and its dual, but I don't even know what that would mean in the two-sided case, since a two-sided fibration is a span rather than a morphism.

Mike Shulman (Dec 19 2024 at 19:39):

Discrete morphisms are the right class of some kind of factorization system, that's the "modulated" approach.