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Stream: theory: category theory

Topic: left leg is fibration, right leg is opfibration

Matteo Capucci (he/him) (Nov 08 2022 at 11:35):

Is there a name for an endospan of, say, functors which is a fibration on the left span and an opfibration on the right?
The prime example is given by any finitely complete category $\cal C$, giving $\cal C \overset{cod}\leftarrow C^\downarrow \overset{dom}\to \cal C$, and more generally, all categories with [[display maps]] $\cal D \subseteq C$, including cartesian monoidal categories: $\cal C \overset{\pi_2}\leftarrow C \times C \overset{\times}\to \cal C$.

Matteo Capucci (he/him) (Nov 08 2022 at 11:37):

Uhm this might just be a [[bifibration]]? Though I'm curious you can characterize the coherence of a bifibration by talking about spans alone

Amar Hadzihasanovic (Nov 08 2022 at 11:55):

It's a [[two-sided fibration]]!

Amar Hadzihasanovic (Nov 08 2022 at 11:55):

Well, the special case where both codomains are the same...

Matteo Capucci (he/him) (Nov 08 2022 at 11:56):

Ooh, is it the same??

Amar Hadzihasanovic (Nov 08 2022 at 11:58):

Yeah, a two-sided fibration is a span (not necessarily an endo-span) where one leg is a fibration and the other an opfibration. And yes the codomain/domain span is the prototypical one.

Matteo Capucci (he/him) (Nov 08 2022 at 11:58):

This seems to be stronger than just asking q to be opfibration: image.png

Dylan Braithwaite (Nov 08 2022 at 12:00):

3bb8ea3e-7277-499e-a996-fbf7f9afc9dc.png

Dylan Braithwaite (Nov 08 2022 at 12:00):

It seems like it's nearly the same but with an extra compatibility condition between the lifts

Amar Hadzihasanovic (Nov 08 2022 at 12:01):

Oh, yes, you are right, there is an extra-compatibility between lifts that's required.

Mike Shulman (Nov 08 2022 at 16:41):

But I would be surprised if there were interesting examples where the compatibility doesn't hold. Note that a two-sided discrete fibration (which the examples in the OP are) is equivalent to a profunctor, and the example of $C^{\downarrow}$ corresponds to the identity (hom) profunctor.

Matteo Capucci (he/him) (Nov 08 2022 at 18:18):

Mike Shulman said:

But I would be surprised if there were interesting examples where the compatibility doesn't hold. Note that a two-sided discrete fibration (which the examples in the OP are) is equivalent to a profunctor, and the example of $C^{\downarrow}$ corresponds to the identity (hom) profunctor.

Oh I see! I think two-sided discrete fibration nails it

dusko (Jan 05 2023 at 08:42):

Matteo Capucci (he/him) said:

Is there a name for an endospan of, say, functors which is a fibration on the left span and an opfibration on the right?
The prime example is given by any finitely complete category $\cal C$, giving $\cal C \overset{cod}\leftarrow C^\downarrow \overset{dom}\to \cal C$, and more generally, all categories with [[display maps]] $\cal D \subseteq C$, including cartesian monoidal categories: $\cal C \overset{\pi_2}\leftarrow C \times C \overset{\times}\to \cal C$.

a span of a discrete fibration and a discrete opfibration is the category of elements of a profunctor (distributor). the prime example that you mention corresponds to the profunctor $hom$ of the category $\cal C$. benabou presented the bicategory of profunctors using such discrete fibrations...

if you drop the discreteness, then the span lives in a tricategory of (what was it called again?) probifunctors... i think gordon and power and ross street started from such examples to figure out the coherences for tricategories...

((this is like time travel: i am commenting about a thread from october to refer to papers from the 90s and from the 60s. it may have something to do with my age :)))

dusko (Jan 05 2023 at 08:51):

oh sorry now i see that mike shulman said more or less the same just above. sorry about that. i always say that nowadays everyone writes papers so noone has time to read them. it seems it's not just papers.