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Stream: theory: category theory

Topic: fibration over loose morphisms of a double category

Nathaniel Virgo (Sep 08 2024 at 07:48):

I seem to have a double category in which for each pair of objects, the category of loose morphisms between them is a fibration. (Loose means the dimension in which morphisms don't compose strictly.) It probably obeys some nice equations that I haven't worked out yet, which relate each of these fibrations to each other.

Do structures like this exist in the literature?

Mike Shulman (Sep 08 2024 at 16:01):

What do you mean by saying that a category "is" a fibration? Being a fibration is a property of a functor.

Nathaniel Virgo (Sep 09 2024 at 04:58):

Right, sorry, that was a bit muddled. What I mean is, I have two double categories $\mathbb{E}$ and $\mathbb{B}$ that share the same set of objects, with an identity-on-objects strict (I think) double functor $p:\mathbb{E\to B}$, which I think of as forgetting information about the loose maps. This functor has the property that the induced functor $\mathbb{E}_{loose}(X,Y)\to \mathbb{B}_{loose}(X,Y)$ is a fibration for each pair of objects $X,Y$. (It's actually an opfibration rather an a fibration in my case, but I guess that doesn't make a lot of difference.)

But probably $p$ has other nice properties, such that we might want to call $p$ some kind of fibration itself. I guess I'm just being lazy and asking if anything like that exists in the literature already, so I can just go through and check it has the right properties without needing to work out what they should be.

$\mathbb{B}$ is an equipment if that makes a difference, and I expect that $\mathbb{E}$ is also. They both have a monoidal structure as well.

Matteo Capucci (he/him) (Sep 09 2024 at 08:01):

This should be a special case of [[double fibration]] (special in being identity in the tight direction), since in those the category of loose arrows of $\mathbb E$ are fibred over the analogous category for $\mathbb B$. So this gives you a fibration $p_{loose}:\mathbb E_{loose} \to \mathbb B_{loose}$. Then you know the double functor respects boundaries of loose arrows so if you have a square between loose arrows with trivial tight boundaries, so a map $\alpha : f \Rightarrow p(f'): X \nrightarrow p(Y')$, you get a lift which is necessarily a loose arrow $\alpha^*f' : \alpha^*X \nrightarrow Y'$, meaning that $p_{loose}$ restricts to local fibrations as you want.