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Stream: theory: category theory

Topic: name this thing

Matteo Capucci (he/him) (Jan 23 2021 at 10:03):

Suppose I have a bicategory $E$ together with a pseudofunctor $F :E \to C$ such that $F$ restricts to a fibration on every hom-category, but it's not 'globally' a (2-)fibration.
I think the most natural name for this thing is a 'local fibration', though 'local' seems to be a bit overloaded in CT. In particular there seems to be already a thing called 'local fibration'.
Is this thing already known? If yes, how? If no, what could be a good name?
Another proposal: $F$ is 'locally fibrational'.

Mike Shulman (Jan 25 2021 at 13:36):

Yes, I've called that a "local fibration", e.g. Def 6.3 here. I don't think there's much danger of confusion with the model-categorical usage. Bakovic talks about functors being "locally a fibration", while Buckley says that such a functor is "locally fibred".

Matteo Capucci (he/him) (Jan 25 2021 at 13:37):

Thanks Mike! Weird, I skimmed Bakovic's paper and didn't notice he defined it.

Mike Shulman (Jan 25 2021 at 13:41):

It's just a passing remark in the second bullet of Def 4.7. (I found it by "Ctrl+F"-ing for "local".)

Matteo Capucci (he/him) (Jan 25 2021 at 13:41):

Seems like a good strategy