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Stream: theory: category theory

Topic: reference on discrete fibrations wanted

John Baez (Dec 22 2020 at 23:40):

What's a good introduction to Grothendieck fibrations that explains the connection between discrete fibrations and functors $F : \mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$? I need one for my paper with @Jade Master, @Mike Shulman and @Fabrizio Genovese

John Baez (Dec 22 2020 at 23:41):

I don't see anything at all about discrete fibrations in Borceux's book or Bart Jacobs' book, oddly enough.

The Elephant talks about discrete fibrations but just mentions in passing that they're equivalent to presheaves; he doesn't prove it or even sketch why it's true.

John Baez (Dec 22 2020 at 23:42):

@Jade Master checked Sheaves in Geometry and Logic and couldn't find it there either. It was mentioned but nothing else.

John Baez (Dec 22 2020 at 23:42):

She didn't find it in Yau and Johnson's book on Two-Dimensional Categories either.

Christian Williams (Dec 22 2020 at 23:45):

I think Section 2.4 of Riehl's Category Theory in Context is good.

John Baez (Dec 22 2020 at 23:47):

Okay, thanks - I'll check it out!

John Baez (Dec 22 2020 at 23:50):

Alas, it doesn't say much about fibrations. It has an exercise, 2.4viii, which defines "discrete fibration". It tells you to show that given a presheaf on $C$, the forgetful functor from its category of elements to $C$ is a discrete fibration.

John Baez (Dec 22 2020 at 23:51):

So that's good, that's half the battle: the other half is taking a discrete fibration and getting a presheaf. In the end you'd want to prove these two procedures give an equivalence of categories between presheaves on $C$ and discrete fibrations with base $C$.

Nathanael Arkor (Dec 22 2020 at 23:58):

Loregian–Riehl's Categorical notions of fibration covers both the equivalence between discrete fibrations and presheaves, and the 2-equivalence in the non-discrete setting.

John Baez (Dec 22 2020 at 23:59):

Thanks, I'll look there.

John Baez (Dec 23 2020 at 00:01):

Yay, the thing I want is mentioned in the abstract, and it's Theorem 2.1.2., and it's actually proved. :thumbs_up:

Thanks!!!