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Stream: theory: algebraic topology

Topic: computing sheaf cohomology using simplicial homotopy groups

Matteo Capucci (he/him) (Apr 12 2023 at 14:14):

I don't even know if the title is right :S by simplicial homotopy groups I mean the homotopy groups of a simplicial set.

Let $F$ be a sheaf on a finitely complete site $C$, $H$ a locally trivial cover of $1 \in C$.

1. Construct the nerve of the cover H, i.e. turn it into a simplicial object in C whose n-faces are given by (n+1)-wise intersections of 'opens' in H (see [[Cech nerve]])
2. Precompose F with the functor $\Delta \to C$ singling out the simplicial object we just constructed. This yields a simplicial set $F_H \in \bf sSet$.

Now what is the relation between the homotopy type of $F_H$ and the sheaf cohomology of $F$?

Matteo Capucci (he/him) (Apr 12 2023 at 14:17):

My ultimate goal is to understand whether this is the way the 'obstructions to glueing' interpretation of non-trivial cocyles in sheaf cohomology can be related to the intuition in these notes by @John Baez, where cohomology is instead intepreted in 'Galois-theoretic' terms, i.e. non-trivial cocycles are ways a 'small thing can sit inside a big thing', here the small thing being a local section and big thing being a global one (I guess)!

Reid Barton (Apr 12 2023 at 18:38):

F is a sheaf of abelian groups?

Reid Barton (Apr 12 2023 at 18:39):

The Cech nerve of H is a simplicial object. If you evaluate F on it, you will get a cosimplicial abelian group.

John Baez (Apr 12 2023 at 18:41):

Maybe Matteo meant to say $F : \Delta^{\rm{op}} \to C$, since he calls it a simplicial object.

John Baez (Apr 12 2023 at 18:42):

I also suspect he's dealing with a sheaf of sets rather than a sheaf of abelian groups... just because he's that kind of guy.

John Baez (Apr 12 2023 at 18:43):

But if you can cook up a simplicial set in any way, you can then define its homotopy groups (which he's mentioned), its homology groups, and its cohomology groups.

Reid Barton (Apr 12 2023 at 18:51):

I would have assumed it was a sheaf of sets too except that I don't know any way to take a cover, and a sheaf, and get out a simplicial set (rather than a cosimplicial something).

John Baez (Apr 12 2023 at 19:01):

Aha.

Tim Hosgood (Apr 13 2023 at 12:09):

it's quite common that what you actually want to precompose with is the opposite of the Čech nerve (which would solve this cosimplicial vs simplicial problem)

Tim Hosgood (Apr 13 2023 at 12:09):

by "quite common" I mean "this is how you can define the space of e.g. principal bundles using simplicial presheaves and Čech nerves"