Do we have non-abelian sheaf cohomology? - MathOverflow

Do we have non-abelian sheaf cohomology?

2010-08-23T17:05:33Z

Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:

$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative structure.

Can we define $H^i(X,F)$ ? Note that for $N=1$, this would be just $H^i(X,O_X^*)$.

(Please give reference for your claims)

Answer by Charles Siegel for Do we have non-abelian sheaf cohomology?

2010-08-23T17:11:22Z

Look up Giraud's "Cohomologie Non-abelienne", it should answer most of your questions. And, for the record, for $i=1$, by an analagous argument, you get a cohomology set (only a group if abelian) which is the moduli of vector bundles of rank $n$, or more generally principal $G$-bundles, if you replace $GL(N,\mathbb{C})$ with another group $G$.

Edit: "answer most of your questions" is somewhat stronger than what I actually meant. It's a place to start looking, and to see why the answer is far from trivial. I've heard vaguely that Lurie has some answers to this question (perhaps in Higher Topos Theory?) but I haven't had a chance to go through in detail and look at it.

Answer by José Figueroa-O'Farrill for Do we have non-abelian sheaf cohomology?

2010-08-23T17:12:07Z

Chapter 4 of Jean-Luc Brylinski's Loop spaces, characteristic classes and geometric quantization talks about non-abelian sheaf cohomology in degree 1 (i.e., $i=1$ in the notation in the question). I heard that there was a generalisation for $i>1$ due to Giraud, but I never studied it and cannot give you a reference.

Answer by David Roberts for Do we have non-abelian sheaf cohomology?

2010-08-24T00:09:09Z

The quick reply is: not really for $i \gt 2$, and not in the way you perhaps expect for $i=2$, see below.

The comment on Charles' answer about 'teaching you never to ask that question again' is partly true, partly not. The lesson to learn from Giraud is that really one does not use groups for coefficients of higher cohomology. For a start, Giraud's $H^2(X,G)$ is not functorial with respect to group homomorphisms $G\to H$! One also does not get the exact sequences that one expects (this is due to the lack of functoriality). But this is not a problem with his definition of the cohomology set, but a problem with what category you believe the coefficients lie in. This is because the coefficient object of Giraud's cohomology is actually the crossed module $AUT(G) = (G \to Aut(G))$, and the assignment $G \mapsto AUT(G)$ is not functorial. (Aside: Giraud contains lots of other important things on stacks and gerbes and sites and so on, so the book is not a waste of time by any means)

But little-known work by Debremaeker[1-3] from the 1970s fixed this up and showed that really the Giraud cohomology was functorial with respect to morphisms of crossed modules. This has been recently extended by Aldrovandi and Noohi [4] by showing that it is functorial with respect to weak maps of crossed modules aka butterflies/papillion.

It was realised by John E. Roberts (no relation) and Ross Street that the most general nonabelian cohomology has as coefficient objects higher categories. In fact, we now know that the coefficients of $n^{th}$ degree cohomology is an $n$-category (usually an $n$-groupoid, though), even when we are talking about usual abelian cohomology.

Everything I've talked about is just for groups etc in Set, but it can all be done internal to a topos, i.e. for sheaves of groups, and more generally a Barr-exact category (and probably weaker, but Barr-exact means that the monadic description of cohomology therein due to Duskin (probably going back to Beck) works fine).

[1] R. Debremaeker, Cohomologie a valeurs dans un faisceau de groupes croises sur un site. I, Acad. Roy. Belg. Bull. Cl. Sci. (5), 63, (1977), 758 -- 764.

[2] R. Debremaeker, Cohomologie a valeurs dans un faisceau de groupes croises sur un site. II, Acad. Roy. Belg. Bull. Cl. Sci. (5), 63, (1977), 765 -- 772.

[3] R. Debremaeker, Non abelian cohomology, Bull. Soc. Math. Belg., 29, (1977), 57 -- 72.

[4] E. Aldrovandi and B. Noohi, Butterflies I: Morphisms of 2-group stacks, Advances in Mathematics, 221, (2009), 687 -- 773.