# Does the presence of cocycle conditions indicate the existence of an underlying cohomology theory?

Motivation: We have two examples:

(Abelian) Kummer theory (resp. Artin-Schreier theory) has a hidden cohomology theory given by Galois cohomology. The cocycle conditions become clear when you look at the multiplicative (resp. additive) form of Hlbert's theorem 90.

Descent theory for sheaves and stacks: In the case of sheaves, the cocycle condition is clear when we write $U_{ij}:=U_i \times_U U_j$ and look at the descent sequence. For stacks, the situation is even more obvious when you look at the coherence isomorphisms, which satisfy an explicit cocycle condition. The underlying cohomology theory here is Čech cohomology.

Question: Is this a general phenomenon in mathematics, that the presence of cocycle conditions is a good indicator that there is a cohomology theory determining things from behind the scenes, or is it just a coincidence and I just happened to see two very interesting and special cases?

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The cocycle condition is about gluing sheaves/bundles/.... And when you have sheaves you have cohomology. I don't think there's much more to say. Indeed, you basically gave the answer yourself. –  JBorger Mar 24 '10 at 8:08
Apologies for my ignorance. I was under the impression that when you glue bundles you end up having something that looks like a Cech H^1, but it's just a formal notation as there isn't any obvious way to get an H^2. Is there something I'm missing? –  babubba Mar 24 '10 at 8:52

I had lots of thoughts on that kind of question, and feel uneasy to speak as my answer can range from a tautology, through systematic and positive, but somewhat ignorant toward not-well understood cases, to mere impressions and (seeming?) "counterexample" oriented answer. The basic question is what you mean by a cocycle. Usually one talks on expressions of some higher categorical coherence, or about some notion of homotopy behind it. In such cases the answer is normally yes: the equivalent or homotopic cocycles will form cohomology classes and this can be in all understood cases done naturally and systematically. Higher nonabelian cohomology can be done for all $n$, as now many frameworks know (Brown, Jardine, Toen, Street...) and cohomology boils down to take homotopy classes into certain suspension of the coefficient object. For one recent framework we can advertise our own work (pdf).

I slightly believe anyway that some algebraic cases can be outside of the current homotopy categorical framework and I discussed that much on the n-category cafe, nforum and elsewhere. Namely model categories treat on equal footing homology and cohomology, while the minimal conditions on a setup to be able to do cohomology of homology is less than both simultaneously (cf. work of Rosenberg on "right exact structures" on a category, pdf).

Finally, we can imagine more complicated category-like structures where one can do much of the usual combinatorics but can not properly do the equivalence classes when needed for cohomology. There is one example which is maybe repairable, due Shahn Majid, namely he has a notion of bialgebra cocycles for a noncommutative and noncocommutative bialgebra. Now in special cocommutative or commutative cases like Lie algebras and/or abelian coefficients he recovers some known cohomology theories like Chevalley-Eilenberg cohomology for Lie algebras. In low dimensional cases he also gets some interesting nonabelian cocycles of much usage like Drinfel'd 2-twist and Drinfel'd 3-associator which are used in the study of monoidal categories, CFT, knot theory and quantum groups. In this example the differential and cocycles are defined for every $n$ but 17 years after the discovery, there is still no known way to define well the cohomology classes, for dimension 3 or more, for general bialgebra, despite the special cases and despite the cocycles and the differential. See the nlab page bialgebra cocycle for the basics (and the references therein).

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My point of view on this is that cocycle conditions always come from applying a functor to some kind of cobar construction. In the case of sheaf cohomology, for example, you start with a cover $V \to U$ (if it's a classical topology, $V = \coprod U_i$ for open subsets $U_i \subset U$) and the cobar construction is $C^n = V \times_U V \times_U \dots \times_U V$ ($n$ times). If you pick $V$ acyclic for your functor then taking the cohomology of the chain complex associated to the functor applied to this cosimplicial object is a cohomology theory.

This is assuming that your functor (in the case of sheaf cohomology: global sections) takes values in an abelian category. If it takes values in groups, you can still define $H^0$ and $H^1$, but it stops there. If it takes values in pointed sets, you can only define $H^0$. More generally, if it takes values in $E_n$-algebras (coherent commutativity up to level $n$ in some homotopical category), you'll be able to define $H^i$ for $i\leq n$.

So no, not every cocycle condition comes from a cohomology theory that's defined for every integer. But it does for abelian functors.

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I have a stupid (and somewhat vague) comment: in the 'abelian' cases the answer is usually 'yes' by trivial reasons. If you check some cocycle condition (of any natuer) on objects of some abelian category, it is usually expressed by the object in question lying in the kernel of something. So, you have a left-exact functor of abelian categories; then you can consider its derived functors!:)

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