nLab homotopical cohomology theory

Idea

Under certain conditions on a given homotopy theory on a category $C$ (namely if $C$ satisfies the axioms of a category of fibrant objects), the morphisms in the homotopy category $Ho(C)$ are represented already by spans

$$\array{ \hat X &\to& A \\ \downarrow^\simeq \\ X }$$

(ana-morphisms) instead of longer sequences of zig-zags.

In such a case it makes sense to address such a span as a nonabelian cocycle on $X$ with coefficients in $A$ and to regard

$$H(X,A) := Ho(C)(X,A)$$

as the cohomology of $X$ with coefficients in $A$. From the point of view of generalized sheaf cohomology this can be understood by recognizing the replacement object $\hat X$ appearing here as a codescent object.

Literature

The fundamental ideas and facts are given in

• Kenneth S. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (BrownAHT)

This develops the theory of localization in categories of fibrant objects and then applies it to cohomology with coefficients in sheaves with values in combinatorial spectra.

Homotopical cohomology theory in the context of the homotopy theory of simplicial presheaves (presheaves with values in simplicial sets) has been much developed by Jardine. For instance

• J. Jardine, Cocycle categories (arXiv)