nLab
homotopical cohomology theory

Idea

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

X^ A X \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 XX with coefficients in AA and to regard

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

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

Literature

The fundamental ideas and facts are given in

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)

Revised on June 30, 2009 10:20:54 by David Corfield (81.158.202.139)