group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Let $C$ be a stable (∞,1)-category (or (∞,1)-topos) and $A \in C$ a stable coefficient object. For $X \in C$ and $n \in \mathbb{Z}$ or ($n \in \mathbb{N}$) write
for the cohomology of $X$ with coefficients in $A$ in degree $n$.
Say a morphism in $C$ is $A$-local if it induces isomorphisms on all these cohomology groups. Let $W$ be the class of all such morphisms.
Then the $A$-cohomology localization of $C$ is – if it exists – the localization of an (∞,1)-category of $C$ at $W$.
Set theoretic issues in cohomology localization – and their solution using large cardinal axioms such as Vopenka's principle, is discussed in
Carles Casacuberta, Dirk Scevenels, Jeff Smith, Implications of large-cardinal principles in homotopical localization Advances in Mathematics Volume 197, Issue 1, 20 October 2005, Pages 120-139
Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiri Rosicky, Definable orthogonality classes in accessible categories are small (arXiv:1101.2792)