nLab cohomology localization

Definition

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

$$H^n(X,A) := \pi_0 C(X, \mathbf{B}^n A)$$

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 $W$.

Examples

• At function algebras on ∞-stacks is discussed a class of examples of cohomology localizations of (∞,1)-sheaf (∞,1)-toposes over sites of algebras over an abelian Lawvere theory at cohomology with coefficients in the canonical line object $\mathbb{A}^1$.

Related concepts

• homology localization

References

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. Journal of the European Mathematical Society (2015). Volume: 017, Issue: 3, page 549-589, (arXiv:1101.2792)