nLab
cohomology localization

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Let CC be a stable (∞,1)-category (or (∞,1)-topos) and ACA \in C a stable coefficient object. For XCX \in C and nn \in \mathbb{Z} or (nn \in \mathbb{N}) write

H n(X,A):=π 0C(X,B nA) H^n(X,A) := \pi_0 C(X, \mathbf{B}^n A)

for the cohomology of XX with coefficients in AA in degree nn.

Say a morphism in CC is AA-local if it induces isomorphisms on all these cohomology groups. Let WW be the class of all such morphisms.

Then the AA-cohomology localization of CC is – if it exists – the localization of an (∞,1)-category of CC at WW.

Examples

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)

Last revised on August 31, 2019 at 09:27:42. See the history of this page for a list of all contributions to it.