nLab
cohomology localization

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

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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

Revised on September 3, 2012 18:09:46 by Urs Schreiber (131.174.188.82)