nLab
groupoid cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems


Contents

Idea

Groupoid cohomology is the cohomology specifically of ordinary groupoids, more generally that of internal groupoids.

Groupoid cohomology generalizes group cohomology, which is the cohomology of delooping groupoids of groups. Analogously to abelian and nonabelian group cohomology there is abelian and nonabelian groupoid cohomology.

Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types.

A common special case of groupoid cohomology is the cohomology of action groupoids: this is (Borel)-equivariant cohomology.

General groupoid cohomology may be regarded as a generalization of equivariant cohomology exactly analogous to the passge from global action groupoids to orbifolds.

Details

Let H=\mathbf{H} = ∞Grpd be the (∞,1)-topos of ∞-groupoids. Let XHX \in \mathbf{H} be an ordinary 1-groupoid. Let AHA \in \mathbf{H} be an arbitrary \infty-groupoid. Then the cohomology of XX with coefficients in AA is

H(X,A):=π 0H(X,A). H(X,A) := \pi_0 \mathbf{H}(X,A) \,.

Concrete formulas for this work exactly as described in detail at group cohomology, only wherever there we have the unique object \bullet, now there may be arbitrary objects.

Examples

Revised on September 12, 2010 10:50:00 by Urs Schreiber (188.20.66.18)