nLab groupoid cohomology

• group cohomology

  • nonabelian group cohomology, groupoid cohomology

• group extension

• Lie group cohomology

  • ∞-Lie groupoid cohomology

---

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

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

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

• The Liouville cocycle is a 1-cocycle on the action groupoid of the action of conformal rescalings on the space of Riemannian metrics on a surface.