nLab ∞-group cohomology

Idea

For $\mathbf{H}$ an ∞-topos and for $G \in Grp(\mathbf{H})$ an ∞-group in $\mathbf{H}$, the higher analog of group cohomology of $G$ is the cohomology of the delooping object $\mathbf{B}G \in \mathbf{H}$.

Given a cocycle $\mathbf{B}G \longrightarrow \mathbf{B}^{n+1} A$ in the $\infty$-group cohomology of $G$, then its homotopy fiber (the principal ∞-bundle over $\mathbf{B}G$ that it modulates) is the corresponding ∞-group extension.

Examples

• ∞-Lie groupoid cohomology

Related concepts

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$	coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$	trivial representation
dependent product along $\mathbf{B}G \to \ast$	homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)