nLab
∞-group cohomology

Contents

Contents

Idea

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

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

Examples

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

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

Last revised on January 1, 2014 at 08:58:19. See the history of this page for a list of all contributions to it.