Contents

# Contents

## 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

homotopy type theoryrepresentation 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)

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