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.

**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) |

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