homotopy invariance

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

A functor on spaces (e.g. some cohomology functor) is called “homotopy invariant” if it does not distinguish between a space $X$ and the space $X \times I$, where $I$ is an interval; equivalently if it takes the same value on morphisms which are related by a (left) homotopy.

The term *homotopy invariant* may also refer to the refinement of invariants to homotopy theory, hence to homotopy fixed points.

A generalized (Eilenberg-Steenrod) cohomology-functor is by definition homotopy invariant, but for instance its refinement to differential cohomology in general no longer is.

**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 April 20, 2017 at 10:27:27. See the history of this page for a list of all contributions to it.