nLab homotopy invariance (changes)

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Context

Homotopy theory

Idea
Examples
Related concepts

Idea

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.

Examples

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

Related concepts

homeomorphism invariant

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 14:27:27. See the history of this page for a list of all contributions to it.