nLab homotopy invariance

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

• characteristic class

• generalized (Eilenberg-Steenrod) cohomology

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)