nLab homotopy invariance



Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




A functor on spaces (e.g. some cohomology functor) is called “homotopy invariant” if it does not distinguish between a space XX and the space X×IX \times I, where II 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 theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum 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.