Contents

# Contents

## 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.

homotopy type theoryrepresentation 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.