Homotopy extension property


In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

Definition

Let be a topological space, and let.
We say that the pair has the homotopy extension property if, given a homotopy and a map such that, there exists an extension of to a homotopy such that
That is, the pair has the homotopy extension property if any map
can be extended to a map .
If the pair has this property only for a certain codomain, we say that has the homotopy extension property with respect to.

Visualisation

The homotopy extension property is depicted in the following diagram
If the above diagram commutes, then pair has the homotopy extension property if there exists a map which makes the diagram commute. By currying, note that a map is the same as a map.
Note that this diagram is dual to that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.

Properties

If has the homotopy extension property, then the simple inclusion map is a cofibration.
In fact, if you consider any cofibration, then we have that is homeomorphic to its image under. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.