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.

be a topological space, and let

We say that the pair

has the homotopy extension property if, given a homotopy

and a map

∘ ι =

then there exists an extension of

∘ ι =

has the homotopy extension property if any map

can be extended to a map

agree on their common domain).

If the pair has this property only for a certain codomain

has the homotopy extension property with respect to

The homotopy extension property is depicted in the following diagram If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map

which makes the diagram commute.

By currying, note that homotopies expressed as maps

are in natural bijection with expressions as maps

Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.

has the homotopy extension property, then the simple inclusion map

ι :

is a cofibration.

In fact, if

ι :

is a cofibration, then

is homeomorphic to its image under

ι

This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.