Generally, given a category with a concept of path space object and hence of right homotopy, then a morphism is said to have the right homotopy extension property with respect to an object if it has the left lifting property against one of the two path endpoint evaluation maps , i.e. if in every commuting square as below, a diagonal lift exists:
Here the top morphism exhibits a right homotopy between two morphisms , and the diagonal filler is an extension of the top morphism along and exhibiting a compatible right homotopy between morphisms , whence the terminology.
The concept of homotopy extension property is the Eckmann-Hilton dual of that of (left) homotopy lifting property, where instead one considers the presence of a cylinder object, hence a notion of left homotopy, and lifting in diagrams of the form
In situations where both path space objects as well as cylinder objects exist and are compatible, so that the concepts of left and right homotopy coincide, one may equivalently rephrase right homotopy extension in terms of left homotopy (and left homotopy lifting in terms of right homotopy).
The resulting definition is necessarily less transparent (def. 1 below), but it happens to be more commonly used in the literature. Specifically in the archetypical case that the ambient category is that of topological spaces and cylinders and path space objects are induced from the standard topological interval object, a Hurewicz cofibration (often just called cofibration ) is a continuous function that satisfies the left homotopy extension property with respect to all topological spaces.
A continuous function of topological spaces is said to satisfy the (left) homotopy extension property (HEP) with respect to a space if for any map and a homotopy such that , a homotopy exists such that .
If we write
then this is expressed by means of the commutative diagram
Here we denote , so that . The map is sometimes said to be the initial condition of a homotopy extension problem. is the extension of the homotopy with given initial condition which itself extends .
Of course it is superfluous to write the arrow : if we erase it, the commutativity of the remaining square just surrounding its position is saying ; however it is conceptually nice to think of as extending some .
One can instead of the diagram above write a diagram involving adjoint maps. In other words, instead of any homotopy we use the exponential law to write where the correspondence is given by the formula . Then the homotopy lifting property is the existence of the diagonal map in the diagram:
where is the map of evaluation at zero . This is an instance of right lifting property with respect to maps .
A map is a Hurewicz cofibration if it satisfies the homotopy extension property with respect to all spaces.
If a map has the homotopy extension property with respect to a space , then for any map , the pushout has the homotopy extension property with respect to the space .
This is a general statement about classes of morphisms defined by a left lifting property, see at injective and projective morphisms – closure properties
We would like to find to complete the commutative diagram
Consider the external square obtained by composing the horizontal arrows:
By the assumption on , there is a as in the diagram, such that both triangles commute, i.e. and .
If is satisfying the HEP with respect to then there is a diagonal in that external square which is some map . This map together with , by the universal property of pushout, determines a unique map such that and . We need to show only that as holds by the construction of as stated.
By the definition of and the commutativity of the original double square diagram, and . This is almost what we wanted except that we precompose the wanted identity with both maps into the pushout. Thus by the uniqueness part of the universal property of pushout it follows that .