The idea of a cofibration will be fundamental as we talk further about homotopy theory, as will its dual idea of a fibration. The point of the cofibration condition is the following. Oftentimes, we have a subspace {A \subset X} and a map {A \stackrel{f}{\rightarrow} Y}. We’d like to know when we can extend this map over all of {X}. One useful criterion can be given by algebraic functors. For instance, singular homology can be used to show that the map {1:S^{n} \rightarrow S^n} does not extend over {D^{n+1}}.

Many of the things we care about in algebraic topology are homotopy invariant, though. As a result, it would be nice to know when the question of how {A \rightarrow Y} extends depends only on the homotopy class of {A}. This is precisely the definition of a cofibration. Dualizing gives the definition of a fibration, which I will talk about some other time.

One can, not surprisingly, approach the business of fibrations and cofibrations in an axiomatic manner. This is the theory of model categories, due to Quillen. I don’t know anything about that though.