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 $latex {A subset X}&fg=000000$ and a map $latex {A stackrel{f}{rightarrow} Y}&fg=000000$. We’d like to know when we can extend this map over all of $latex {X}&fg=000000$. One useful criterion can be given by algebraic functors. For instance, singular homology can be used to show that the map $latex {1:S^{n} rightarrow S^n}&fg=000000$ does not extend over $latex {D^{n+1}}&fg=000000$.

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 $latex {A rightarrow Y}&fg=000000$ extends depends only on the homotopy class of $latex {A}&fg=000000$. This is precisely the definition of a cofibration. Dualizing gives the definition of a…

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