## Cofibrations Climbing Mount Bourbaki

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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