In mathematics, in particular homotopy theory, a continuous mapping


where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. This definition is dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces. This duality is informally referred to as Eckmann–Hilton duality.

A more general notion of cofibration is developed in the theory of model categories.

Basic theorems

One then decomposes into the composite of a cofibration and a homotopy equivalence. That is, can be written as the map
with , when is the inclusion, and on and on .



The homotopy colimit generalizes the notion of a cofibration.


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