# Cofibration

In mathematics, in particular homotopy theory, a continuous mapping

${\displaystyle i\colon A\to X}$,

where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines fibrations. For a more general notion of cofibration see the article about model categories.

## Basic theorems

• For Hausdorff spaces a cofibration is a closed inclusion (injective with closed image); for suitable spaces, a converse holds
• Every map can be replaced by a cofibration via the mapping cylinder construction
• There is a cofibration (A, X), if and only if there is a retraction from
${\displaystyle X\times I}$
to
${\displaystyle (A\times I)\cup (X\times \{0\})}$,

since this is the pushout and thus induces maps to every space sensible in the diagram.