Cofibration

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In mathematics, in particular homotopy theory, a continuous mapping

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[edit]

  • 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
 X \times I
to
 (A \times I) \cup (X \times \{0\}),

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

Examples[edit]

  • Cofibrations are preserved under push-outs and composition, as one sees from the definition via diagram-chasing.
  • A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if (X, A) is a CW pair, then A \to X is a cofibration). This follows from the previous fact since S^{n-1} \to D^n is a cofibration for every n.

References[edit]