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Definition. Homotopy extension property (HEP) [math-000O]

A map i:A→Xi : A \to X of spaces has the homotopy extension property for a space YY if

  1. for each homotopy H:A×I→YH : A \times I \to Y
  2. and for each map f:X→Yf : X \to Y with f(i(a))=H(a,0)f(i(a)) = H(a, 0) for all a∈Aa \in A

there is a homotopy H′:X×I→YH' : X \times I \to Y such that

H′(i(a),t)=H(a,t)H′(x,0)=f(x)\begin{align*} &H'(i(a), t) &= &H(a, t) \\ &H'(x, 0) &= &f(x) \end{align*}

for all a∈Aa \in A, x∈Xx \in X and t∈It \in I. The idea can be expressed in the following commutative diagram:

figure tex2120

The homotopy H′H' is called the extension of HH with initial condition ff.