Cofibration
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces i : A → X {\displaystyle i:A\to X} is a cofibration if it has the homotopy extension property with respect to all topological spaces S {\displaystyle S} . That is, i {\displaystyle i} is a cofibration if for each topological space S {\displaystyle S} , and for any continuous maps f , f ′ : A → S {\displaystyle f,f':A\to S} and g : X → S {\displaystyle g:X\to S} with g ∘ i = f {\displaystyle g\circ i=f} , for any homotopy h : A × I → S {\displaystyle h:A\times I\to S} from f {\displaystyle f} to f ′ {\displaystyle f'} , there is a continuous map g ′ : X → S {\displaystyle g':X\to S} and a homotopy h ′ : X × I → S {\displaystyle h':X\times I\to S} from g {\displaystyle g} to g ′ {\displaystyle g'} such that h ′ ( i ( a ) , t ) = h ( a , t ) {\displaystyle h'(i(a),t)=h(a,t)} for all a ∈ A {\displaystyle a\in A} and t ∈ I {\displaystyle t\in I} .