In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.
Let
X
A\subsetX
(X,A)
f\bullet\colonA → YI
\tilde{f}0\colonX → Y
f\bullet
\tilde{f}\bullet\colonX → YI
\tilde{f}\bullet\circ\iota=\left.\tilde{f}\bullet\right|A=f\bullet
That is, the pair
(X,A)
G\colon((X x \{0\})\cup(A x I)) → Y
G'\colonX x I → Y
G
G'
Y
(X,A)
Y
The homotopy extension property is depicted in the following diagram
If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map
\tilde{f}\bullet
\tilde{f}\bullet\colonX\toYI
\tilde{f}\bullet\colonX x I\toY
Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.
X
A
X
(X,A)
(X,A)
(X x \{0\}\cupA x I)
X x I.
If
(X,A)
\iota\colonA\toX
\iota\colonY\toZ
Y
\iota
. Allen Hatcher . 2002 . Algebraic Topology . Cambridge University Press . 0-521-79540-0 .