Composite of homotopies

Definition

Definition assuming all homotopies take time $1$

Suppose:

• $f_{1},f_{2},f_{3}:X\to Y$ are continuous maps
• $I$ is the closed unit interval
• $F_{12}:X\times I\to Y$ is a homotopy from $f_{1}$ to $f_{2}$
• $F_{23}:X\times I\to Y$ is a homotopy from $f_{2}$ to $f_{3}$

Then, the composite of homotopies $F_{13}=F_{12}*F_{23}$ is a homotopy from $f_{1}$ to $f_{3}$ given as follows:

$F_{13}(x,t)=\lbrace {\begin{array}{rl}F_{12}(x,2t),&0\leq t<1/2\\F_{23}(x,2t-1),&1/2\leq t\leq 1\end{array}}$