Definition 3.2.1.3. Let $f_0, f_1: X \rightarrow Y$ be a pair of morphisms of simplicial sets and let $h: \Delta ^1 \times X \rightarrow Y$ be a homotopy from $f_0$ to $f_1$. If $K \subseteq X$ is a simplicial subset, we say that $h$ is constant along $K$ if the restriction $h|_{ \Delta ^1 \times K}$ factors through the projection map $\Delta ^1 \times K \twoheadrightarrow K$.
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