Kerodon

Example 2.4.1.7. Let $X$ and $Y$ be topological spaces and let $f,g: X \rightarrow Y$ be continuous functions, which we regard as morphisms in the simplicial category $\operatorname{Top}_{\bullet }$ of Example 2.4.1.5. Then a homotopy from $f$ to $g$ in the sense of Definition 2.4.1.6 is a homotopy in the usual sense: a continuous function $h: [0,1] \times X = | \Delta ^1 | \times X \rightarrow Y$ satisfying $h(0,x) = f(x)$ and $h(1,x) = g(x)$ for all $x \in X$.