Definition 4

Let $(C, d)$ and $(C', d')$ be two complexes, $f$ and $g$ two morphisms from $C$ to $C'$. A homotopy connecting $f$ to $g$ is any graded A-homomorphism $s$ of degree 1 from $C$ to $C'$ such that $g - f = d' \circ s + s \circ d$. We say that $f$ and $g$ are homotopic if there exists a homotopy connecting $f$ to $g$. If $h$ is a third morphism from $C$ to $C'$ and if $s$ (resp. $t$) is a homotopy connecting $f$ to $g$ (resp. $g$ to $h$), then $s + t$ is a homotopy connecting $f$ to $h$; consequently, the relation « $f$ and $g$ are two homotopic morphisms from $C$ to $C'$ » is an equivalence relation, whose classes are called the homotopy classes of morphisms from $C$ to $C'$.

Given two topological spaces $X$ and $Y$, and a continuous mapping $f : X \to Y$, one defines a linear mapping $f_*$ from the singular complex (cf. No. 3) $C(X, A)$ into $C(Y, A)$ by setting $f_*(e_s) = e_{f \circ s}$ for $s \in \Sigma_n(X)$. This mapping is a morphism of complexes. Two continuous mappings $f$ and $g$ from $X$ into $Y$ are said to be topologically homotopic if there exists a continuous mapping $h$ from $[0, 1] \times X$ into $Y$ such that $h(0, x) = f(x)$ and $h(1, x) = g(x)$ for all $x \in X$. One shows that, if $f$ and $g$ are topologically homotopic, the morphisms $f_*$ and $g_*$ are homotopic in the sense of Definition 4 above. It is this fact that is at the origin of the terminology used in algebra.