Proposition 3

*Let $C, \tilde{C}, C', \tilde{C}'$ be four $A$-complexes, $u : \tilde{C} \to C, v : \tilde{C} \to C, u' : C' \to \tilde{C}'$ and $v' : C' \to \tilde{C}'$ four morphisms of complexes.

a) If $u$ and $u'$ are homotopic to $v$ and $v'$ respectively, then the two morphisms $\mathrm{Homgr}_A(u, u')$ and $\mathrm{Homgr}_A(v, v')$ from $\mathrm{Homgr}_A(C, C')$ to $\mathrm{Homgr}_A(\tilde{C}, \tilde{C}')$ are homotopic.

b) If $u$ and $u'$ are homotopisms, $\mathrm{Homgr}_A(u, u')$ is a homotopism.

c) If $C$ or $C'$ is homotopic to zero, $\mathrm{Homgr}_A(C, C')$ is homotopic to zero.*

Let us denote by the same letter $d$ the differentials of the complexes $C$, $C_1$, $C'$, $C'_1$, and by $D$ the differentials of $\mathrm{Homgr}_A(C,C')$ and $\mathrm{Homgr}_A(C_1,C'_1)$. If $u$ (resp. $u'$) is homotopic to $v$ (resp. $v'$), there exists a graded homomorphism of degree $1$,

such that

Let $W:\mathrm{Homgr}_A(C,C')\to\mathrm{Homgr}_A(C_1,C'_1)$ be the graded homomorphism of degree $1$ such that, for $f\in\mathrm{Homgr}_A(C,C')_n$, $n\in\mathbf{Z}$, one has

We then have

This is written as $DW+WD=\mathrm{Homgr}_A(u,u')-\mathrm{Homgr}_A(v,v')$, whence a).

Let us prove b). If $u$ and $u'$ are homotopy equivalences, let $\alpha:C\to\widetilde C$ and $\alpha':\widetilde C'\to C'$ be morphisms of complexes such that $u\circ\alpha$, $\alpha\circ u$, $u'\circ\alpha'$, $\alpha'\circ u'$ are homotopic respectively to $\mathrm{Id}_C$, $\mathrm{Id}_{\widetilde C}$, $\mathrm{Id}_{\widetilde C'}$, $\mathrm{Id}_{C'}$. Then $\mathrm{Homgr}(u,u')\circ\mathrm{Homgr}(\alpha,\alpha')$, which is equal to $\mathrm{Homgr}(\alpha\circ u,u'\circ\alpha')$, is homotopic by a) to

analogously $\mathrm{Homgr}(\alpha,\alpha')\circ\mathrm{Homgr}(u,u')$ is homotopic to $\mathrm{Id}_{\mathrm{Homgr}(\widetilde C,\widetilde C')}$, whence b).

Finally c) follows from b) (applied to the case where $\widetilde C$ or $\widetilde C'$ is zero).