Kerodon

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Construction 2.5.0.8 (The Homotopy Category of Chain Complexes). Let $\operatorname{\mathcal{A}}$ be an additive category. We define a category $\operatorname{hCh}(\operatorname{\mathcal{A}})$ as follows:

  • The objects of $\operatorname{hCh}(\operatorname{\mathcal{A}})$ are chain complexes with values in $\operatorname{\mathcal{A}}$.

  • If $C_{\ast }$ and $D_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then $\operatorname{Hom}_{ \operatorname{hCh}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )$ is the quotient of $\operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )$ by the relation of chain homotopy equivalence. If $f: C_{\ast } \rightarrow D_{\ast }$ is a chain map, we denote its equivalence class by $[f] \in \operatorname{Hom}_{\operatorname{hCh}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast } )$.

  • If $C_{\ast }$, $D_{\ast }$, and $E_{\ast }$ are chain complexes with values in $\operatorname{\mathcal{A}}$, then the composition law

    \[ \circ : \operatorname{Hom}_{\operatorname{hCh}(\operatorname{\mathcal{A}})}( D_{\ast }, E_{\ast }) \times \operatorname{Hom}_{ \operatorname{hCh}(\operatorname{\mathcal{A}})}( C_{\ast }, D_{\ast }) \rightarrow \operatorname{Hom}_{ \operatorname{hCh}(\operatorname{\mathcal{A}})}( C_{\ast }, E_{\ast } ) \]

    is uniquely determined by the requirement that $[g] \circ [f] = [g \circ f]$ for every pair of chain maps $f: C_{\ast } \rightarrow D_{\ast }$ and $g: D_{\ast } \rightarrow E_{\ast }$ (this operation is well-defined by virtue of Remark 2.5.0.7).

We will refer to $\operatorname{hCh}(\operatorname{\mathcal{A}})$ as the homotopy category of $\operatorname{Ch}(\operatorname{\mathcal{A}})$.