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Construction (The Homotopy Category of a Differential Graded Category). Let $\operatorname{\mathcal{C}}$ be a differential graded category. We define a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, we define

    \[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y) = \mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]

    If $f$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$, let $[f]$ denote its image in the homology group $\mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y)$.

  • For each object $X \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the identity morphism from $X$ to itself in the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by $[ \operatorname{id}_ X ]$, where $\operatorname{id}_ X$ is the identity morphism from $X$ to itself in $\operatorname{\mathcal{C}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law

    \[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( Y,Z ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Y) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, Z) \]

    is characterized by the formula $[g] \circ [f] = [g \circ f]$ for $f \in \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$ and $g \in \mathrm{Z}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) )$ (this composition law is well-defined by virtue of Proposition

We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the homotopy category of the differential graded category $\operatorname{\mathcal{C}}$.