Definition 2.5.4.1. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and let $f$ and $f'$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. A homotopy from $f$ to $f'$ is a $1$-chain $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{1}$ satisfying $\partial (h) = f'-f$. We will say that $f$ and $f'$ are homotopic if there exists a homotopy from $f$ to $f'$: that is, if we have an equality $[f] = [f']$ in the homology group $\mathrm{H}_{0}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$.
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