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Proposition 2.5.4.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing a pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$. Let $f$ and $g$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ which are homotopic. Then:
- $(a)$
For any object $W \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and any $0$-cycle $u \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{0}$, the composite cycles $f \circ u$ and $g \circ u$ are homotopic.
- $(b)$
For any object $Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$ and any $0$-cycle $v \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Z)_{0}$, the composite cycles $v \circ f$ and $v \circ g$ are homotopic.
Proof.
By virtue of Remarks 2.5.4.3 and 2.5.4.4, we can regard Proposition 2.5.4.5 as a special case of Proposition 1.4.4.7. However, it is easy to prove directly. If $h \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{1}$ is a homotopy from $f$ to $g$ and $u$ is a $0$-cycle in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{0}$, then the calculation
\begin{eqnarray*} \partial ( h \circ u ) & = & ((\partial h) \circ u) - (h \circ (\partial u)) \\ & = & (\partial h) \circ u \\ & = & (g-f) \circ u \\ & = & (g \circ u) - (f \circ u) \end{eqnarray*}
shows that $(h \circ u) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{1}$ is a homotopy from $f \circ u$ to $g \circ u$. This proves $(a)$, and $(b)$ follows from a similar argument.
$\square$