Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.5.4.3. 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 $g$ be $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. Then giving a homotopy from $f$ to $g$ in the sense of Definition 2.5.4.1 is equivalent to giving a homotopy from $f$ to $g$ as morphisms in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ (Definition 1.3.3.1): this follows from Example 2.5.3.4. In particular, $f$ and $g$ are homotopic in the sense of Definition 2.5.4.1 if and only if they are homotopic in the sense of Definition 1.3.3.1.