Kerodon

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

Example 7.6.1.16 (Products in a Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\{ q_ i: Y \rightarrow Y_ i \} $ be a collection of morphisms in the underlying category of $\operatorname{\mathcal{C}}$ (that is, each $q_ i$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Y_ i)_{\ast }$). Using Example 4.6.5.14 (together with Exercise 3.2.2.18), we see that the following conditions are equivalent:

$(1)$

The morphisms $q_ i$ exhibit $Y$ as a product of the collection $\{ Y_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

$(2)$

For every object $X \in \operatorname{\mathcal{C}}$, the map of chain complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow \prod _{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y_ i)_{\ast } \]

induces an isomorphism on homology in degrees $\geq 0$.