Kerodon

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Remark 2.5.3.8 (Comparison with the Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{\mathcal{C}}^{\circ }$ denote its underlying ordinary category (Construction 2.5.2.4). Suppose that $\sigma$ is an $n$-simplex of the nerve $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\circ } )$, consisting of objects $\{ X_ i \} _{0 \leq i \leq n}$ and $0$-cycles $\{ f_{ji} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_ i, X_ j )_0 \}$ satisfying $f_{ii} = \operatorname{id}_{X_ i}$ and $f_{ki} = f_{kj} \circ f_{ji}$ for $0 \leq i \leq j \leq k \leq n$. We can then construct an $n$-simplex $U(\sigma )$ of the differential graded nerve $\operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$, given by

$U( \sigma ) = ( \{ X_ i \} _{0 \leq i \leq n}, \{ f_ I \} ) \quad \quad f_{I} = \begin{cases} f_{ji} & \text{ if } I = \{ j > i \} \\ 0 & \text{ otherwise.} \end{cases}$

The construction $\sigma \mapsto U(\sigma )$ determines a map of simplicial sets $U: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\circ } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}( \operatorname{\mathcal{C}})$. This map is a monomorphism, whose image is the simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ spanned by those $n$-simplices $( \{ X_ i \} _{0 \leq i \leq n}, \{ f_ I \} )$ with the property that $f_{I} = 0$ for $|I| > 2$.