Kerodon

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Remark 2.5.3.9. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $K_{\bullet }$ be a simplicial set. To give a map of simplicial sets $f: K_{\bullet } \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$, one must supply the following data:

  • For each vertex $x$ of $K_{\bullet }$, an object $f(x)$ of the differential graded category $\operatorname{\mathcal{C}}$.

  • For each $k > 0$ and each $k$-simplex $\sigma : \Delta ^ k \rightarrow K_{\bullet }$ with initial vertex $x = \sigma (0)$ and final vertex $y = \sigma (k)$, a $(k-1)$-chain $f(\sigma ) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f(x), f(y) )_{k-1}$.

Moreover, this data must satisfy the following conditions:

  • If $e$ is a degenerate edge of $K_{\bullet }$ connecting a vertex $x$ to itself, then $f(e)$ is the identity morphism $\operatorname{id}_{f(x)} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( f(x), f(x) )_{0}$.

  • If $\sigma $ is a degenerate simplex of $K_{\bullet }$ having dimension $\geq 2$, then $f(\sigma ) = 0$.

  • Let $k > 0$ and let $\sigma : \Delta ^ k \rightarrow K_{\bullet }$ be an $k$-simplex of $K_{\bullet }$. For $0 < b < k$, let $\sigma _{\leq b}: \Delta ^{b} \hookrightarrow K_{\bullet }$ denote the composition of $\sigma $ with the inclusion map $\Delta ^{b} \hookrightarrow \Delta ^{k}$ (which is the identity on vertices), and let $\sigma _{\geq b}: \Delta ^{k-b} \hookrightarrow K_{\bullet }$ denote the composition of $\sigma $ with the map $\Delta ^{k-b} \hookrightarrow \Delta ^{k}$ given on vertices by $i \mapsto i+b$. Then we have

    \begin{eqnarray*} \partial f(\sigma ) = \sum _{b=1}^{k-1} (-1)^{k-b} (f( \sigma _{\geq b}) \circ f( \sigma _{\leq b} ) - f( d_ b \sigma ) ) \end{eqnarray*}