# Kerodon

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Notation 2.5.9.9. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ denote the underlying simplicial category (Construction 2.5.9.2). Let $n \geq 0$ be a nonnegative integer and let $\sigma$ be a nondegenerate $(n+1)$-simplex of the homotopy coherent nerve $\operatorname{N}^{\operatorname{hc}}_{\bullet }(\operatorname{\mathcal{C}}^{\Delta })$, which we will identify with a simplicial functor $\sigma : \operatorname{Path}[n+1]_{\bullet } \rightarrow \operatorname{\mathcal{C}}^{\Delta }_{\bullet }$. Set $X = \sigma (0)$ and $Y = \sigma (n+1)$, and $I = \{ 1, 2, \cdots , n \}$, so that Remark 2.4.5.4 supplies a morphism of simplicial sets

$\operatorname{\raise {0.1ex}{\square }}^{I} \simeq \operatorname{Hom}_{\operatorname{Path}[n+1]}(0, n+1)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}( X, Y)_{\bullet } = \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ),$

which we can identify with a chain map $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. For any choice of ordering of $I$, this map carries the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ of Construction 2.5.9.6 to an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_ n$, which we will denote by $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n} ])$.