Kerodon

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

Example 4.6.6.14 (Pinched Morphism Spaces in the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category (Definition 2.5.2.1), let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the differential graded nerve of $\operatorname{\mathcal{C}}$ (Definition 2.5.3.7), and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$ (which we also view as objects of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$), and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ denote the chain complex of morphisms from $X$ to $Y$. For $n \geq 0$, we can identify $n$-simplices of the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ with $(n+1)$-simplices $\sigma : \Delta ^{n+1} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ for which $\sigma (0) = X$ and $d_0(\sigma )$ is the constant $n$-simplex with the value $Y$ (Remark 4.6.6.2). Concretely, such a simplex can be described as a datum $I \mapsto f_{I}$, defined for each subset $I = \{ i_0 > i_1 > i_2 > \cdots > i_ k > i_{k+1} \} \subseteq [n+1]$ having at least two elements, with the following properties:

$(1)$

If $i_{k+1} > 0$, then $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{k}$, which is equal to $\operatorname{id}_ Y$ in the case $k=0$ and vanishes for $k > 0$.

$(2)$

If $i_{k+1} = 0$, then $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{k}$ which satisfies the identity

\[ \partial f_{I} = \sum _{a=1}^{k} (-1)^{a} ( f_{ \{ i_0 > i_1 > \cdots > i_ a \} } \circ f_{ \{ i_ a > \cdots > i_{k+1} \} } - f_{I \setminus \{ i_ a \} } ). \]

Note that, by virtue of $(1)$, we can rewrite this identity as

4.36
\begin{equation} \label{equation:pinched-morphism-in-dg-nerve} \partial f_{I} = \begin{cases} 0 & \textnormal{ if } k=0 \\ \sum _{a=0}^{k} (-1)^{a+1} f_{I \setminus \{ i_ a\} } & \textnormal{ if } k > 0. \end{cases} \end{equation}

Let $J = \{ j_0 < j_1 < \cdots < j_ k \} $ be a nonempty subset of $[n]$. For $\{ f_ I \} $ as above, define $g_{J} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{k}$ by the formula $g_ J = (-1)^{k(k-1)/2} f_{ \{ j_ k + 1 > j_{k-1} + 1 > \cdots > 0 \} }$. We can then rewrite the identity (4.36) as

\begin{eqnarray*} \partial g_{J} & = & \sum _{b = 0}^{k} (-1)^{b} g_{J \setminus \{ j_ b\} }. \end{eqnarray*}

The construction $J \mapsto g_{J}$ can then be identified with a morphism from the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^ n)$ of Construction 2.5.5.9 to the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. This identification depends functorially on $n$, and therefore determines an isomorphism of simplicial sets

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X, Y) \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ), \]

where $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ denotes the Eilenberg-MacLane space associated to the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ (Construction 2.5.6.3). In particular, the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X, Y)$ has the structure of a simplicial abelian group.