Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Theorem 2.5.3.10. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is an $\infty$-category.

Proof. Suppose we are given $0 < j < n$ and a map of simplicial sets $\sigma _0: \Lambda ^{n}_{j} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. Using Remark 2.5.3.9, we see that $\sigma _0$ can be identified with the data of a pair $( \{ X_ i \} _{0 \leq i \leq n}, \{ f_ I \} )$, where $\{ X_ i \} _{0 \leq i \leq n}$ is a collection of objects of $\operatorname{\mathcal{C}}$ and $f_{I} \in \operatorname{Map}_{\operatorname{\mathcal{C}}}( X_{i_0}, X_{i_ k} )_{k-1}$ is defined for every subset $I = \{ i_ k > i_{k-1} > \cdots > i_0 \} \subseteq [n]$ for which $k > 0$ and $[n] \neq I \neq [n] \setminus \{ j\}$, satisfying the identity

2.25
\begin{eqnarray} \label{equation:checking-dg-nerve-is-infty-category} \partial f_{I} = \sum _{0 < a < k} (-1)^{a} f_{ I \setminus \{ i_ a \} } + (-1)^{k(a+1)} f_{ \{ i_ k > \cdots > i_ a \} } \circ f_{ \{ i_{a} > \cdots > i_0 \} }. \end{eqnarray}

We wish to show that $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. To give such an extension, we must supply chains $f_{[n]} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_ n)_{n-1}$ and $f_{[n] \setminus \{ j\} } \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_ n)_{n-2}$ which satisfy (2.25) in the cases $I = [n]$ and $I = [n] \setminus \{ j\}$. We claim that there is a unique such extension which also satisfies $f_{[n]} = 0$. Applying (2.25) in the case $I = [n]$, we deduce that $f_{[n] \setminus \{ j\} }$ is necessarily given by

$(-1)^{j+1} f_{ [n] \setminus \{ j\} } = \sum _{ 0 < b < n} (-1)^{n(b+1)} (f_{ \{ n > \cdots > b \} } \circ f_{ \{ b > \cdots > 0\} }) + \sum _{ 0 < b < n, b \neq j} (-1)^{b} f_{[n] \setminus \{ b\} }.$

To complete the proof, it will suffice to verify that this prescription also satisfies (2.25) in the case $I = [n] \setminus \{ j\}$. In what follows, for $0 \leq a < b \leq n$, let us write $[ba]$ for the set $\{ b > b-1 > \cdots > a \}$. We now compute

\begin{eqnarray*} (-1)^{j+1} \partial f_{[n] \setminus \{ j\} } & = & \sum _{ 0 < b < n} (-1)^{n(b+1)} \partial (f_{ [nb]} f_{[b]}) + \sum _{ 0 < b < n, b \neq j} (-1)^{b} \partial f_{[n] \setminus \{ b\} } \\ & = & \sum _{0 < b < n} (-1)^{n(b+1)} (\partial f_{[nb]}) f_{[b]} - \sum _{0 < b < n} (-1)^{(n+1)b} f_{[nb]} (\partial f_{[b]}) \\ & & + \sum _{ 0 < b < n, b \neq j} (-1)^{b} \partial f_{ [n] \setminus \{ b\} } \\ & = & \sum _{ 0< b < c < n} (-1)^{nb+n+c-b} (f_{[nb] \setminus \{ c\} } f_{[b]}) + \sum _{ 0 < b < c < n} (-1)^{(n-b)c} f_{ [nc] } f_{[cb]} f_{[b] } \\ & & - \sum _{ 0 < a < b < n } (-1)^{nb+b-a} ( f_{[nb]} f_{ [b] \setminus \{ a\} } ) - \sum _{ 0 < a < b < n } (-1)^{(n-a)b} (f_{[nb]} f_{[ba]} f_{[a] } ) \\ & & + \sum _{ 0 < a < b < n, b \neq j} (-1)^{b+a} f_{ [n] \setminus \{ a,b\} } - \sum _{ 0 < a < b < n, b \neq j} (-1)^{na+n+b-a} f_{[na] \setminus \{ b\} } f_{[a]} \\ & & - \sum _{ 0 < b < c < n, b \neq j} (-1)^{b+c} f_{ [n] \setminus \{ b,c\} } + \sum _{ 0 < b < c < n, b \neq j} (-1)^{nc+c-b} f_{ [nc] } f_{ [c] \setminus \{ b\} } \end{eqnarray*}

Here the second and fourth terms cancel, the sixth term cancels with first except for those summands with $c=j$, the eighth term cancels with the third except for those summands with $a = j$, and the fifth term cancels the seventh except for those terms with $a = j$ and $c = j$, respectively. After multiplying by $(-1)^{j+1}$, we can rewrite this identity as

\begin{eqnarray*} \partial f_{[n] \setminus \{ j\} } & = & \sum _{ 0 < b < j } (-1)^{(n-1)(b+1)} (f_{ [nb] \setminus \{ j \} } \circ f_{ [b] }) + \sum _{ j < b < n} (-1)^{(n-1)b} (f_{[nb]} \circ f_{[b] \setminus \{ j\} } ) \\ & & + \sum _{ j < b < n} (-1)^{b-1} f_{[n] \setminus \{ b,j\} } + \sum _{ 0 < b < j } (-1)^{b} f_{ [n] \setminus \{ b,j \} }, \end{eqnarray*}

which recovers equation (2.25) in the case $I = [n] \setminus \{ j\}$. $\square$