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2.5.3 The Differential Graded Nerve

We now explain how to associate to each differential graded category $\operatorname{\mathcal{C}}$ an $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$, which we will refer to as the differential graded nerve of $\operatorname{\mathcal{C}}$. We begin by describing the simplices of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

Construction 2.5.3.1. Let $\operatorname{\mathcal{C}}$ be a differential graded category. For $n \geq 0$, we let $\operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the collection of all ordered pairs of all ordered pairs $( \{ X_ i \} _{0 \leq i \leq n}, \{ f_ I \} )$, where:

  • Each $X_ i$ is an object of the differential graded category $\operatorname{\mathcal{C}}$.

  • For every subset $I = \{ i_ k > i_{k-1} > \cdots > i_0 \} \subseteq [n]$ having at least two elements, $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{i_{0}}, X_{i_ k} )_{k-1}$ which satisfies the identity

    \begin{eqnarray*} \partial f_{I} = \sum _{0 < a < k} (-1)^{a} f_{ I \setminus \{ i_ a \} } + (-1)^{k(a+1)} f_{ \{ i_ k > i_{m-1} > \cdots > i_ a \} } \circ f_{ \{ i_ a > \cdots > i_1 > i_0 \} } \end{eqnarray*}

Example 2.5.3.2 (Vertices of the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then $\operatorname{N}_{0}^{\operatorname{dg}}( \operatorname{\mathcal{C}})$ can be identified with the collection $\operatorname{Ob}(\operatorname{\mathcal{C}})$ of objects of $\operatorname{\mathcal{C}}$.

Example 2.5.3.3 (Edges of the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then $\operatorname{N}_{1}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ can be identified with the collection of all triples $(X_0, X_1, f)$ where $X_0$ and $X_1$ are objects of $\operatorname{\mathcal{C}}$ and $f$ is a $0$-cycle in the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_1)_{\bullet }$. In other words, $\operatorname{N}_{1}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is the collection of all morphisms in the the underlying category $\operatorname{\mathcal{C}}^{\circ }$ of Construction 2.5.2.4.

Example 2.5.3.4 ($2$-Simplices of the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then an element of $\operatorname{N}_{2}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ is given by the following data:

  • A triple of objects $X_{0}, X_1, X_2 \in \operatorname{Ob}(\operatorname{\mathcal{C}})$.

  • A triple of $0$-cycles

    \[ f_{10} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1)_{0} \quad \quad f_{20} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2)_{0} \quad \quad f_{21} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_1, X_2)_{0}. \]
  • A $1$-chain $f_{210} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2)_{1}$ satisfying the identity

    \[ \partial (f_{210}) = (f_{21} \circ f_{10}) - f_{20}. \]

Here the $1$-chain $f_{210}$ can be regarded as a witness to the assertion that that the $0$-cycles $f_{20}$ and $f_{21} \circ f_{10}$ are homologous: that is, they represent the same element of the homology group $\mathrm{H}_0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_2) )$. We can present this data graphically by the diagram

\[ \xymatrix@C =50pt@R=50pt{ & X_1 \ar [dr]^{f_{21}} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{f_{210}} & \\ X_0 \ar [ur]^{f_{10}} \ar [rr]_{f_{20}} & & X_2. } \]

We now explain how to organize the collection $\{ \operatorname{N}_{n}^{\operatorname{dg}}( \operatorname{\mathcal{C}}) \} _{n \geq 0}$ into a simplicial set.

Proposition 2.5.3.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Let $m, n \geq 0$ be nonnegative integers and let $\alpha : [n] \rightarrow [m]$ be a nondecreasing function. Then the construction

\[ ( \{ X_ i \} _{0 \leq i \leq m}, \{ f_{I} \} ) \mapsto ( \{ X_{ \alpha (j) } \} _{ 0 \leq j \leq n}, \{ g_{J} \} ), \]
\[ g_{J} = \begin{cases} f_{ \alpha (J) } & \text{ if } \alpha |_{J} \text{ is injective } \\ \operatorname{id}_{ X_ i } & \text{ if } J = \{ j_{1} > j_{0} \} \text{ with } \alpha (j_{1}) = i = \alpha (j_{0}) \\ 0 & \text{ otherwise. } \end{cases} \]

determines a map of sets $\alpha ^{\ast }: \operatorname{N}_{m}^{\operatorname{dg}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

Proof. Let $( \{ X_ i \} _{0 \leq i \leq m}, \{ f_ I \} )$ be an element of $\operatorname{N}_{m}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. For each subset $J \subseteq [n]$ with at least two elements, define $g_{J}$ as in the statement of Proposition 2.5.3.5. We wish to show that $( \{ X_{ \alpha (j) } \} _{ 0 \leq j \leq n}, \{ g_{J} \} )$ is an element of $\operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$. For this, we must show that for each subset

\[ J = \{ j_{k} > j_{k-1} > \cdots > j_1 > j_{0} \} \subseteq [n] \]

having at least two elements, we have an equality

2.22
\begin{eqnarray} \label{equation:functoriality-of-dg-nerve} \partial g_{J} = \sum _{1 < a < k} (-1)^{a} g_{J \setminus \{ j_ a \} } + (-1)^{k(a+1)} g_{ \{ j_{k} > j_{k-1} > \cdots > j_ a \} } \circ g_{ \{ j_ a > \cdots > j_1 > j_{0} \} }.\end{eqnarray}

We distinguish three cases:

  • Suppose that the restriction $\alpha |_{J}$ is injective. In this case, we can rewrite (2.22) as an equality

    \begin{eqnarray*} \partial f_{ \alpha (J)} = \sum _{0 < a < k} (-1)^{a} f_{\alpha (J) \setminus \{ \alpha (j_ a) \} } + (-1)^{k(a+1)} f_{ \{ \alpha (j_{k}) > \ldots > \alpha (j_ a) \} } \circ f_{ \{ \alpha (j_ a) > \cdots > \alpha (j_{0} )\} },\end{eqnarray*}

    which follows from our assumption that $( \{ X_ i \} _{0 \leq i \leq m}, \{ f_ I \} )$ is an element of $\operatorname{N}_{m}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

  • Suppose that $J = \{ j_{1} > j_{0} \} $ is a two-element set satisfying $\alpha (j_{1}) = i = \alpha (j_{0} )$ for some $0 \leq i \leq m$. In this case, we can rewrite $(\ref{equation:functoriality-of-dg-nerve})$ as an equality $\partial (\operatorname{id}_{ X_ i}) = 0$, which follows from Remark 2.5.2.2.

  • Suppose that $J = \{ j_{k} > j_{k-1} > \cdots > j_1 > j_{0} \} $ has at least three elements and that $\alpha |_{J}$ is not injective, so that $g_{J} = 0$. We now distinguish three (possibly overlapping) cases:

    • The map $\alpha $ is not injective because $\alpha (j_{1}) = i = \alpha ( j_{0} )$ for some $0 \leq i \leq m$. In this case, the expressions $g_{ J \setminus \{ j_ a \} }$ and $g_{ \{ j_{a} > \cdots > j_1 > j_{0} \} \} }$ vanish for $1 < a < k$. We can therefore rewrite (2.22) as an an equality

      \[ g_{J \setminus \{ j_1 \} } = g_{ \{ j_ k > \cdots > j_1 \} } \circ g_{ \{ j_{1} > j_{0} \} }, \]

      which follows from the identities $g_{ J \setminus \{ j_1 \} } = g_{ \{ j_ k > \cdots > j_1 \} }$ and $g_{ \{ j_1 > j_0 \} } = \operatorname{id}_{ X_ i }$.

    • The map $\alpha $ is not injective because $\alpha (j_{k} ) = i = \alpha ( j_{k-1} )$ for some $0 \leq i \leq m$. In this case, the expressions $g_{J \setminus \{ j_ a \} }$ and $g_{ \{ j_{k} > \cdots > j_ a \} }$ vanish for $0 < a < k-1$. We can therefore rewrite (2.22) as an an equality

      \[ g_{J \setminus \{ j_{k-1} \} } = g_{ \{ j_ k > j_{k-1} \} } \circ g_{ \{ j_{k-1} > \cdots > j_0 \} }, \]

      which follows from the identities $g_{J \setminus \{ j_{k-1} \} } = g_{ \{ j_{k-1} > \cdots > j_0 \} }$ and $g_{ \{ j_ k > j_{k-1} \} } = \operatorname{id}_{ X_ i }$.

    • The map $\alpha $ is not injective because we have $\alpha ( j_{b} ) = \alpha ( j_{b+1} )$ for some $0 < b < k-1$. In this case, the chains $g_{ J \setminus \{ j_{a} \} }$ vanish for $a \notin \{ b, b+1\} $, and the compositions $g_{ \{ j_ k > \cdots > j_ a \} } \circ g_{ \{ j_ a > \cdots > j_0 \} }$ vanish for all $0 < a < k$. We can therefore rewrite (2.22) as an an equality $g_{J \setminus \{ j_ b \} } = g_{ J \setminus \{ j_{b+1} \} }$, which is clear.

$\square$

Exercise 2.5.3.6. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Suppose we are given a pair of nondecreasing functions $\alpha : [k] \rightarrow [m]$ and $\beta : [m] \rightarrow [n]$. Show that the function $(\beta \circ \alpha )^{\ast }$ of Proposition 2.5.3.5 coincides with the composition $\alpha ^{\ast } \circ \beta ^{\ast }$.

Definition 2.5.3.7. Let $\operatorname{\mathcal{C}}$ be a differential graded category. We let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the simplicial set whose value on an object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ is the set $\operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ of Construction 2.5.3.1, and whose value on a nondecreasing function $\alpha : [n] \rightarrow [m]$ is the function $\alpha ^{\ast }: \operatorname{N}_{m}^{\operatorname{dg}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{n}^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ of Proposition 2.5.3.5. We will refer to $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ as the differential graded nerve of $\operatorname{\mathcal{C}}$.

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 $|I| > 2$.

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

    \begin{eqnarray*} \partial f(\sigma ) = \sum _{0 < a < k} (-1)^{a} f( d_ a \sigma ) + (-1)^{k(a+1)} f( \sigma _{\geq a}) \circ f( \sigma _{\leq a} ). \end{eqnarray*}

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\} $. $\square$

Remark 2.5.3.11. The theory of differential graded categories can be regarded as a special case of the more general theory of $A_{\infty }$-categories (see [MR1925734]). Definition 2.5.3.7 and Theorem 2.5.3.10 have been extended to the setting of $A_{\infty }$-categories by Faonte; we refer the reader to [MR3607208] for details.