Kerodon

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

2.5.9 Comparison with the Homotopy Coherent Nerve

Throughout this section, we maintain the notational convention of §2.5.8, denoting the tensor product of chain complexes $X_{\ast }$ and $Y_{\ast }$ by $X_{\ast } \boxtimes Y_{\ast }$. According to Proposition 2.5.8.7, the Alexander-Whitney homomorphisms

\[ \mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]

determine a colax monoidal structure on the normalized Moore complex functor $\mathrm{N}_{\ast }: \operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$. Applying Remark 2.1.5.12, we deduce that the right adjoint functor $\mathrm{K}: \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{ Ab }_{\Delta }$ inherits the structure of a lax monoidal functor. Composing with the (lax monoidal) forgetful functor $\operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Set_{\Delta }}$, we obtain the following:

Proposition 2.5.9.1. The functor $\mathrm{K}: \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Set_{\Delta }}$ admits a lax monoidal structure, which associates to each pair of chain complexes $X_{\ast }$ and $Y_{\ast }$ a map of simplicial sets

\[ \mu _{X_{\ast }, Y_{\ast }}: \mathrm{K}( X_{\ast } ) \times \mathrm{K}(Y_{\ast } ) \rightarrow \mathrm{K}( X_{\ast } \boxtimes Y_{\ast } ) \]

which can be described concretely as follows:

  • Let $\sigma $ and $\tau $ be $n$-simplices of $\mathrm{K}(X_{\ast })$ and $\mathrm{K}(Y_{\ast })$, respectively, which we identify with chain maps

    \[ \sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow X_{\ast } \quad \quad \tau : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow Y_{\ast }. \]

    Then $\mu _{X_{\ast }, Y_{\ast }}( \sigma , \tau ) \in \mathrm{K}_{n}( X_{\ast } \boxtimes Y_{\ast } )$ is the composite map

    \[ \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{N}_{\ast }( \Delta ^ n \times \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow {\sigma \boxtimes \tau } X_{\ast } \boxtimes Y_{\ast }. \]

Applying the general construction described in Remark 2.1.7.4 to the lax monoidal functor $\mathrm{K}: \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Set_{\Delta }}$, we obtain the following:

Construction 2.5.9.2. Let $\operatorname{\mathcal{C}}$ be a differential graded category. We define a simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\Delta }_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet }$ is the generalized Eilenberg-MacLane space $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$. More concretely, the $n$-simplices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet }$ are chain maps $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}^{\Delta }_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$ and every nonnegative integer $n \geq 0$, the composition law

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(Y,Z)_{n} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}( X,Y)_{n} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}( X,Z)_{n} \]

    carries a pair $(\sigma , \tau )$ to the $n$-simplex of $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } )$ given by the composite map

    \[ \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{N}_{\ast }( \Delta ^ n \times \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow {\sigma \boxtimes \tau } X_{\ast } \boxtimes Y_{\ast }. \]

We will refer to $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ as the underlying simplicial category of the differential graded category $\operatorname{\mathcal{C}}$.

Remark 2.5.9.3. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{\mathcal{C}}^{\circ }$ denote its underlying category (in the sense of Construction 2.5.2.4). Then $\operatorname{\mathcal{C}}^{\circ }$ is isomorphic to the underlying ordinary category $\operatorname{\mathcal{C}}^{\Delta }_{0}$ of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }$ (in the sense of Example 2.4.1.4). Both of these categories can be described concretely as follows:

  • The objects of $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_0$ are the objects of $\operatorname{\mathcal{C}}$.

  • For objects $X,Y \in \operatorname{\mathcal{C}}$, the morphisms from $X$ to $Y$ in the category $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_0$ are given by $0$-cycles in the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.

Remark 2.5.9.4. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then the underlying simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ is locally Kan (Definition 2.4.1.8). This follows from the observation that each of the simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\Delta }}(X,Y)_{\bullet } = \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ has the structure of a simplicial abelian group, and is therefore automatically a Kan complex (Proposition 1.1.9.9).

Remark 2.5.9.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $f,g: X \rightarrow Y$ be morphisms from $X$ to $Y$ in the underlying category $\operatorname{\mathcal{C}}^{\circ }$ (that is, $0$-cycles of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$). Then $f$ and $g$ are homotopic as morphisms of the differential graded category $\operatorname{\mathcal{C}}$ (in the sense of Definition 2.5.4.1) if and only if they are homotopic as morphisms of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ (Remark 2.4.1.9); see Example 2.5.6.6. It follows that the isomorphism of underlying categories $\operatorname{\mathcal{C}}^{\circ } \simeq \operatorname{\mathcal{C}}^{\Delta }_{0}$ of Remark 2.5.9.3 induces an isomorphism from the homotopy $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (given by Construction 2.5.4.6) to the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\Delta }$ (given by Construction 2.4.6.1).

Our goal in this section is to establish a refinement of Remark 2.5.9.5. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ denote the underlying simplicial category. Then $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$ is locally Kan (Remark 2.5.9.4), so the homotopy coherent nerve $\operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ is an $\infty $-category (Theorem 2.4.5.1). Similarly, the differential graded nerve $\operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.5.3.10). The $\infty $-categories $\operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ and $\operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ are generally not isomorphic as simplicial sets. However, we will construct a comparison map $\operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ and show that it is a trivial Kan fibration (and therefore an equivalence of $\infty $-categories; see Proposition ). We begin with some auxiliary remarks.

Construction 2.5.9.6 (The Fundamental Chain of a Cube). Let $I$ be a finite set of cardinality $n$, and let $\operatorname{\raise {0.1ex}{\square }}^{I} = \prod _{i \in I} \Delta ^1$ denote the associated cube (Notation 2.4.5.2), which we will identify with the nerve of the partially ordered set of all subsets of $I$. Using this identification, we obtain a bijective correspondence

\[ \{ \text{Linear orderings of $I$} \} \simeq \{ \text{Nondegenerate $n$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{I}$} \} , \]

which carries a linear ordering $\{ i_1 < i_2 < \cdots < i_ n \} $ to the chain of subsets

\[ \emptyset \subset \{ i_1 \} \subset \{ i_1, i_2 \} \subset \cdots \subset \{ i_1, \ldots , i_{n-1} \} \subset I. \]

In particular, the symmetric group $\Sigma _{I}$ of permutations of $I$ acts simply transitively on the set of nondegenerate $n$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{I}$.

Fix a linear ordering of $I$, corresponding to a nondegenerate $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\raise {0.1ex}{\square }}^{I}$. We let $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ denote the alternating sum $\sum _{\pi \in \Sigma _{I} } (-1)^{\pi } \pi (\sigma )$, which we regard as an $n$-chain of the normalized chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$. We will refer to $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ as the fundamental chain of the cube $\operatorname{\raise {0.1ex}{\square }}^{I}$. We will be particularly interested in the special case where $I$ is the set $\{ 1, 2, \cdots , n \} $, endowed with its usual ordering; in this case, we denote the cube $\operatorname{\raise {0.1ex}{\square }}^{I}$ by $\operatorname{\raise {0.1ex}{\square }}^{n}$ and its fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ by $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$.

Remark 2.5.9.7. Let $n$ be a nonnegative integer. Then the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ of Construction 2.5.9.6 is given by the iterated shuffle product

\[ [ \Delta ^1] \triangledown [ \Delta ^1 ] \triangledown \cdots \triangledown [\Delta ^1] \in \mathrm{N}_{n}( \Delta ^1 \times \Delta ^1 \times \cdots \times \Delta ^1; \operatorname{\mathbf{Z}}) \simeq \mathrm{N}_{n}( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \]

(see §2.5.7); here $[ \Delta ^1 ]$ denotes the generator of the group $\mathrm{N}_{1}( \Delta ^1; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}$ (which is also the fundamental chain of the $1$-dimensional cube $\operatorname{\raise {0.1ex}{\square }}^1$).

Warning 2.5.9.8. The simplicial set $\operatorname{\raise {0.1ex}{\square }}^{I}$ and its normalized chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ depend only on the choice of the finite set $I$. However, the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ of Construction 2.5.9.6 is a priori ambiguous up to a sign. One can resolve this ambiguity by choosing a linear ordering on the set $I$ (as in Construction 2.5.9.6), which will be sufficient for our purposes in this section. However, less is needed: one needs only an orientation on the set $I$ (or equivalently an orientation of the topological manifold-with-boundary $| \operatorname{\raise {0.1ex}{\square }}^{I} | \simeq [0,1]^{I}$).

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 $Y = \sigma (n+1)$, so that $\sigma $ induces a map of simplicial sets

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

which we can identify with a chain map $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. This map carries the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ to an $n$-chain of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ carries the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ 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 $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n} ])$.

Proposition 2.5.9.10. Let $\operatorname{\mathcal{C}}$ be a differential graded category. Then there is a unique functor of $\infty $-categories $\mathfrak {Z}: \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ with the following properties:

  • On $0$-simplices the functor $\mathfrak {Z}$ is the identity: that is, it carries each object of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }$ to the corresponding object of the differential graded category $\operatorname{\mathcal{C}}$.

  • Let $n \geq 0$ and let $\sigma $ be an $(n+1)$-simplex of $\operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$. Set $X = \sigma (0)$ and $Y = \sigma (n+1)$. Then the value of $\mathfrak {Z}(\sigma )$ on $\{ n+1 > n > \cdots > 0 \} $ is the chain $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n} ] ) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ of Notation 2.5.9.9.

The proof of Proposition 2.5.9.10 will require an elementary property of Construction 2.5.9.6.

Notation 2.5.9.11. Let $I$ be a finite linearly ordered set of cardinality $n > 0$ and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the corresponding simplicial cube. For each element $i \in I$, the linear ordering on $I$ restricts to linear ordering on the subset $I \setminus \{ i\} $, which determines a fundamental chain

\[ [ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ; \operatorname{\mathbf{Z}}). \]

We will write $[ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ for the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]$ under the inclusion of simplicial sets

\[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \hookrightarrow \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \operatorname{\raise {0.1ex}{\square }}^{I}. \]

Similarly, we write $[ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] \in \mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$ for the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]$ under the inclusion

\[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \hookrightarrow \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \simeq \operatorname{\raise {0.1ex}{\square }}^{I}. \]

Lemma 2.5.9.12. Let $n$ be a nonnegative integer and let $I$ denote the linearly ordered set $\{ 1 < 2 < \cdots < n \} $. Then we have an equality

\[ \partial [ \operatorname{\raise {0.1ex}{\square }}^{I} ] = \sum _{i=1}^{n} (-1)^{i} ([ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{ I \setminus \{ i\} } ] - [ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{ I \setminus \{ i\} } ] ) \]

in the abelian group $\mathrm{N}_{n-1}( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}})$.

Remark 2.5.9.13. Lemma 2.5.9.12 is a homological incarnation of the following topological assertion: the geometric realization $| \operatorname{\raise {0.1ex}{\square }}^{I} | \simeq [0,1]^{I}$ is a manifold, whose boundary can be written as a union of the faces $\{ 0\} \times [0,1]^{I \setminus \{ i\} }$ and $\{ 1\} \times [0,1]^{I \setminus \{ i\} }$.

Proof of Lemma 2.5.9.12. Using the description of $[ \operatorname{\raise {0.1ex}{\square }}^{I} ]$ as a shuffle product (Remark 2.5.9.7) and the fact that the shuffle product satisfies the Leibniz rule (Proposition 2.5.7.10), we compute

\begin{eqnarray*} \partial [ \operatorname{\raise {0.1ex}{\square }}^{I} ] & = & \partial ( [ \Delta ^1 ] \triangledown \cdots \triangledown [ \Delta ^1] ) \\ & = & \sum _{i=1}^{n} (-1)^{i-1} [ \operatorname{\raise {0.1ex}{\square }}^{ i-1} ] \triangledown \partial ( [ \Delta ^1] ) \triangledown [\operatorname{\raise {0.1ex}{\square }}^{n-i}] \\ & = & \sum _{i=1}^{n} (-1)^{i} [ \operatorname{\raise {0.1ex}{\square }}^{ i-1} ] \triangledown (d_1[ \Delta ^1] - d_0[ \Delta ^1]) \triangledown [\operatorname{\raise {0.1ex}{\square }}^{n-i}] \\ & = & \sum _{i=1}^{n} (-1)^{i} ([ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{ I \setminus \{ i\} } ] - [ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{ I \setminus \{ i\} } ] ). \end{eqnarray*}
$\square$

Remark 2.5.9.14. Let $n$ be a nonnegative integer. It follows from Lemma 2.5.9.12 that the boundary $\partial [ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ belongs to the subcomplex $\mathrm{N}_{\ast }( \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \subset \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$. In other words, the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ in the relative chain complex

\[ \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}, \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) = \mathrm{N}_{\ast }(\operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) / \mathrm{N}_{\ast }( \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \]

is a cycle. In fact, one can be more precise: the construction $1 \mapsto [ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ determines a quasi-isomorphism of chain complexes $u_ n: \operatorname{\mathbf{Z}}[n] \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}, \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$. To prove this, we proceed by induction on $n$: the case $n=0$ is trivial, and the inductive step follows by identifying $u$ with the composition

\begin{eqnarray*} \operatorname{\mathbf{Z}}[n] & \simeq & \operatorname{\mathbf{Z}}[1] \boxtimes \operatorname{\mathbf{Z}}[n-1] \\ & \xrightarrow {\operatorname{id}\boxtimes u_{n-1}} & \operatorname{\mathbf{Z}}[1] \boxtimes \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n-1}, \operatorname{\partial \raise {0.1ex}{\square }}^{n-1}; \operatorname{\mathbf{Z}}) \\ & \simeq & \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{1}, \operatorname{\partial \raise {0.1ex}{\square }}^{1}; \operatorname{\mathbf{Z}}) \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n-1}, \operatorname{\partial \raise {0.1ex}{\square }}^{n-1}; \operatorname{\mathbf{Z}}) \\ & \xrightarrow { \mathrm{EZ} }& \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}, \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \end{eqnarray*}

where $\mathrm{EZ}$ denotes the Eilenberg-Zilber map of Variant 2.5.7.17 (which is a quasi-isomorphism by virtue of Theorem 2.5.7.14. Note that this property characterizes the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n}]$ up to sign (since the quotient map $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}, \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$ is an isomorphism in degree $n$).

Lemma 2.5.9.15. Let $I$ be a finite linearly ordered set which is a union of disjoint subsets $I_{-}, I_{+} \subseteq I$. Then the Alexander-Whitney homomorphism $\mathrm{AW}: \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; Z) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}}; \operatorname{\mathbf{Z}}) \times \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}})$ satisfies

\[ \mathrm{AW}( [ \operatorname{\raise {0.1ex}{\square }}^{I} ]) = (-1)^{d} [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \boxtimes [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}} ], \]

where $d$ denotes the cardinality of the set $\{ (i,j) \in I_{-} \times I_{+}: i > j \} $.

Proof. Using Remark 2.5.9.7 (and the graded-commutativity of the shuffle product; see Proposition 2.5.7.10), we observe that the shuffle product map

\[ \triangledown : \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}}; \operatorname{\mathbf{Z}}) \times \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I_{-}} \times \operatorname{\raise {0.1ex}{\square }}^{I_{+}}; \operatorname{\mathbf{Z}}) \simeq \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{I}; \operatorname{\mathbf{Z}}) \]

satisfies $[ \operatorname{\raise {0.1ex}{\square }}^{I} ] = (-1)^{d} [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \triangledown [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}}] $. Applying the Alexander-Whitney homomorphism and invoking Proposition 2.5.8.9, we obtain the identity

\[ \mathrm{AW}( [ \operatorname{\raise {0.1ex}{\square }}^{I} ] ) = (-1)^{d} \mathrm{AW}([ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \triangledown [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}}] ) = [ \operatorname{\raise {0.1ex}{\square }}^{I_{-}} ] \boxtimes [ \operatorname{\raise {0.1ex}{\square }}^{I_{+}} ]. \]
$\square$

Proof of Proposition 2.5.9.10. Fix an integer $n \geq 0$, and let $\sigma $ be an $(n+1)$-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 let us identify the simplicial set $\operatorname{Hom}_{ \operatorname{Path}[n+1]}( 0, n+1)_{\bullet }$ with the cube $\operatorname{\raise {0.1ex}{\square }}^{n}$. By virtue of Remark 2.5.3.9, it will suffice to verify the following three assertions:

$(a)$

If $n=0$ and $\sigma $ is the degenerate edge of $\operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ determined by the object $X \in \operatorname{\mathcal{C}}$, then $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n}] ) = \operatorname{id}_ X$.

$(b)$

If $n > 0$ and $\sigma $ is degenerate, then $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n}] ) = 0$.

$(c)$

If $n \geq 0$, then

\begin{eqnarray*} \partial \sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n}] ) = \sum _{i=1}^{n} (-1)^{i} d_ i(\sigma )( [ \operatorname{\raise {0.1ex}{\square }}^{n-1} ]) + (-1)^{(n+1)(i+1)} \sigma _{\geq i}( [\operatorname{\raise {0.1ex}{\square }}^{n-i} ] ) \circ \sigma _{\leq i}( [ \operatorname{\raise {0.1ex}{\square }}^{i-1}] ) ); \end{eqnarray*}

Assertion $(a)$ is immediate from the definition. To prove $(b)$, we observe that $\sigma $ determines a map of simplicial sets

\[ \operatorname{Hom}_{\operatorname{Path}[n+1]}( 0, n+1)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\Delta } }( X,Y )_{\bullet } \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)), \]

which we can identify with a chain map $u: \mathrm{N}_{\ast }( \operatorname{Hom}_{\operatorname{Path}[n+1]}( 0, n+1); \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. If $\sigma $ is degenerate, then (as a simplicial functor) it factors as a composition

\[ \operatorname{Path}[n+1]_{\bullet } \rightarrow \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}^{\Delta }_{\bullet }, \]

where $\rho $ is a simplicial functor satisfying $\rho (0) = 0$ and $\rho (n+1) = n$. For $n > 0$, it follows that the chain map $u$ factors through the complex $\mathrm{N}_{\ast }( \operatorname{Hom}_{ \operatorname{Path}[n]}( 0, n); \operatorname{\mathbf{Z}}) \simeq \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n-1}; \operatorname{\mathbf{Z}})$. Since $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ is a simplicial set of dimension $\leq n-1$, the chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n-1}; \operatorname{\mathbf{Z}})$ vanishes in degrees $\geq n$ (see Example 2.5.5.13). In particular, the map $u$ vanishes in degree $n$, so that $\sigma ( [ \operatorname{\raise {0.1ex}{\square }}^{n} ] ) = 0$.

We now prove $(c)$. Set $I = \{ 1, 2, \ldots , n \} $. Using Lemma 2.5.9.12, we obtain the identity

\[ \partial \sigma ([ \operatorname{\raise {0.1ex}{\square }}^{I} ]) = \sum _{i=1}^{n} (-1)^{i} (\sigma ([ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]) - \sigma ([ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} }] ). \]

It will therefore suffice to show that, for each $1 \leq i \leq n$, we have equalities

\[ \sigma ( [ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] ) = d_ i(\sigma )( [ \operatorname{\raise {0.1ex}{\square }}^{n-1} ] ) \]

\[ \sigma ([ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} })] ) = (-1)^{(i-1)(n-i)} \sigma _{\geq i}( [\operatorname{\raise {0.1ex}{\square }}^{n-i} ] ) \circ \sigma _{\leq i}( [ \operatorname{\raise {0.1ex}{\square }}^{i-1}] ) ) \]

in the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n-1}$. The first of these identities follows immediately from the definition of $d_ i(\sigma )$. To prove the second, we note that the inclusion $\{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \hookrightarrow \operatorname{\raise {0.1ex}{\square }}^{I} \simeq \operatorname{Hom}_{ \operatorname{Path}[n+1]}(0,n+1)_{\bullet }$ factors as a composition

\begin{eqnarray*} \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } & \simeq & \operatorname{\raise {0.1ex}{\square }}^{n-i} \times \operatorname{\raise {0.1ex}{\square }}^{i-1} \\ & \simeq & \operatorname{Hom}_{\operatorname{Path}[n+1]}( i, n+1)_{\bullet } \times \operatorname{Hom}_{ \operatorname{Path}[n+1]}( 0, i)_{\bullet } \\ & \xrightarrow {\circ } & \operatorname{Hom}_{ \operatorname{Path}[n+1]}( 0, n+1)_{\bullet }. \end{eqnarray*}

Set $Z = \sigma (i)$. Using the fact that $\sigma $ is a simplicial functor (and the definition of the simplicial category $\operatorname{\mathcal{C}}^{\Delta }_{\bullet }$), we see that $\sigma ([ \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] )$ is the image of the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]$ under the composite map

\begin{eqnarray*} \mathrm{N}_{\ast }( [ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ]; \operatorname{\mathbf{Z}}) & \xrightarrow { \mathrm{AW} } & \mathrm{N}_{\ast }( [ \operatorname{\raise {0.1ex}{\square }}^{n-i} ]; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( [ \operatorname{\raise {0.1ex}{\square }}^{i-1} ]; \operatorname{\mathbf{Z}}) \\ & \xrightarrow { \sigma _{\geq i} \boxtimes \sigma _{\leq i} } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z,Y)_{\ast } \boxtimes \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } \\ & \xrightarrow {\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }. \end{eqnarray*}

The desired result now follows from the identity $\mathrm{AW}( [ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ] ) = (-1)^{(i-1)(n-i)} [ \operatorname{\raise {0.1ex}{\square }}^{n-i} ] \boxtimes [ \operatorname{\raise {0.1ex}{\square }}^{i-1} ]$ supplied by Lemma 2.5.9.15. $\square$

Exercise 2.5.9.16. Let $\operatorname{\mathcal{C}}$ be a differential graded category, and let $\mathfrak {Z}: \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ be the functor of $\infty $-categories supplied by Proposition 2.5.9.10. Show that $\mathfrak {Z}$ is bijective on simplices of dimension $n \leq 2$ (for the case $n=2$, this is essentially the content of Remark 2.5.4.4).

The functor $\mathfrak {Z}: \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ is generally not bijective on simplices of dimension $n \geq 3$. Nevertheless, we have the following:

Theorem 2.5.9.17. Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\mathfrak {Z}: \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \rightarrow \operatorname{N}^{\operatorname{dg}}_{\bullet }(\operatorname{\mathcal{C}})$ be the functor of $\infty $-categories supplied by Proposition 2.5.9.10. Then $\mathfrak {Z}$ is a trivial Kan fibration of simplicial sets.

Proof. Fix an integer $n \geq 0$ and a diagram of simplicial sets

\[ \xymatrix { \partial \Delta ^{n+1} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } ) \ar [d]^{ \mathfrak {Z} } \\ \Delta ^{n+1} \ar [r]^-{\tau } \ar@ {-->}[ur]^{\sigma } & \operatorname{N}^{\operatorname{dg}}_{\bullet }( \operatorname{\mathcal{C}}); } \]

we wish to show that the map $\sigma _0$ admits an extension $\sigma : \Delta ^ n \rightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ as indicated, rendering the diagram commutative. Let us abuse notation by identifying $\sigma _0$ with a simplicial functor from $\operatorname{Path}[ \partial \Delta ^{n+1} ]_{\bullet }$ to $\operatorname{\mathcal{C}}^{\Delta }$. Set $X = \sigma _0(0)$ and $Y = \sigma _0(n+1)$, so that $\sigma _0$ determines a map of simplicial sets

\[ u_0: \operatorname{\partial \raise {0.1ex}{\square }}^{n} \simeq \operatorname{Hom}_{\operatorname{Path}[ \partial \Delta ^{n+1} ]}( 0, n+1) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\Delta }}( X, Y)_{\bullet } = \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \]

(see Proposition 2.4.6.11), which we will identify with a chain map $f_0: \mathrm{N}_{\ast }( \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$. By virtue of Corollary 2.4.6.12, choosing an extension of $\sigma _0$ to a map $\sigma : \Delta ^{n+1} \rightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ is equivalent to choosing an extension of $u_0$ to a map of simplicial sets $u: \operatorname{\raise {0.1ex}{\square }}^{n} \rightarrow \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) )$, or an extension of $f_0$ to a chain map $f: \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$.

Note that the boundary $\partial [ \operatorname{\raise {0.1ex}{\square }}^{n} ]$ belongs to the subcomplex $\mathrm{N}_{\ast }( \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \subset \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$ (see Lemma 2.5.9.12). Unwinding the definitions, we see that $\tau $ supplies a chain $z \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n}$ satisfying $\partial (z) = f_0( \partial [ \operatorname{\raise {0.1ex}{\square }}^{n}] ) \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{n-1}$. Let $M_{\ast }$ denote the subcomplex of $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$ generated by $\mathrm{N}_{\ast }( \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$ together with the fundamental chain $[ \operatorname{\raise {0.1ex}{\square }}^{n} ]$, so that $f_0$ extends uniquely to a chain map $f_1: M_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ satisfying $f_1( [ \operatorname{\raise {0.1ex}{\square }}^{n} ] ) = z$. Unwinding the definitions, we see that if $f: \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ is a map of chain complexes extending $f_0$, then the corresponding extension $\sigma : \Delta ^{n+1} \rightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )$ of $\sigma _0$ satisfies $\mathfrak {Z} \circ \sigma = \tau $ if and only if $f|_{ M_{\ast } } = f_1$. We will complete the proof by showing that $M_{\ast }$ is a direct summand of $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$ (so that any map $f_1: M_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ can be extended to $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}})$). To prove this, note that we have an exact sequence of chain complexes

\[ 0 \rightarrow \operatorname{\mathbf{Z}}[n] \xrightarrow { [ \operatorname{\raise {0.1ex}{\square }}^{n}] } \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^ n, \operatorname{\partial \raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) / M_{\ast } \rightarrow 0, \]

where the first map is a quasi-isomorphism (Variant 2.5.7.17). It follows that the chain complex $\mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) / M_{\ast }$ is acyclic and free in each degree, so that the exact sequence

\[ 0 \rightarrow M_{\ast } \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }( \operatorname{\raise {0.1ex}{\square }}^{n}; \operatorname{\mathbf{Z}}) / M_{\ast } \rightarrow 0 \]

splits by virtue of Proposition 2.5.1.10. $\square$