2.4.3 The Homotopy Coherent Nerve
Let $\operatorname{Top}$ denote the category of topological spaces and let $\operatorname{N}_{\bullet }(\operatorname{Top})$ denote its nerve (Construction 1.3.1.1). Then $\operatorname{N}_{\bullet }(\operatorname{Top})$ is a simplicial set whose $2$-simplices can be identified with diagrams of topological spaces $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ & X_{1} \ar [dr]^{ f_{21} } & \\ X_{0} \ar [ur]^{ f_{10} } \ar [rr]_{ f_{20} } & & X_{2} } \]
which commute in the sense that $f_{21} \circ f_{10}$ is equal to $f_{20}$. In the study of algebraic topology, one often encounters diagrams which commute in the weaker sense that the composition $f_{21} \circ f_{10}$ homotopic to $f_{20}$. By definition, this means that there exists a continuous function $h: [0,1] \times X_{0} \rightarrow X_{2}$ which satisfies the boundary conditions
\[ h|_{ \{ 0\} \times X_0} = f_{21} \circ f_{10} \quad \quad h|_{ \{ 1\} \times X_0 } = f_{20}. \]
In this case, we say that the function $h$ is a homotopy from $f_{21} \circ f_{10}$ to $f_{20}$, and that $h$ is a witness to the homotopy commutativity of the diagram $\sigma $. In this section, we will introduce an enlargement $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Top})$ of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{Top})$, whose $2$-simplices are given by pairs $(\sigma , h )$ where $\sigma $ is a (possibly noncommutative) diagram as above, and $h$ is a witness to the homotopy commutativity of $\sigma $. This is a special case of a general construction (Definition 2.4.3.5) which can be applied to any simplicial category.
Notation 2.4.3.1 (Simplicial Path Categories). Let $(Q, \leq )$ be a partially ordered set, and let $\operatorname{Path}_{(2)}[Q]$ denote the path $2$-category of $Q$ (Construction 2.3.5.1). We let $\operatorname{Path}[Q]_{\bullet }$ denote the simplicial category obtained from the strict $2$-category $\operatorname{Path}_{(2)}[Q]$ by applying the construction of Example 2.4.2.8. More concretely, we can describe the simplicial category $\operatorname{Path}[Q]_{\bullet }$ as follows:
The objects of $\operatorname{Path}[Q]_{\bullet }$ are the elements of the partially ordered set $Q$.
If $x$ and $y$ are elements of $Q = \operatorname{Ob}(\operatorname{Path}[Q]_{\bullet } )$, then $\operatorname{Hom}_{\operatorname{Path}[Q]}( x, y)_{\bullet }$ is the nerve of the partially ordered set of finite linearly ordered subsets $\{ x = x_0 < x_1 < \cdots < x_ m = y \} \subseteq Q$ with least element $x$ and largest element $y$, ordered by reverse inclusion.
For each element $x \in Q = \operatorname{Ob}(\operatorname{Path}[Q]_{\bullet } )$, the identity morphism $\operatorname{id}_{x}$ is the singleton $\{ x\} \in \operatorname{Hom}_{\operatorname{Path}[Q]}(x,x)_0$.
For $x,y,z \in Q = \operatorname{Ob}(\operatorname{Path}[Q] )$, the composition law
\[ \operatorname{Hom}_{\operatorname{Path}[Q]}(y,z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{Path}[Q]}( x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Path}[Q]}(x,z)_{\bullet } \]
is given on vertices by the construction $(S, T) \mapsto S \cup T$.
In the special case where $Q = [n] = \{ 0 < 1 < \cdots < n \} $, we denote the simplicial category $\operatorname{Path}[Q]_{\bullet }$ by $\operatorname{Path}[n]_{\bullet }$.
Definition 2.4.3.5 (The Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the simplicial set given by the construction
\[ ([n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}) \mapsto \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[n]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) = \{ \text{Simplicial functors $\operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$} \} . \]
We will refer to $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ as the homotopy coherent nerve of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.
Example 2.4.3.9 (Vertices and Edges of the Homotopy Coherent Nerve). In the cases $Q = [0]$ and $Q = [1]$, the map $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ is an equivalence of simplicial categories (since a path in $Q$ is uniquely determined by its endpoints). It follows that for every simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, the comparison map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ of Remark 2.4.3.8 is bijective on vertices and edges. In particular:
Vertices of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with objects $X$ of the underlying category $\operatorname{\mathcal{C}}$.
Edges of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be identified with morphisms $f: X \rightarrow Y$ of the underlying category $\operatorname{\mathcal{C}}$.
The face operators $d^{1}_{0}, d^{1}_{1}: \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carry a morphism $f: X \rightarrow Y$ to its target $Y = d^{1}_0(f)$ and source $f = d^{1}_1(f)$, respectively.
The degeneracy operator $s^{0}_0: \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carries an object $X \in \operatorname{\mathcal{C}}$ to the identity morphism $\operatorname{id}_{X}: X \rightarrow X$.
Example 2.4.3.10 ($2$-Simplices of the Homotopy Coherent Nerve). Let $Q = \{ x_0 < x_1 < x_2 \} $ be a linearly ordered set with three elements. Then the map $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ is not an equivalence of simplicial categories. In the underlying category $\operatorname{Path}[Q]$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ & x_1 \ar [dr]^{ \{ x_1 < x_2 \} } & \\ x_0 \ar [ur]^{ \{ x_0 < x_1 \} } \ar [rr]_{ \{ x_0 < x_2 \} } & & x_2 } \]
does not commute: the composition of the diagonal maps is the path $\{ x_0 < x_1 < x_2 \} $. However, it commutes in a weak sense: there is an edge of the simplicial set $\operatorname{Hom}_{\operatorname{Path}[Q]}(x_0,x_2)_{\bullet }$ having source $\{ x_0 < x_1 < x_2 \} $ and target $\{ x_0 < x_2 \} $. It follows that for any simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, a choice of $2$-simplex
\[ \sigma \in \operatorname{N}_{2}^{\operatorname{hc}}(\operatorname{\mathcal{C}}) = \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[2]_{\bullet } , \operatorname{\mathcal{C}}_{\bullet }) \simeq \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[Q]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) \]
determines a (possibly non-commutative) diagram $\sigma _0$:
\[ \xymatrix@R =50pt@C=50pt{ & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]^{ f_{20} } & & X_2, } \]
in $\operatorname{\mathcal{C}}$, together with a homotopy $h$ from $f_{21} \circ f_{10}$ to $f_{20}$ (in the sense of Definition 2.4.1.6). Conversely, every choice of homotopy from $f_{21} \circ f_{10}$ to $f_{20}$ determines a unique $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ (see Proposition 2.4.6.10).
Example 2.4.3.11 (Comparison with the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a strict $2$-category and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.8). For any partially ordered set $Q$, Remark 2.4.2.9 and Theorem 2.3.5.6 supply bijections
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[Q]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) & \simeq & \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{Str}}}( \operatorname{Path}_{(2)}[Q], \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{ULax}}}( Q, \operatorname{\mathcal{C}}). \end{eqnarray*}
Restricting to partially ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is the homotopy coherent nerve of Definition 2.4.3.5 and $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is the Duskin nerve of Construction 2.3.1.1.
Example 2.4.3.12 (The Case of an Ordinary Category). Let $\operatorname{\mathcal{C}}$ be an ordinary category, regarded as a constant simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ via the construction of Example 2.4.2.4. Combining Examples 2.3.1.3 and Examples 2.4.3.11, we obtain isomorphisms
\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{\operatorname{\mathcal{C}}} ). \]
Unwinding the definitions, we see that the composite isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{hc}}_{\bullet }( \underline{\operatorname{\mathcal{C}}})$ is the comparison map of Remark 2.4.3.8. In other words, when restricted to constant simplicial categories, the homotopy coherent nerve of Definition 2.4.3.5 reduces to the classical nerve of Construction 1.3.1.1.