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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 1.2.1.1). Then $\operatorname{N}_{\bullet }(\operatorname{Top})$ is a simplicial set whose $2$-simplices can be identified with diagrams of topological spaces $\sigma :$

\[ \xymatrix { & X_{1} \ar [dr]^{ f_{21} } & \\ X_{0} \ar [ur]^{ f_{10} } \ar [rr]^{ f_{20} } & & X_{2} } \]

which commute in the sense that $f_{20}$ is equal to the composition $f_{21} \circ f_{10}$. In the study of algebraic topology, one often encounters diagrams which commute in the weaker sense that $f_{20}$ is homotopic to the composition $f_{21} \circ f_{10}$. By definition, this means that there exists a continuous function $h: [0,1] \times X_{0} \rightarrow X_{2}$ which satisfies the boundary conditions

\[ h|_{ \{ 0\} \times X_0 } = f_{20} \quad \quad h|_{ \{ 1\} \times X_0} = f_{21} \circ f_{10}. \]

In this case, we say that the function $h$ is a homotopy from $f_{20}$ to $f_{21} \circ f_{10}$, 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 2.3.6.1). 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 2.4.2.6. 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 $y$.

  • 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 $\operatorname{Path}[n]_{\bullet }$.

Remark 2.4.3.2. Let $Q$ be a partially ordered set. The simplicial category $\operatorname{Path}[Q]_{\bullet }$ can be regarded as a “thickened version” of $Q$. For every pair of elements $x,y \in Q$, the simplicial set $\operatorname{Hom}_{\operatorname{Path}[Q]}(x,y)_{\bullet }$ is empty if $x \nleq y$, and weakly contractible (see ) if $x \leq y$ (since it is the nerve of a partially ordered set with a least element $\{ x,y\} $). In particular, there is a unique simplicial functor $\pi : \operatorname{Path}[Q]_{\bullet } \rightarrow Q$ which is the identity on objects (where we abuse notation by identifying $Q$ with the associated constant simplicial category of Example 2.4.2.2). The simplicial functor $\pi $ is a prototypical example of a weak equivalence in the setting of simplicial categories (see ).

Remark 2.4.3.3. A topologically enriched variant of $\operatorname{Path}[Q]_{\bullet }$ appears in the work of Leitch ([MR0365560]); see appendix B of [MR0353298] for a related construction.

Remark 2.4.3.4 (Relationship with Ordinary Path Categories). Let $Q$ be a partially ordered set and let $\mathrm{Gr}(Q)$ denote the associated directed graph, given concretely by

\[ \operatorname{Vert}( \mathrm{Gr}(Q) ) = Q \quad \quad \operatorname{Edge}( \mathrm{Gr}(Q) ) = \{ (x,y) \in Q: x < y \} . \]

Then the path category $\operatorname{Path}[ \mathrm{Gr}(Q) ]$ of Construction 1.2.6.1 is the underlying category of the simplicial category $\operatorname{Path}[Q]_{\bullet }$ of Notation 2.4.3.1 (see Remark 2.3.6.2). In other words, we can regard $\operatorname{Path}[Q]_{\bullet }$ as a simplicially enriched version of $\operatorname{Path}[ \mathrm{Gr}(Q) ]$. Beware that the simplicial enrichment is nontrivial in general: that is, the simplicial mapping sets $\operatorname{Hom}_{\operatorname{Path}[Q]}( x, y)_{\bullet }$ are usually not constant.

Definition 2.4.3.5 (The Homotopy Coherent Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ 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 $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ as the homotopy coherent nerve of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.

Remark 2.4.3.6. The homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ was introduced by Cordier in [MR648798] (motivated by earlier work of Vogt on the theory of homotopy coherence; see [MR0331376]).

Remark 2.4.3.7. The homotopy coherent nerve of Definition 2.4.3.5 determines a functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\bullet )$ from the category $\operatorname{Cat_{\Delta }}$ of simplicial categories (Definition 2.4.1.11) to the category $\operatorname{Set_{\Delta }}$ of simplicial sets (Definition 1.1.1.12. This is a special case of the general construction described in Variant 1.1.7.6, associated to the cosimplicial object of $\operatorname{Cat_{\Delta }}$ given by

\[ \operatorname{{\bf \Delta }}\rightarrow \operatorname{Cat_{\Delta }}\quad \quad [n] \mapsto \operatorname{Path}[n]_{\bullet }. \]

Remark 2.4.3.8 (Comparison with the Nerve). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ denote the underlying ordinary category. For every partially ordered set $Q$, composition with the simplicial functor $\operatorname{Path}[Q]_{\bullet } \rightarrow Q$ of Remark 2.4.3.2 induces a monomorphism

\[ \{ \text{Ordinary functors $Q \rightarrow \operatorname{\mathcal{C}}$} \} \hookrightarrow \{ \text{Simplicial functors $\operatorname{Path}[Q]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$} \} . \]

Restricting this construction to partially ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain a monomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$, where $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the nerve of Construction 1.2.1.1 and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet } )$ is the homotopy coherent nerve of Definition 2.4.3.5.

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 $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ of Remark 2.4.3.8 is bijective on vertices and edges. In particular:

  • Vertices of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ can be identified with objects $X$ of the underlying category $\operatorname{\mathcal{C}}$.

  • Edges of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ can be identified with morphisms $f: X \rightarrow Y$ of the underlying category $\operatorname{\mathcal{C}}$.

  • The face maps $d_{0}, d_{1}: \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet }) \rightarrow \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ carry a morphism $f: X \rightarrow Y$ to its codomain $Y = d_0(f)$ and domain $f = d_1(f)$, respectively.

  • The degeneracy map $s_0: \operatorname{N}_{0}^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet }) \rightarrow \operatorname{N}_{1}^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ 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 $2$ 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

\[ \xymatrix { & x_1 \ar [dr]^{ \{ x_1 < x_2 \} } & \\ x_0 \ar [ur]^{ \{ x_0 < x_1 \} } \ar [rr]^{ \{ x_0 < x_2 \} } & & x_2 } \]

does not commute: the composition of the diagonal maps is the path $\{ x_0 < x_1 < x_2 \} $. However, it commutes in a weaker sense: there is an edge joining $\{ x_0 < x_2 \} $ to $\{ x_0 < x_1 < x_2 \} $ in the simplicial set $\operatorname{Hom}_{\operatorname{Path}[Q]}(x,z)_{\bullet }$. It follows that for any simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, a choice of $2$-simplex

\[ \sigma \in \operatorname{N}_{2}^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[2]_{\bullet } , \operatorname{\mathcal{C}}_{\bullet }) \simeq \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[Q]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) \]

determines a (possibly non-commutative) diagram $\sigma _0$:

\[ \xymatrix { & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]^{ f_{20} } & & X_2, } \]

in $\operatorname{\mathcal{C}}$, together with a homotopy $h$ from $f_{20}$ to $f_{21} \circ f_{10}$ (in the sense of Definition 2.4.1.6). In §2.4.6, we will prove the converse: every choice of homotopy from $f_{20}$ to $f_{21} \circ f_{10}$ determines a unique $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{\bullet } )$ (Proposition 2.4.6.9).

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 2.4.2.6). For any partially ordered set $Q$, Remark 2.4.2.7 and Theorem 2.3.6.7 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{Str}}}( \operatorname{Path}_{(2)}^{\operatorname{c}}[Q], \operatorname{\mathcal{C}}^{\operatorname{c}} ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{ULax}}}( Q, \operatorname{\mathcal{C}}^{\operatorname{c}} ); \end{eqnarray*}

here $\operatorname{\mathcal{C}}^{\operatorname{c}}$ denotes the conjugate $2$-category of $\operatorname{\mathcal{C}}$ (Construction 2.2.3.4). Restricting these bijections to the special case where $Q$ has the form $[n] = \{ 0 < 1 < \cdots < n \} $, we obtain an isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{\bullet } ) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}^{\operatorname{c}} ), \]

where $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}_{\bullet })$ is the homotopy coherent nerve of Definition 2.4.3.5 and $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}^{\operatorname{c}})$ 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 2.4.2.2. Combining Examples 2.3.1.3 and Examples 2.4.3.11, we obtain isomorphisms

\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) = \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{C}}^{\operatorname{c}} ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{\operatorname{\mathcal{C}}}_{\bullet } ). \]

Unwinding the definitions, we see that the composite isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{hc}}_{\bullet }( \underline{\operatorname{\mathcal{C}}}_{\bullet })$ 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 1.2.1.1.