2.4 Simplicial Categories
Let $\operatorname{Top}$ denote the category of topological spaces. By definition, a morphism in the category $\operatorname{Top}$ is a continuous function $f: X \rightarrow Y$. In homotopy theory, one is fundamentally concerned not only with continuous functions themselves, but also with homotopies between them: that is, continuous functions $h: [0,1] \times X \rightarrow Y$. More generally, for each $n \geq 0$, one can consider the set
\[ \operatorname{Hom}_{\operatorname{Top}}(X,Y)_{n} = \{ \text{Continuous functions $\sigma : | \Delta ^ n | \times X \rightarrow Y$} \} ; \]
here $| \Delta ^ n |$ denotes the topological simplex of dimension $n$. The sets $\{ \operatorname{Hom}_{\operatorname{Top}}(X,Y)_{n} \} _{n \geq 0}$ can be assembled into a simplicial set $\operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$, and the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{Top}}(X,Y)_{\bullet }$ endows $\operatorname{Top}$ with the structure of a simplicial category: that is, a category which is enriched over simplicial sets, in the sense of Definition 2.1.7.1. Much as the singular simplicial set $\operatorname{Sing}_{\bullet }(X) = \operatorname{Hom}_{\operatorname{Top}}( \ast , X )_{\bullet }$ can be regarded as a combinatorial encoding of the homotopy type of an individual topological space $X$, the simplicial enrichment of $\operatorname{Top}$ can be regarded as a combinatorial encoding of the homotopy theory of topological spaces.
Our goal in this section is to provide an introduction to the theory of simplicial categories. We begin in §2.4.1 by defining the notion of simplicial category (Definition 2.4.1.1). The collection of (small) simplicial categories can itself be organized into a category $\operatorname{Cat_{\Delta }}$, in which the morphisms are given by simplicial functors (Definition 2.4.1.11). In §2.4.2 we provide many examples of how simplicial categories arise in nature: in particular, we explain that $\operatorname{Cat_{\Delta }}$ can be regarded as an enlargement of the usual category $\operatorname{Cat}$ of small categories (Example 2.4.2.4), and also of the category $\operatorname{2Cat}_{\operatorname{Str}}$ of strict $2$-categories (Example 2.4.2.8).
Recall that to every category $\operatorname{\mathcal{C}}$ we can associate a simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ called the nerve of $\operatorname{\mathcal{C}}$ (Construction 1.3.1.1). In §2.4.3, we describe a generalization of this construction (due to Cordier) which associates to each simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ a simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ called the homotopy coherent nerve of $\operatorname{\mathcal{C}}_{\bullet }$ (Definition 2.4.3.5). This construction specializes to the ordinary nerve in the case where $\operatorname{\mathcal{C}}_{\bullet }$ is an ordinary category (and to the Duskin nerve in the case where $\operatorname{\mathcal{C}}_{\bullet }$ arises from a strict $2$-category: see Example 2.4.3.11). It is particularly well-behaved in the special case where $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan (meaning that simplicial $\operatorname{Hom}$-sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ are Kan complexes): in this case, a theorem of Cordier and Porter asserts that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1).
In §2.4.4, we show that the homotopy coherent nerve functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint (Corollary 2.4.4.4). This left adjoint carries each simplicial set $S_{\bullet }$ to a simplicial category $\operatorname{Path}[ S ]_{\bullet }$ which we will refer to as the (simplicial) path category of $S_{\bullet }$. The construction $S_{\bullet } \mapsto \operatorname{Path}[ S ]_{\bullet }$ is a generalization of the classical path category studied in §1.3.7: when $S_{\bullet }$ is the $1$-dimensional simplicial set associated to a directed graph $G$, the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ can be identified with the ordinary category $\operatorname{Path}[G]$ of Construction 1.3.7.1 (see Proposition 2.4.4.7). For a general simplicial set $S_{\bullet }$, the path category $\operatorname{Path}[ S ]_{\bullet }$ is a complicated object. However, in each fixed simplicial degree $m$ it is relatively simple: the ordinary category $\operatorname{Path}[ S ]_{m}$ can be identified with the classical path category of a certain directed graph $G_ m$ which can be described concretely in terms of the combinatorics of $S_{\bullet }$ (Theorem 2.4.4.10). We will exploit this description in §2.4.5 to carry out the proof of Theorem 2.4.5.1, and again in §2.4.6 to compare the homotopy category of a (locally Kan) simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ to the homotopy category of its associated $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (Proposition 2.4.6.9).
Warning 2.4.0.1. The ordinary nerve functor $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding from the category $\operatorname{Cat}$ of small categories to the category $\operatorname{Set_{\Delta }}$ of simplicial sets (Proposition 1.3.3.1). However, the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ is not fully faithful when regarded as a functor of ordinary categories. Phrased differently, the adjoint functors
\[ \xymatrix@1{ \operatorname{Set_{\Delta }} \ar@ <.4ex>[r]^-{ \operatorname{Path}[ - ]_{\bullet } } & \operatorname{Cat_{\Delta }} \ar@ <.4ex>[l]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }} \]
associate to each simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ a counit map $v: \operatorname{Path}[ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}_{\bullet })]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$, which is almost never an isomorphism of simplicial categories. However, we will see later that $v$ is a weak equivalence of simplicial categories whenever $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan (
). Moreover, the construction $\operatorname{\mathcal{C}}_{\bullet } \mapsto \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ establishes an equivalence from the homotopy theory of (locally Kan) simplicial categories $\operatorname{\mathcal{C}}_{\bullet }$ with the homotopy theory of $\infty $-categories ().
Structure
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Subsection 2.4.1: Simplicial Enrichment
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Subsection 2.4.2: Examples of Simplicial Categories
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Subsection 2.4.3: The Homotopy Coherent Nerve
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Subsection 2.4.4: The Path Category of a Simplicial Set
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Subsection 2.4.5: From Simplicial Categories to $\infty $-Categories
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Subsection 2.4.6: The Homotopy Category of a Simplicial Category
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Subsection 2.4.7: Example: Braid Monoids