Example 2.4.2.1 (Simplicial Sets). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets. Then $\operatorname{Set_{\Delta }}$ can be regarded as (the underlying ordinary category of) a simplicial category, which we will also denote by $\operatorname{Set_{\Delta }}$: given a pair of simplicial sets $X_{\bullet }$ and $Y_{\bullet }$, we define $\operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } )_{\bullet }$ to be the simplicial set $\operatorname{Fun}( X_{\bullet }, Y_{\bullet })$ parametrizing morphisms from $X_{\bullet }$ to $Y_{\bullet }$ (see Construction 1.5.3.1).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
2.4.2 Examples of Simplicial Categories
We now supply some examples of simplicial categories.
Example 2.4.2.2 (Functor Categories). Let $\operatorname{\mathcal{C}}$ be a category and let $Y: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every simplicial set $K$, we let $Y^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor given on objects by the formula $Y^{K}(C) = \operatorname{Fun}( K, Y(C) )$. If $X: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is another functor, we let $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( X, Y)_{\bullet }$ denote the simplicial set given by the functor Together with the evident composition maps this construction endows $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ with the structure of a simplicial category.
\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad [n] \mapsto \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( X, Y^{ \Delta ^ n} ). \]
\[ \circ : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( Y, Z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( X, Z)_{\bullet }, \]
Example 2.4.2.3 (Delooping). Let $M_{\bullet }$ be a simplicial monoid. We let $BM_{\bullet }$ denote the simplicial category having a single object $X$, with $\operatorname{Hom}_{BM}(X,X)_{\bullet } = M_{\bullet }$ and the composition law is given by the monoid structure on $M_{\bullet }$. We will refer to $BM_{\bullet }$ as the delooping of the simplicial monoid $M_{\bullet }$. Note that the construction $M_{\bullet } \mapsto BM_{\bullet }$ induces an equivalence of categories
\[ \{ \text{Simplicial Monoids} \} \simeq \{ \text{Simplicial Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X\} $} \} . \]
We can produce many more examples using the construction of Remark 2.1.7.4. If $\operatorname{\mathcal{A}}$ is a monoidal category equipped with a (lax) monoidal functor $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{Set_{\Delta }}$, then every $\operatorname{\mathcal{A}}$-enriched category can also be regarded as a simplicial category. We now consider four instances of this construction:
We can take $F: \operatorname{Set}\hookrightarrow \operatorname{Set_{\Delta }}$ to be the functor which carries each set $S$ to the associated constant simplicial set (Construction 1.1.5.2).
We can take $F: \operatorname{Cat}\hookrightarrow \operatorname{Set_{\Delta }}$ to be the functor which carries each category $\operatorname{\mathcal{C}}$ to its nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Construction 1.3.1.1).
We can take $F: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ to be the functor which carries each simplicial set $S_{\bullet }$ to the opposite simplicial set $S_{\bullet }^{\operatorname{op}}$.
We can take $F: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$ to be the functor which carries each topological space to the singular simplicial set $\operatorname{Sing}_{\bullet }(\operatorname{\mathcal{C}})$ (Construction 1.2.2.2).
Example 2.4.2.4 (Ordinary Categories as Simplicial Categories). Let $\operatorname{\mathcal{C}}$ be an ordinary category. We define a simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ as follows:
The objects of $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{Ob}( \underline{\operatorname{\mathcal{C}}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, $\operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}} }( X, Y )_{\bullet }$ is the constant simplicial set associated to the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.1.5.2).
For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law
on $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ is determined by the composition law $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ on $\operatorname{\mathcal{C}}$.
\[ c_{Z,Y,X}: \operatorname{Hom}_{\underline{\operatorname{\mathcal{C}}}}( Y,Z)_{\bullet } \times \operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}} }( X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \underline{\operatorname{\mathcal{C}}}}( X, Z )_{\bullet } \]
We will refer to $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ as the constant simplicial category associated to $\operatorname{\mathcal{C}}$. Under the fully faithful embedding $\operatorname{Cat_{\Delta }}\hookrightarrow \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Cat})$ of Remark 2.4.1.12, it corresponds to the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \{ \operatorname{\mathcal{C}}\} \hookrightarrow \operatorname{Cat}$ (see Construction 1.1.5.2). In particular, the underlying category of $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ (in the sense of Example 2.4.1.4) is the ordinary category $\operatorname{\mathcal{C}}$.
Remark 2.4.2.5. It follows from Corollary 1.1.5.9 and Remark 2.4.1.12 that the construction of Example 2.4.2.4 is fully faithful. Its essential image consists of those simplicial categories $\operatorname{\mathcal{E}}_{\bullet }$ having the property that, for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{E}}_{\bullet })$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }$ is discrete (Definition 1.1.5.10). We will sometimes abuse notation by not distinguishing between the ordinary category $\operatorname{\mathcal{C}}$ and the constant simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$.
\[ \operatorname{Cat}\rightarrow \operatorname{Cat_{\Delta }}\quad \quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{\mathcal{C}}}_{\bullet } \]
Remark 2.4.2.6. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{\mathcal{D}}_{\bullet }$ be a simplicial category. Applying Proposition 1.1.5.5 (and Remark 2.4.1.12), we deduce that the restriction map is bijective. In other words, the fully faithful embedding of Remark 2.4.2.5 is left adjoint to the forgetful functor of Example 2.4.1.4.
\[ \{ \text{Simplicial functors $\underline{\operatorname{\mathcal{C}}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$} \} \simeq \{ \text{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_0$} \} , \]
\[ \operatorname{Cat}\hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{\mathcal{C}}}_{\bullet } \]
\[ \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Cat}\quad \quad \operatorname{\mathcal{D}}_{\bullet } \mapsto \operatorname{\mathcal{D}}_0. \]
Remark 2.4.2.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then the simplicial category $\underline{\operatorname{\mathcal{C}}}_{\bullet }$ of Example 2.4.2.4 is locally Kan (since constant simplicial sets are Kan complexes; see Example 1.2.5.6).
Example 2.4.2.8 (Strict $2$-Categories as Simplicial Categories). Let $\operatorname{\mathcal{C}}$ be strict $2$-category (Definition 2.2.0.1). Then we can associate to $\operatorname{\mathcal{C}}$ a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ as follows:
The objects of $\operatorname{\mathcal{C}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the simplicial set $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is the nerve of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}( \operatorname{\mathcal{C}}_{\bullet } ) = \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition law
of $\operatorname{\mathcal{C}}_{\bullet }$ is given by the nerve of the composition functor $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$.
\[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( Y,Z )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Z)_{\bullet } \]
Remark 2.4.2.9. In the situation of Example 2.4.2.8, we will generally abuse notation by identifying the strict $2$-category $\operatorname{\mathcal{C}}$ with the associated simplicial category $\operatorname{\mathcal{C}}_{\bullet }$. Note that the underlying category of $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Example 2.4.1.4) agrees with the underlying category of $\operatorname{\mathcal{C}}$ (in the sense of Remark 2.2.0.3). Moreover, since the nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$ is fully faithful (Proposition 1.3.3.1), the construction of Example 2.4.2.8 supplies a fully faithful embedding where $\operatorname{2Cat}_{\operatorname{Str}}$ denotes the category of strict $2$-categories (see Definition 2.2.5.5).
\[ \operatorname{2Cat}_{\operatorname{Str}} \hookrightarrow \operatorname{Cat_{\Delta }}\quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}_{\bullet }, \]
Remark 2.4.2.10. Let $\operatorname{\mathcal{C}}$ be an ordinary category, regarded as a strict $2$-category having only identity $2$-morphisms (Example 2.2.0.6). Then the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ associated to $\operatorname{\mathcal{C}}$ by Example 2.4.2.8 agrees with the simplicial category associated to $\operatorname{\mathcal{C}}$ by Example 2.4.2.4.
Remark 2.4.2.11. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category. Then the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ of Example 2.4.2.8 is locally Kan if and only if every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible: that is, if and only if $\operatorname{\mathcal{C}}$ is a $(2,1)$-category (in the sense of Definition 2.2.8.5). This follows from Proposition 1.3.5.2.
Example 2.4.2.12 (The Conjugate of a Simplicial Category). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We define a new simplicial category $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ as follows:
The objects of $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ are the objects of $\operatorname{\mathcal{C}}_{\bullet }$.
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, we have an equality of simplicial sets
here the right hand side denotes the opposite of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ (Construction 1.4.2.2).
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }) = \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the composition law
on $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ is obtained from the composition law on $\operatorname{\mathcal{C}}_{\bullet }$ by passing to opposite simplicial sets.
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}} }( X, Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\operatorname{op}}; \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{c}}}(X,Z)_{\bullet } \]
We will refer to $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ as the conjugate of the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$.
Remark 2.4.2.13. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ denote the conjugate simplicial category (Example 2.4.2.12). Then, when regarded as a simplicial object of $\operatorname{Cat}$, the conjugate simplicial category $\operatorname{\mathcal{C}}^{\operatorname{c}}_{\bullet }$ is given by the functor here $\mathrm{Op}$ denotes the involution of $\operatorname{{\bf \Delta }}$ described in Notation 1.4.2.1. In particular, the underlying ordinary categories of $\operatorname{\mathcal{C}}_{\bullet }$ and $\operatorname{\mathcal{C}}_{\bullet }^{\operatorname{c}}$ are the same.
\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { \mathrm{Op}} \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { [n] \mapsto \operatorname{\mathcal{C}}_{n} } \operatorname{Cat}; \]
Remark 2.4.2.14. 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). Then the conjugate simplicial category $(\operatorname{\mathcal{C}}_{\bullet })^{\operatorname{c}}$ can be identified with the simplicial category $(\operatorname{\mathcal{C}}^{\operatorname{c}})_{\bullet }$ associated to the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ of Construction 2.2.3.4. In particular, if $\operatorname{\mathcal{C}}$ is an ordinary category, then we have a canonical isomorphism $\operatorname{\mathcal{C}}_{\bullet }^{\operatorname{c}} \simeq \operatorname{\mathcal{C}}_{\bullet }$.
Remark 2.4.2.15. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. Then $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan if and only if the conjugate simplicial category $\operatorname{\mathcal{C}}_{\bullet }^{\operatorname{c}}$ (Example 2.4.2.12) is locally Kan.
Example 2.4.2.16 (Topologically Enriched Categories). Let $\operatorname{Top}$ denote the category of topological spaces. The formation of singular simplicial sets (Construction 1.2.2.2) determines a functor which preserves finite products (in fact, it preserves all small limits), and can therefore be regarded as a monoidal functor from $\operatorname{Top}$ to $\operatorname{Set_{\Delta }}$ (where we endow both $\operatorname{Top}$ and $\operatorname{Set_{\Delta }}$ with the cartesian monoidal structure). Applying Remark 2.1.7.4, we see that every topologically enriched category $\operatorname{\mathcal{C}}$ can be regarded as a simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ having the same objects, with morphism simplicial sets given by here $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ denotes the set of morphisms from $X$ to $Y$, endowed with the topology determined by the topological enrichment of $\operatorname{\mathcal{C}}$ (see Example 2.1.7.8).
\[ \operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto \operatorname{Sing}_{\bullet }(X) \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } = \operatorname{Sing}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ); \]
Remark 2.4.2.17. Let $\operatorname{\mathcal{C}}$ be a topologically enriched category, and let $\operatorname{\mathcal{C}}_{\bullet }$ denote the associated simplicial category (Example 2.4.2.16). Then $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan (since the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ of any topological space $X$ is a Kan complex; see Proposition 1.2.5.8).
Warning 2.4.2.18. Let $\operatorname{Top}_{\mathrm{LCH}}$ denote the full subcategory of $\operatorname{Top}$ spanned by the locally compact Hausdorff spaces. Then we can view $\operatorname{Top}_{\mathrm{LCH}}$ as a topologically enriched category, where we endow each of the sets with the compact-open topology, generated by open sets of the form $\{ f \in \operatorname{Hom}_{\operatorname{Top}}(X,Y): f(K) \subseteq U \} $ where $K \subseteq X$ is compact and $U \subseteq Y$ is open. On this subcategory, the simplicial enrichment of Example 2.4.2.16 coincides with the simplicial enrichment of Example 2.4.1.5. Beware that some technical issues arise if we allow spaces which are not locally compact:
Given topological spaces $X$, $Y$, and $Z$, the composition map
need not be continuous (with respect to the compact-open topologies on $\operatorname{Hom}_{\operatorname{Top}}(X,Y)$, $\operatorname{Hom}_{\operatorname{Top}}(Y,Z)$, and $\operatorname{Hom}_{\operatorname{Top}}(X,Z)$) when $Y \notin \operatorname{Top}_{\mathrm{LCH}}$. Consequently, the construction of compact-open topologies does not determine a topological enrichment of $\operatorname{Top}$ (in the sense of Example 2.1.7.8).
Given topological spaces $X$ and $Y$, a function $| \Delta ^ n | \rightarrow \operatorname{Hom}_{\operatorname{Top}}(X,Y)$ which is continuous (for the compact-open topology on $\operatorname{Hom}_{\operatorname{Top}}(X,Y)$) need not correspond to a continuous function $| \Delta ^ n | \times X \rightarrow Y$ when $X \notin \operatorname{Top}_{\mathrm{LCH} }$.
\[ \operatorname{Hom}_{\operatorname{Top}_{\mathrm{LCH} }}( X, Y) = \{ \text{Continuous functions $f: X \rightarrow Y$} \} \]
\[ \operatorname{Hom}_{\operatorname{Top}}(Y,Z) \times \operatorname{Hom}_{\operatorname{Top}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{Top}}(X,Z) \quad \quad (g,f) \mapsto g \circ f \]
One can remedy these difficulties by replacing $\operatorname{Top}$ by the subcategory of compactly generated weak Hausdorff spaces introduced in [MR0251719].