Definition 2.4.1.1 (Simplicial Categories). A simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ consists of the following data:
- $(1)$
A collection $\operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, whose elements we refer to as objects of $\operatorname{\mathcal{C}}_{\bullet }$. We will often abuse notation by writing $X \in \operatorname{\mathcal{C}}_{\bullet }$ to indicate that $X$ is an element of $\operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.
- $(3)$
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a morphism of simplicial sets
\[ c_{Z,Y,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)_{\bullet }, \]which we will refer to as the composition law.
- $(4)$
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a vertex $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{0}$, which we will refer to as the identity morphism of $X$.
These data are required to satisfy the following conditions:
- $(A)$
For every quadruple of objects $W,X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the diagram of simplicial sets
\[ \xymatrix@C =-80pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [dl]_{ \operatorname{id}\times c_{Y,X,W} } \ar [dr]^{ c_{Z,Y,X} \times \operatorname{id}} & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{\bullet } \ar [dr]_{ c_{Z,Y,W} } & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [dl]^{ c_{Z,X,W} } \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z)_{\bullet } & } \]commutes (in other words, the composition law of $(3)$ is associative).
- $(U)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the maps of simplicial sets
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \{ \operatorname{id}_ X \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \xrightarrow { c_{Y,X,X} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \]\[ \{ \operatorname{id}_ Y \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \xrightarrow { c_{Y,Y,X} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \]coincide with the projection maps onto the factor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$.