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Definition 2.4.1.11 (Simplicial Functors). Let $\operatorname{\mathcal{C}}_{\bullet }$ and $\operatorname{\mathcal{D}}_{\bullet }$ be simplicial categories. A simplicial functor $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$ consists of the following data:

$(1)$

For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, an object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}}_{\bullet })$.

$(2)$

For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, a map of simplicial sets $F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet }$.

These data are required to satisfy the following conditions:

  • For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the map of simplicial sets $F_{X,X}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(X) )_{\bullet }$ carries the vertex $\operatorname{id}_ X$ to the vertex $\operatorname{id}_{F(X)}$.

  • For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}_{\bullet })$, the diagram of simplicial sets

    \[ \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r] \ar [d]^{ F_{Y,Z} \times F_{X,Y} } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ F_{X,Z} } \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) )_{\bullet } } \]

    is commutative.

We let $\operatorname{Cat_{\Delta }}$ denote the category whose objects are (small) simplicial categories and whose morphisms are simplicial functors.