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Definition 4.6.6.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a simplicial functor.

  • We say that $F$ is weakly fully faithful if, for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the induced map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )_{\bullet }$ is a weak homotopy equivalence of simplicial sets.

  • We say that $F$ is weakly essentially surjective if the induced functor of homotopy categories $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is essentially surjective (that is, every object of $\operatorname{\mathcal{D}}$ is homotopy equivalent to an object of the form $F(X)$, for some $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$).

  • We say that $F$ is a weak equivalence of simplicial categories if it is weakly fully faithful and weakly essentially surjective.