nLab
fully faithful (infinity,1)-functor

Definition

An (∞,1)-functor $F:C\to D$ is fully faithful if, when modeled as a functor of simplicially enriched categories the induced functor on homotopy categories

$$hF:hC\to hD$$h F : h C \to h D

of ${\mathrm{Ho}}_{\mathrm{Top}}$-enriched categories is fully faithful, i.e. if the morphisms

$$h{F}_{X,Y}:hC(X,Y)\to hD(F(X),F(Y))$$h F_{X,Y} : h C(X,Y) \to h D(F(X), F(Y))

are isomorphisms in ${\mathrm{Ho}}_{\mathrm{Top}}$ for all objects $X,Y$.

Remarks

• See also (∞,1)-essentially surjective functor.