nLab
fully faithful (infinity,1)-functor

Idea

The generalization to the context of (∞,1)-category-theory of the notion of a full and faithful functor in ordinary category theory.

Definition

An (∞,1)-functor $F:C\to D$ is full and faithful if for all objects $x,y\in C$ it induced an equivalence on the hom-∞-groupoids

$$F_{x,y}:{\mathrm{Hom}}_C(x,y)\stackrel{\simeq }{\to }{\mathrm{Hom}}_D(F(x),F(y)).$$F_{x,y} : Hom_C(x,y) \stackrel{\simeq}{\to} Hom_D(F(x), F(y)) \,.

A full and faithful $(\mathrm{\infty },1)$-functor $F:C\to D$ exhibits $C$ as a full sub-(∞,1)-category of $D$ and one tends to write

$$F:C\hookrightarrow D$$F : C \hookrightarrow D

to indicate this.

Properties

A full and faithful $(\mathrm{\infty },1)$-functor is precisely a monomorphism in (∞,1)Cat, hence a (-1)-truncated morphism.

An (∞,1)-functor which is both full and faithful as well as an essentially surjective (∞,1)-functor is an equivalence of (∞,1)-categories.

Related concepts

• full functor, faithful functor, full and faithful functor, essentially surjective functor

• essentially surjective (∞,1)-functor,

• equivalence of (∞,1)-categories

References

This appears as definition 1.2.10 in

• Jacob Lurie, Higher Topos Theory