nLab
fully faithful (infinity,1)-functor

Contents

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:CDF : C \to D is full and faithful if for all objects x,yCx,y \in C it induced an equivalence on the hom-∞-groupoids

F x,y:Hom C(x,y)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 (,1)(\infty,1)-functor F:CDF : C \to D exhibits CC as a full sub-(∞,1)-category of DD and one tends to write

F:CD F : C \hookrightarrow D

to indicate this.

Properties

A full and faithful (,1)(\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.

References

This appears as definition 1.2.10 in

Revised on May 20, 2014 07:02:23 by Toby Bartels (64.89.53.56)