nLab
fully faithful (infinity,1)-functor

Contents

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 induces 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

Every full and faithful (,1)(\infty,1)-functor is a monomorphism in (∞,1)Cat, but being a full and faithful (,1)(\infty,1)-functor is a stronger condition. An (,1)(\infty,1)-functor FF is a monomorphism if and only if it induces a monomorphism on hom-spaces and every equivalence FXFYF X \simeq F Y is in the effective image of FF (see this MathOverflow question).

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

Last revised on November 11, 2019 at 08:20:25. See the history of this page for a list of all contributions to it.