# nLab fully faithful (infinity,1)-functor

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# 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 $F : C \to D$ is full and faithful if for all objects $x,y \in C$ it induces an equivalence on the hom-∞-groupoids

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

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

$F : C \hookrightarrow D$

to indicate this.

## Properties

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