# nLab fully faithful (infinity,1)-functor

### Context

#### $\left(\infty ,1\right)$-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 induced an equivalence on the hom-∞-groupoids

${F}_{x,y}:{\mathrm{Hom}}_{C}\left(x,y\right)\stackrel{\simeq }{\to }{\mathrm{Hom}}_{D}\left(F\left(x\right),F\left(y\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$F_{x,y} : Hom_C(x,y) \stackrel{\simeq}{\to} Hom_D(F(x), F(y)) \,.

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

$F:C↪D$F : C \hookrightarrow D

to indicate this.

## Properties

A full and faithful $\left(\infty ,1\right)$-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 12, 2012 04:07:28 by Mike Shulman (71.136.230.118)