#
nLab

fully faithful (infinity,1)-functor

### Context

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

**(∞,1)-category theory**

## Background

## Basic concepts

## Universal constructions

## Local presentation

## Theorems

## Models

# 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

A full and faithful $(\infty,1)$-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

Last revised on September 20, 2017 at 03:43:34.
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