#
nLab

faithful functor

Contents
### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Definition

### In 1-category theory

A functor $F: C \to D$ from the category $C$ to the category $D$ is *faithful* if for each pair of objects $x, y \in C$, the function

$F : C(x,y) \to D(F(x), F(y))$

between hom sets is injective.

More abstractly, we may say a functor is faithful if it is $2$-surjective – or loosely speaking, ‘surjective on equations between given morphisms’.

### In higher category theory

See also *faithful morphism* for a generalization to an arbitrary 2-category.

And see 0-truncated morphism for generalization to (∞,1)-categories (see there).

## Properties

(The simple proof is spelled out for instance at *epimorphism*.)

Last revised on September 9, 2018 at 07:33:17.
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