Category theory; homological algebra Definition

Full and faithful functors

Builds on Functor Full subcategory Group Group homomorphism Injective function Surjective function

Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor. For objects $A$ , $B$ of $\mathcal{C}$ , let $F_{A, B} \colon \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{D}}(F(A), F(B))$ be the map $f \mapsto F(f)$ .

$F$ is faithful if $F_{A, B}$ is injective for all objects $A$ , $B$ .
$F$ is full if $F_{A, B}$ is surjective for all objects $A$ , $B$ .
$F$ is fully faithful if $F_{A, B}$ is bijective for all objects $A$ , $B$ .

Remarks

A faithful functor may still identify two morphisms with different domains or codomains, and it may send different objects to the same object. Example: the forgetful functor from groups to sets is faithful, since two homomorphisms with the same underlying map are equal. It is not full: the constant map $\Z \to \Z$ with value $1$ is a function between the underlying sets but not a homomorphism. The inclusion of a full subcategory is fully faithful. Non-example: the functor from groups to the category with one object and one morphism is not faithful. The two homomorphisms $\Z \to \Z$ given by $x \mapsto 0$ and $x \mapsto x$ are sent to the same morphism. A fully faithful functor reflects isomorphisms: if $F(f)$ is an isomorphism, then so is $f$ .

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`3GZ99NCX22`
URI	`m2://3GZ99NCX22`
Kind	definition
Writing	W0 stub
MSC	18A22
Defines	Full and faithful functors
Uses	Functor, Full subcategory, Group, Group homomorphism, Injective function, Surjective function
Changed	2 November 2026

Full and faithful functors

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here