Epimorphism

A morphism that is right-cancellative; the categorical analogue of a surjection.

Epimorphism

Let $\mathcal C$ be a category and let $f:A\to B$ be a morphism in $\mathcal C$ .

Definition

The morphism $f$ is an epimorphism (or epi) if it is right-cancellative with respect to composition : for every object $C$ of $\mathcal C$ and all morphisms $g,h:B\to C$ ,

g\circ f = h\circ f \quad \Longrightarrow \quad g=h.

Equivalently, postcomposition with $f$ induces an injective map of hom-sets

\mathrm{Hom}_{\mathcal C}(B,C)\;\longrightarrow\;\mathrm{Hom}_{\mathcal C}(A,C),\qquad g\mapsto g\circ f,

for every object $C$ .

Basic properties

Every isomorphism is an epimorphism.
Epimorphisms are stable under composition: if $f:A\to B$ and $g:B\to C$ are epis, then $g\circ f$ is an epi.
Duality: $f$ is an epi in $\mathcal C$ iff $f$ is a monomorphism in the opposite category $\mathcal C^{\mathrm{op}}$ .

Examples

$\mathbf{Set}$ : A function $f:A\to B$ is an epimorphism in $\mathbf{Set}$ iff it is a surjective function .
$\mathbf{Grp}$ , $\mathbf{Ab}$ , $R\mathbf{-Mod}$ : A homomorphism (or $R$ -linear map) is an epimorphism iff it is surjective on the underlying sets.
In particular, the quotient map $G\to G/N$ and the canonical projection $M\to M/N$ are epis.
$\mathbf{Top}$ : A continuous map $f:X\to Y$ is an epimorphism in $\mathbf{Top}$ iff it is surjective as a map of underlying sets.

Remark

Not every category has the property “epis are surjective on underlying sets” (for instance, in some algebraic categories epimorphisms can fail to be surjective). The definition above is the one that makes sense in an arbitrary category .