Epimorphism

In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g₁, g₂: Y → Z,

${\displaystyle g_{1}\circ f=g_{2}\circ f\implies g_{1}=g_{2}.}$

Some authors use the adjective epi (an epimorphism is a morphism which is epi). Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts. The dual of an epimorphism is a monomorphism (i.e. an epimorphism in a category C is a monomorphism in the dual category C^op).

Epimorphism can be a subtly weaker condition than surjectivity. For example, in the category of rings, the inclusion ${\displaystyle \mathbb {Z} \to \mathbb {Q} }$ of integers into rational numbers is an epimorphism, since the images of integers under a homomorphism also determine the images of quotients of integers. In the category of Hausdorff spaces, an epimorphism is precisely a continuous function with dense image, since the image of a Cauchy sequence determines the image of its limit point: for example the inclusion ${\displaystyle \mathbb {Q} \to \mathbb {R} }$ of the metric space of rational numbers into the real number line.

Many authors in abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see § Terminology below.