epimorphism

An epimorphism, abbreviated epi, is a morphism f: A → B in a category C that is right-cancellable: for every object X and every pair of morphisms g, h: B → X, whenever g ∘ f = h ∘ f, it follows that g = h. This abstract definition generalizes the notion of surjectivity beyond sets and functions.

In many familiar categories, epimorphisms coincide with surjective morphisms. For example, in the category of sets, an epimorphism is exactly a surjective function. The same holds in the categories of groups and vector spaces over a field, where surjective homomorphisms or linear maps are precisely the epimorphisms.

However, epimorphisms do not always coincide with surjectivity in every category. In the category of topological spaces, epimorphisms are maps whose images are dense. A standard example is the inclusion of the rationals into the reals, Q → R, which has dense image and is an epimorphism in Top, yet is not surjective.

Epimorphisms are related to other categorical notions such as monomorphisms (the dual concept, right-cancellability vs left-cancellability) and coequalizers. A morphism that is a coequalizer of some pair is always an epimorphism, and in many categories surjections arise as coequalizers.

Overall, the concept of epimorphism captures the idea of “being able to distinguish targets from their images” via composition, and its concrete meaning depends on the ambient category.