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, 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; for example, the inclusion

Every morphism in a concrete category whose underlying function is surjective is an epimorphism.

As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context.

This in turn is equivalent to the induced natural transformation being a monomorphism in the functor category SetC.

For example, the map from the half-open interval [0,1) to the unit circle S1 (thought of as a subspace of the complex plane) that sends x to exp(2πix) (see Euler's formula) is continuous and bijective but not a homeomorphism since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the category Top.

Another example is the embedding Q → R in the category Haus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism.

Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are epimorphisms in the modern sense.

It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept.

In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.