Editing Epimorphism (section)

== Terminology ==
The companion terms ''epimorphism'' and ''[[monomorphism]]'' were first introduced by [[Nicolas Bourbaki|Bourbaki]]. Bourbaki uses ''epimorphism'' as shorthand for a [[surjective function]]. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections.  Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.  [[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.  However, this distinction never caught on.

It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept.  Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of rings.  In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.