Editing Surjective function (section)

===Surjections as epimorphisms===
A function {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective if and only if it is [[right-cancellative]]:<ref>{{Cite book
|first=Robert
|last=Goldblatt
|title=Topoi, the Categorial Analysis of Logic
|url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|access-date=2009-11-25
|edition=Revised
|year=2006
|orig-year=1984
|publisher=[[Dover Publications]]
|isbn=978-0-486-45026-1}}</ref> given any functions {{Nowrap|''g'',''h'' : ''Y'' → ''Z''}}, whenever {{Nowrap begin}}''g'' <small>o</small> ''f'' = ''h'' <small>o</small> ''f''{{Nowrap end}}, then {{Nowrap begin}}''g'' = ''h''{{Nowrap end}}. This property is formulated in terms of functions and their [[function composition|composition]] and can be generalized to the more general notion of the [[morphism]]s of a [[category (mathematics)|category]] and their composition. Right-cancellative morphisms are called [[epimorphism]]s. Specifically, surjective functions are precisely the epimorphisms in the [[category of sets]]. The prefix ''epi'' is derived from the Greek preposition ''ἐπί'' meaning ''over'', ''above'', ''on''.

Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse ''g'' of a morphism ''f'' is called a [[section (category theory)|section]] of ''f''. A morphism with a right inverse is called a [[split epimorphism]].