Editing Category theory (section)

=== Morphisms ===
Relations among morphisms (such as {{nowrap|1=''fg'' = ''h''}}) are often depicted using [[commutative diagram]]s, with "points" (corners) representing objects and "arrows" representing morphisms.

[[Morphism]]s can have any of the following properties. A morphism {{nowrap|1=''f'' : ''a'' → ''b''}} is:
* a [[monomorphism]] (or ''monic'') if {{nowrap|1=''f'' ∘ ''g''<sub>1</sub> = ''f'' ∘ ''g''<sub>2</sub>}} implies {{nowrap|1=''g''<sub>1</sub> = ''g''<sub>2</sub>}} for all morphisms {{nowrap|1=''g''<sub>1</sub>, ''g<sub>2</sub>'' : ''x'' → ''a''}}.
* an [[epimorphism]] (or ''epic'') if {{nowrap|1=''g''<sub>1</sub> ∘ ''f'' = ''g''<sub>2</sub> ∘ ''f''}} implies {{nowrap|1=''g<sub>1</sub>'' = ''g<sub>2</sub>''}} for all morphisms {{nowrap|1=''g''<sub>1</sub>, ''g''<sub>2</sub> : ''b'' → ''x''}}.
* a ''bimorphism'' if ''f'' is both epic and monic.
* an [[isomorphism]] if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} such that {{nowrap|1=''f'' ∘ ''g'' = 1<sub>''b''</sub> and ''g'' ∘ ''f'' = 1<sub>''a''</sub>}}.{{efn|A morphism that is both epic and monic is not necessarily an isomorphism. An elementary counterexample: in the category consisting of two objects ''A'' and ''B'', the identity morphisms, and a single morphism ''f'' from ''A'' to ''B'', ''f'' is both epic and monic but is not an isomorphism.}}
* an [[endomorphism]] if {{nowrap|1=''a'' = ''b''}}. end(''a'') denotes the class of endomorphisms of ''a''.
* an [[automorphism]] if ''f'' is both an endomorphism and an isomorphism. aut(''a'') denotes the class of automorphisms of ''a''.
* a [[retract (category theory)|retraction]] if a right inverse of ''f'' exists, i.e. if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} with {{nowrap|1=''f'' ∘ ''g'' = 1<sub>''b''</sub>}}.
* a [[section (category theory)|section]] if a left inverse of ''f'' exists, i.e. if there exists a morphism {{nowrap|1=''g'' : ''b'' → ''a''}} with {{nowrap|1=''g'' ∘ ''f'' = 1<sub>''a''</sub>}}.

Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
* ''f'' is a monomorphism and a retraction;
* ''f'' is an epimorphism and a section;
* ''f'' is an isomorphism.