Editing Category theory (section)

== Categories, objects, and morphisms ==
{{Main|Category (mathematics)|Morphism}}

=== Categories ===
A ''category'' <math>\mathcal{C}</math> consists of the following three mathematical entities:
* A [[Class (set theory)|class]] <math>\text{ob}(\mathcal{C})</math>, whose elements are called ''objects'';
* A class <math>\text{hom}(\mathcal{C})</math>, whose elements are called [[morphism]]s or [[Map (mathematics)|maps]] or ''arrows''. <p>Each morphism '''<math>f</math>''' has a ''source object '' '''<math>a</math>''' and ''target object'' '''<math>b</math>'''.</p><p>The expression <math>f:a \rightarrow b</math> would be verbally stated as "<math>f</math> is a morphism from {{mvar|a}} to {{mvar|b}}".</p><p>The expression <math>\text{hom}(a, b)</math> – alternatively expressed as <math>\text{hom}_\mathcal{C}(a, b)</math>, <math>\text{mor}(a, b)</math>, or <math>\mathcal{C}(a, b)</math> – denotes the ''hom-class'' of all morphisms from <math>a</math> to <math>b</math>.{{efn|The name "hom" derives from the fact that the notion of morphism is a generalisation of the notion of [[homomorphism]]. But even in categories whose objects have no notion of homomorphism or where the morphisms are explicitly not (or not precisely) homomorphisms, the classes <math>\text{hom}(a, b)</math> are still referred to as hom-classes.}}</p>
* A [[binary operation]] <math>\circ</math>, called ''composition of morphisms'', such that for any three objects ''{{mvar|a}}'', ''{{mvar|b}}'', and ''{{mvar|c}}'', we have<math display="block">\circ : \text{hom}(b, c) \times \text{hom}(a, b) \mapsto \text{hom}(a, c)</math>The composition of <math>f : a \rightarrow b</math> and <math>g: b \rightarrow c</math> is written as <math>g \circ f</math> or <math>gf</math>,{{efn|Some authors compose in the opposite order, writing ''fg'' or {{nowrap|1=''f'' ∘ ''g''}} for {{nowrap|1=''g'' ∘ ''f''}}. Computer scientists using category theory very commonly write {{nowrap|1=''f'' ; ''g''}} for {{nowrap|1=''g'' ∘ ''f''}}}} governed by two axioms:
*# [[Associativity]]: If <math>f: a \rightarrow b</math>, <math>g: b \rightarrow c</math>, and <math>h: c \rightarrow d</math> then <math display="block">h \circ (g \circ f) = (h \circ g) \circ f</math>
*# [[Identity (mathematics)|Identity]]: For every object {{mvar|x}}, there exists a morphism <math>1_x : x \rightarrow x</math> (also denoted as <math>\text{id}_x</math>) called the ''[[identity morphism]] for {{mvar|x}}'', such that for every morphism <math>f: a \rightarrow b</math>, we have<math display="block">1_b \circ f = f = f \circ 1_a</math><p>From the axioms, it can be proved that there is exactly one [[identity morphism]] for every object.</p>

====Examples====
* The category '''[[category of sets|Set]]'''
** As the class of objects <math>\text{ob} (\text{Set})</math>, we choose the class of all sets.
** As the class of morphisms <math>\text{hom} (\text{Set})</math>, we choose the class of all [[function (mathematics)|function]]s. Therefore, for two objects {{mvar|A}} and {{mvar|B}}, i.e. sets, we have <math>\text{hom} (A,B)</math> to be the class of all functions {{tmath|f}} such that {{tmath|f:A \rightarrow B}}.
** The composition of morphisms {{tmath|\circ}} is simply the usual [[function composition]], i.e. for two morphisms {{tmath|f:A \rightarrow B}} and {{tmath|g:B \rightarrow C}}, we have {{tmath|g \circ f:A \rightarrow C}}, <math>(g \circ f)(x) = g(f(x))</math>, which is obviously associative. Furthermore, for every object {{mvar|A}} we have the identity morphism <math>\text{id}_A</math> to be the identity map <math>\text{id}_A : A \rightarrow A</math>, <math>\text{id}_A (x) = x</math> on {{mvar|A}}

=== 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.