Editing Category (mathematics) (section)

==Examples==
The [[class (set theory)|class]] of all sets (as objects) together with all [[function (mathematics)|function]]s between them (as morphisms), where the composition of morphisms is the usual [[function composition]], forms a large category, '''[[category of sets|Set]]'''. It is the most basic and the most commonly used category in mathematics. The category '''[[category of relations|Rel]]''' consists of all [[Set (mathematics)|sets]] (as objects) with [[binary relation]]s between them (as morphisms).  Abstracting from [[Relation (mathematics)|relations]] instead of functions yields [[Allegory (category theory)|allegories]], a special class of categories.

Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called [[discrete category|discrete]]. For any given [[Set (mathematics)|set]] ''I'', the ''discrete category on I'' is the small category that has the elements of ''I'' as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category.

Any [[Preorder|preordered set]] (''P'', ≤) forms a small category, where the objects are the members of ''P'', the morphisms are arrows pointing from ''x'' to ''y'' when ''x'' ≤ ''y''. Furthermore, if ''≤'' is [[antisymmetric relation|antisymmetric]], there can be at most one morphism between any two objects. The existence of identity morphisms and the composability of the morphisms are guaranteed by the [[reflexive relation|reflexivity]] and the [[transitive relation|transitivity]] of the preorder. By the same argument, any [[partially ordered set]] and any [[equivalence relation]] can be seen as a small category. Any [[ordinal number]] can be seen as a category when viewed as an [[total order|ordered set]].

Any [[monoid]] (any [[algebraic structure]] with a single [[associative]] [[binary operation]] and an [[identity element]]) forms a small category with a single object ''x''. (Here, ''x'' is any fixed set.) The morphisms from ''x'' to ''x'' are precisely the elements of the monoid, the identity morphism of ''x'' is the identity of the monoid, and the categorical composition of morphisms is given by the monoid operation.  Several definitions and theorems about monoids may be generalized for categories.

Similarly any [[group (mathematics)|group]] can be seen as a category with a single object in which every morphism is ''invertible'', that is, for every morphism ''f'' there is a morphism ''g'' that is both [[Morphism#Some specific morphisms|left and right inverse]] to ''f'' under composition. A morphism that is invertible in this sense is called an [[isomorphism]].

A [[groupoid]] is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, [[Group action (mathematics)|group action]]s and [[equivalence relation]]s. Actually, in the view of category the only difference between groupoid and group is that a groupoid may have more than one object but the group must have only one. Consider a topological space ''X'' and fix a base point <math>x_0</math> of ''X'', then <math>\pi_1(X,x_0)</math> is the [[fundamental group]] of the topological space ''X'' and the base point <math>x_0</math>, and as a set it has the structure of group; if then let the base point <math>x_0</math> runs over all points of ''X'', and take the union of all <math>\pi_1(X,x_0)</math>, then the set we get has only the structure of groupoid (which is called as the [[fundamental groupoid]] of ''X''): two loops (under equivalence relation of homotopy) may not have the same base point so they cannot multiply with each other. In the language of category, this means here two morphisms may not have the same source object (or target object, because in this case for any morphism the source object and the target object are same: the base point) so they can not compose with each other.

[[File:Directed.svg|125px|thumb|A directed graph.]]
Any [[directed graph]] [[Generating set|generates]] a small category: the objects are the [[Vertex (graph theory)|vertices]] of the graph, and the morphisms are the paths in the graph (augmented with [[loop (graph theory)|loop]]s as needed) where composition of morphisms is concatenation of paths. Such a category is called the ''[[free category]]'' generated by the graph.

The class of all preordered sets with order-preserving functions (i.e., monotone-increasing functions) as morphisms forms a category, '''[[category of preordered sets|Ord]]'''. It is a [[concrete category]], i.e. a category obtained by adding some type of structure onto '''Set''', and requiring that morphisms are functions that respect this added structure.

The class of all groups with [[group homomorphism]]s as [[morphism]]s and [[function composition]] as the composition operation forms a large category, '''[[Category of groups|Grp]]'''. Like '''Ord''', '''Grp''' is a concrete category. The category '''[[category of abelian groups|Ab]]''', consisting of all [[abelian group]]s and their group homomorphisms, is a [[full subcategory]] of '''Grp''', and the prototype of an [[abelian category]].

The class of all [[graph theory|graphs]] forms another concrete category, where morphisms are graph homomorphisms (i.e., mappings between graphs which send vertices to vertices and edges to edges in a way that preserves all adjacency and incidence relations).

Other examples of concrete categories are given by the following table.

{| class="wikitable"
!Category
!Objects
!Morphisms
|-
|'''[[category of sets|Set]]'''
|[[Set (mathematics)|set]]s
|[[Function (mathematics)|function]]s
|-
|'''[[category of preordered sets|Ord]]'''
|preordered sets
|monotone-increasing functions
|-
|'''[[category of monoids|Mon]]'''
|[[monoids]]
|[[Monoid#Monoid homomorphisms|monoid homomorphisms]]
|-
|'''[[category of groups|Grp]]'''
|[[Group (mathematics)|group]]s
|[[group homomorphism]]s
|-
|'''[[category of graphs|Grph]]'''
|[[graph theory|graph]]s
|graph homomorphisms
|-
|'''[[category of rings|Ring]]'''
|[[ring (mathematics)|ring]]s
|[[ring homomorphism]]s
|-
|'''[[category of fields|Field]]'''
|[[field (mathematics)|field]]s
|[[field homomorphism]]s
|-
|'''[[category of modules|''R''-Mod]]'''
|[[module (mathematics)|''R''-modules]], where ''R'' is a ring
|[[module homomorphism|''R''-module homomorphisms]]
|-
|[[K-Vect|'''Vect'''<sub>''K''</sub>]]
|[[vector space]]s over the [[field (mathematics)|field]] ''K''
|''K''-[[linear map]]s
|-
|'''[[category of metric spaces|Met]]'''
|[[metric space]]s
|[[short map]]s
|-
|'''[[Category of measurable spaces|Meas]]'''
|[[measure space]]s
|[[measurable function]]s
|-
|'''[[category of topological spaces|Top]]'''
|[[topological space]]s
|[[continuous function (topology)|continuous function]]s
|-
|[[category of manifolds|'''Man'''<sup>''p''</sup>]]
|[[smooth manifold]]s
|''p''-times [[continuously differentiable]] maps
|}

[[Fiber bundle]]s with [[bundle map]]s between them form a concrete category.

The category '''[[category of small categories|Cat]]''' consists of all small categories, with [[functor]]s between them as morphisms.