Editing Functor (section)

== Examples ==
; [[Diagram (category theory)|Diagram]]: For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor <math>D \colon J\to C</math>.
; [[Presheaf (category theory)|(Category theoretical) presheaf]]:For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor <math>D \colon C\to J</math>.{{paragraph}}In the special case when J is '''Set''', the category of sets and functions, ''D'' is called a [[Presheaf (category theory)|presheaf]] on ''C''.
; Presheaves (over a topological space): If ''X'' is a [[topological space]], then the [[open set]]s in ''X'' form a [[partially ordered set]] Open(''X'') under inclusion. Like every partially ordered set, Open(''X'') forms a small category by adding a single arrow {{nowrap|''U'' → ''V''}} if and only if <math>U \subseteq V</math>. Contravariant functors on Open(''X'') are called ''[[presheaf|presheaves]]'' on ''X''. For instance, by assigning to every open set ''U'' the [[associative algebra]] of real-valued continuous functions on ''U'', one obtains a presheaf of algebras on ''X''.
; Constant functor: The functor {{nowrap|''C'' → ''D''}} which maps every object of ''C'' to a fixed object ''X'' in ''D'' and every morphism in ''C'' to the identity morphism on ''X''. Such a functor is called a ''constant'' or ''selection'' functor.
; {{term|term=Endofunctor}}{{anchor|name=Endofunctor}}: A functor that maps a category to that same category; e.g., [[polynomial functor]].
; {{term|term=Identity functor}}: In category ''C'', written 1<sub>''C''</sub> or id<sub>''C''</sub>, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
; Diagonal functor: The [[diagonal functor]] is defined as the functor from ''D'' to the functor category ''D''<sup>''C''</sup> which sends each object in ''D'' to the constant functor at that object.
; Limit functor: For a fixed [[index category]] ''J'', if every functor {{nowrap|''J'' → ''C''}} has a [[limit (category theory)|limit]] (for instance if ''C'' is complete), then the limit functor {{nowrap|''C''<sup>''J''</sup> → ''C''}} assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the [[Adjoint functors|right-adjoint]] to the [[diagonal functor]] and invoking the [[Freyd adjoint functor theorem]]. This requires a suitable version of the [[axiom of choice]]. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).
; Power sets functor: The power set functor {{nowrap|''P'' : '''Set''' → '''Set'''}} maps each set to its [[power set]] and each function <math> f \colon X \to Y</math> to the map which sends <math>U \in \mathcal{P}(X)</math> to its image <math>f(U) \in \mathcal{P}(Y)</math>. One can also consider the '''contravariant power set functor''' which sends <math> f \colon X \to Y </math> to the map which sends <math>V \subseteq Y</math> to its [[inverse image]] <math>f^{-1}(V) \subseteq X.</math>{{paragraph}} For example, if <math>X = \{0,1\}</math> then <math>F(X) = \mathcal{P}(X) = \{\{\}, \{0\}, \{1\}, X\}</math>. Suppose <math>f(0) = \{\}</math> and <math>f(1) = X</math>. Then <math>F(f)</math> is the function which sends any subset <math>U</math> of <math>X</math> to its image <math>f(U)</math>, which in this case means <math>\{\} \mapsto f(\{\}) = \{\}</math>, where <math>\mapsto</math> denotes the mapping under <math>F(f)</math>, so this could also be written as <math>(F(f))(\{\})= \{\}</math>. For the other values,<math>
    \{0\} \mapsto f(\{0\})   = \{f(0)\}       = \{\{\}\},\ </math> <math>
    \{1\} \mapsto f(\{1\})   = \{f(1)\}       = \{X\},\ </math> <math>
  \{0,1\} \mapsto f(\{0,1\}) = \{f(0), f(1)\} = \{\{\}, X\}.
</math> Note that <math>f(\{0, 1\})</math> consequently generates the [[trivial topology]] on <math>X</math>. Also note that although the function <math>f</math> in this example mapped to the power set of <math>X</math>, that need not be the case in general.
; {{visible anchor|Dual vector space}}: The map which assigns to every [[vector space]] its [[dual space]] and to every [[linear operator|linear map]] its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed [[field (mathematics)|field]] to itself.
; Fundamental group: Consider the category of [[pointed topological space]]s, i.e. topological spaces with distinguished points. The objects are pairs {{nowrap|(''X'', ''x''<sub>0</sub>)}}, where ''X'' is a topological space and ''x''<sub>0</sub> is a point in ''X''. A morphism from {{nowrap|(''X'', ''x''<sub>0</sub>)}} to {{nowrap|(''Y'', ''y''<sub>0</sub>)}} is given by a [[continuous function (topology)|continuous]] map {{nowrap|''f'' : ''X'' → ''Y''}} with {{nowrap|1=''f''(''x''<sub>0</sub>) = ''y''<sub>0</sub>}}.{{paragraph}} To every topological space ''X'' with distinguished point ''x''<sub>0</sub>, one can define the [[fundamental group]] based at ''x''<sub>0</sub>, denoted {{nowrap|π<sub>1</sub>(''X'', ''x''<sub>0</sub>)}}. This is the [[group (mathematics)|group]] of [[homotopy]] classes of loops based at ''x''<sub>0</sub>, with the group operation of concatenation. If {{nowrap|''f'' : ''X'' → ''Y''}} is a morphism of [[pointed space]]s, then every loop in ''X'' with base point ''x''<sub>0</sub> can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''<sub>0</sub>. This operation is compatible with the homotopy [[equivalence relation]] and the composition of loops, and we get a [[group homomorphism]] from {{nowrap|π(''X'', ''x''<sub>0</sub>)}} to {{nowrap|π(''Y'', ''y''<sub>0</sub>)}}. We thus obtain a functor from the category of pointed topological spaces to the [[category of groups]].{{paragraph}} In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the '''fundamental [[groupoid]]''' instead of the fundamental group, and this construction is functorial.
; Algebra of continuous functions: A contravariant functor from the category of [[topology|topological spaces]] (with continuous maps as morphisms) to the category of real [[associative algebra]]s is given by assigning to every topological space ''X'' the algebra C(''X'') of all real-valued continuous functions on that space. Every continuous map {{nowrap|''f'' : ''X'' → ''Y''}} induces an [[algebra homomorphism]] {{nowrap|C(''f'') : C(''Y'') → C(''X'')}} by the rule {{nowrap|1=C(''f'')(''φ'') = ''φ'' ∘ ''f''}} for every ''φ'' in C(''Y'').
; Tangent and cotangent bundles: The map which sends every [[differentiable manifold]] to its [[tangent bundle]] and every [[smooth map]] to its [[derivative]] is a covariant functor from the category of differentiable manifolds to the category of [[vector bundle]]s. {{paragraph}}Doing this constructions pointwise gives the [[tangent space]], a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, [[cotangent space]] is a contravariant functor, essentially the composition of the tangent space with the [[#Dual vector space|dual space]] above.
; Group actions/representations: Every [[group (mathematics)|group]] ''G'' can be considered as a category with a single object whose morphisms are the elements of ''G''. A functor from ''G'' to '''Set''' is then nothing but a [[Group action (mathematics)|group action]] of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the [[category of vector spaces]], '''Vect'''<sub>''K''</sub>, is a [[linear representation]] of ''G''. In general, a functor {{nowrap|''G'' → ''C''}} can be considered as an "action" of ''G'' on an object in the category ''C''. If ''C'' is a group, then this action is a group homomorphism.
; Lie algebras: Assigning to every real (complex) [[Lie group]] its real (complex) [[Lie algebra]] defines a functor.
; Tensor products: If ''C'' denotes the category of vector spaces over a fixed field, with [[linear operator|linear maps]] as morphisms, then the [[tensor product]] <math>V \otimes W</math> defines a functor {{nowrap|''C'' × ''C'' → ''C''}} which is covariant in both arguments.<ref>{{citation|first1=Michiel|last1=Hazewinkel|author-link1=Michiel Hazewinkel|first2=Nadezhda Mikhaĭlovna|last2=Gubareni|author-link2=Nadezhda Mikhaĭlovna|first3=Nadiya|last3=Gubareni|author-link3=Nadiya Gubareni|first4=Vladimir V.|last4=Kirichenko|author-link4=Vladimir V. Kirichenko|title=Algebras, rings and modules|publisher=Springer|year=2004|isbn=978-1-4020-2690-4}}</ref>
; Forgetful functors: The functor {{nowrap|''U'' : '''Grp''' → '''Set'''}} which maps a [[group (mathematics)|group]] to its underlying set and a [[group homomorphism]] to its underlying function of sets is a functor.{{sfnp|Jacobson|2009|loc=p. 20, ex. 2}}  Functors like these, which "forget" some structure, are termed ''[[forgetful functor]]s''. Another example is the functor {{nowrap|'''Rng''' → '''Ab'''}} which maps a [[ring (algebra)|ring]] to its underlying additive [[abelian group]]. Morphisms in '''Rng''' ([[ring homomorphism]]s) become morphisms in '''Ab''' (abelian group homomorphisms).
; Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor {{nowrap|''F'' : '''Set''' → '''Grp'''}} sends every set ''X'' to the [[free group]] generated by ''X''. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See [[free object]].
; Homomorphism groups: To every pair ''A'', ''B'' of [[group (mathematics)|abelian groups]] one can assign the abelian group Hom(''A'', ''B'') consisting of all [[group homomorphism]]s from ''A'' to ''B''. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor {{nowrap|'''Ab'''<sup>op</sup> × '''Ab''' → '''Ab'''}} (where '''Ab''' denotes the [[category of abelian groups]] with group homomorphisms). If {{nowrap|''f'' : ''A''<sub>1</sub> → ''A''<sub>2</sub>}} and {{nowrap|''g'' : ''B''<sub>1</sub> → ''B''<sub>2</sub>}} are morphisms in '''Ab''', then the group homomorphism {{nowrap|Hom(''f'', ''g'')}}: {{nowrap|Hom(''A''<sub>2</sub>, ''B''<sub>1</sub>) → Hom(''A''<sub>1</sub>, ''B''<sub>2</sub>)}} is given by {{nowrap|''φ'' ↦ ''g'' ∘ ''φ'' ∘ ''f''}}. See [[Hom functor]].
; Representable functors: We can generalize the previous example to any category ''C''. To every pair ''X'', ''Y'' of objects in ''C'' one can assign the set {{nowrap|Hom(''X'', ''Y'')}} of morphisms from ''X'' to ''Y''. This defines a functor to '''Set''' which is contravariant in the first argument and covariant in the second, i.e. it is a functor {{nowrap|''C''<sup>op</sup> × ''C'' → '''Set'''}}. If {{nowrap|''f'' : ''X''<sub>1</sub> → ''X''<sub>2</sub>}} and {{nowrap|''g'' : ''Y''<sub>1</sub> → ''Y''<sub>2</sub>}} are morphisms in ''C'', then the map {{nowrap|Hom(''f'', ''g'') : Hom(''X''<sub>2</sub>, ''Y''<sub>1</sub>) → Hom(''X''<sub>1</sub>, ''Y''<sub>2</sub>)}} is given by {{nowrap|''φ'' ↦ ''g'' ∘ ''φ'' ∘ ''f''}}.{{paragraph}} Functors like these are called [[representable functor]]s. An important goal in many settings is to determine whether a given functor is representable.