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| {{About|the mathematical concept}}
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| In [[mathematics]], a '''functor''' is a type of mapping between [[Category (mathematics)|categories]], which is applied in [[category theory]]. Functors can be thought of as [[homomorphism]]s between categories. In the [[category of small categories]], functors can be thought of more generally as [[morphism]]s.
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| Functors were first considered in [[algebraic topology]], where algebraic objects (like the [[fundamental group]]) are associated to [[topological space]]s, and algebraic homomorphisms are associated to [[continuous function|continuous]] maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areas within mathematics that [[category theory]] can make an abstraction of.
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| The word ''functor'' was borrowed by mathematicians from the [[philosopher]] [[Rudolf Carnap]],<ref>{{citation|first1=Saunders|last1=Mac Lane|authorlink1=Saunders Mac Lane|title=Categories for the Working Mathematician|publisher=Springer-Verlag: New York|year=1971|isbn=978-3-540-90035-1|page=30}}</ref> who used the term in a [[Linguistics|linguistic]] context:<ref>Carnap, The Logical Syntax of Language, p. 13–14, 1937, Routledge & Kegan Paul</ref>
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| see [[function word]].
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| ==Definition==
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| Let ''C'' and ''D'' be [[category (mathematics)|categories]]. A '''functor''' ''F'' from ''C'' to ''D'' is a mapping that<ref>Jacobson (2009), p. 19, def. 1.2.</ref>
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| * associates to each object <math>X \in C</math> an object <math>F(X) \in D</math>,
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| * associates to each morphism <math>f:X\rightarrow Y \in C</math> a morphism <math>F(f):F(X) \rightarrow F(Y) \in D</math> such that the following two conditions hold:
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| ** <math>F(\mathrm{id}_{X}) = \mathrm{id}_{F(X)}\,\!</math> for every object <math>X \in C</math>
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| ** <math>F(g \circ f) = F(g) \circ F(f)</math> for all morphisms <math>f:X \rightarrow Y\,\!</math> and <math>g:Y\rightarrow Z.\,\!</math>
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| That is, functors must preserve [[Morphism#Definition|identity morphisms]] and [[Function composition|composition]] of morphisms.
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| ===Covariance and contravariance===
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| There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a '''contravariant functor''' ''F'' from ''C'' to ''D'' as a mapping that
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| *associates to each object <math>X \in C</math> an object <math>F(X) \in D,</math>
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| *associates to each morphism <math>f:X\rightarrow Y \in C</math> a morphism <math>F(f):F(Y) \rightarrow F(X) \in D</math> such that
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| **<math>F(\mathrm{id}_X) = \mathrm{id}_{F(X)}\,\!</math> for every object <math>X \in C</math>,
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| **<math>F(g \circ f) = F(f) \circ F(g)</math> for all morphisms <math>f:X\rightarrow Y\,\!</math> and <math>g:Y\rightarrow Z.\,\!</math>
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| Note that contravariant functors reverse the direction of composition.
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| Ordinary functors are also called '''covariant functors''' in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a ''covariant'' functor on the [[opposite category]] <math>C^\mathrm{op}</math>.<ref>Jacobson (2009), p. 19–20.</ref> Some authors prefer to write all expressions covariantly. That is, instead of saying <math>F: C\rightarrow D</math> is a contravariant functor, they simply write <math>F: C^{\mathrm{op}} \rightarrow D</math> (or sometimes <math>F:C \rightarrow D^{\mathrm{op}}</math>) and call it a functor.
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| Contravariant functors are also occasionally called ''cofunctors''.
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| ===Opposite functor===
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| Every functor <math>F: C\rightarrow D</math> induces the '''opposite functor''' <math>F^\mathrm{op}: C^\mathrm{op}\rightarrow D^\mathrm{op}</math>, where <math>C^\mathrm{op}</math> and <math>D^\mathrm{op}</math> are the [[opposite category|opposite categories]] to <math>C</math> and <math>D</math>.<ref>{{citation|first1=Saunders|last1=Mac Lane|authorlink1=Saunders Mac Lane|first2=Ieke|last2=Moerdijk|authorlink2=Ieke Moerdijk|title=Sheaves in geometry and logic: a first introduction to topos theory|publisher=Springer|year=1992|isbn=978-0-387-97710-2}}</ref> By definition, <math>F^\mathrm{op}</math> maps objects and morphisms identically to <math>F</math>. Since <math>C^\mathrm{op}</math> does not coincide with <math>C</math> as a category, and similarly for <math>D</math>, <math>F^\mathrm{op}</math> is distinguished from <math>F</math>. For example, when composing <math>F: C_0\rightarrow C_1</math> with <math>G: C_1^\mathrm{op}\rightarrow C_2</math>, one should use either <math>G\circ F^\mathrm{op}</math> or <math>G^\mathrm{op}\circ F</math>. Note that, following the property of [[opposite category]], <math>(F^\mathrm{op})^\mathrm{op} = F</math>.
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| ===Bifunctors and multifunctors===
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| A '''bifunctor''' (also known as a '''binary functor''') is a functor in ''two'' arguments. The [[Hom functor]] is a natural example; it is contravariant in one argument, covariant in the other.
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| Formally, a bifunctor is a functor whose domain is a [[product category]]. For example, the Hom functor is of the type ''C''<sup>op</sup> × ''C'' → '''Set'''.
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| A '''multifunctor''' is a generalization of the functor concept to ''n'' variables. So, for example, a bifunctor is a multifunctor with ''n'' = 2.
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| ==Examples==
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| '''[[Diagram (category theory)|Diagram]]''': For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor <math>D:J\rightarrow C</math>.
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| '''[[Presheaf (category theory)|(Category theoretical) presheaf]]''': For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor <math>D:C\rightarrow J</math>.
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| '''Presheaves:''' 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 ''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''.
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| '''Constant functor:''' The functor ''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.
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| '''Endofunctor''': A functor that maps a category to itself.
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| '''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. Identity functor is an endofunctor.
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| '''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.
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| '''Limit functor''': For a fixed [[index category]] ''J'', if every functor ''J''→''C'' has a [[limit (category theory)|limit]] (for instance if ''C'' is complete), then the limit functor ''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 is covariant).
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| '''Power sets:''' The power set functor ''P'' : '''Set''' → '''Set''' maps each set to its [[power set]] and each function <math> f : X \to Y</math> to the map which sends <math>U \subseteq X</math> to its image <math>f(U) \subseteq Y</math>. One can also consider the contravariant power set functor which sends <math> f : X \to Y </math> to the map which
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| sends <math>V \subseteq Y</math> to its [[inverse image]] <math>f^{-1}(V) \subseteq X.</math>
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| '''{{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.
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| '''Fundamental group:''' Consider the category of [[pointed topological space]]s, i.e. topological spaces with distinguished points. The objects are pairs (''X'', ''x''<sub>0</sub>), where ''X'' is a topological space and ''x''<sub>0</sub> is a point in ''X''. A morphism from (''X'', ''x''<sub>0</sub>) to (''Y'', ''y''<sub>0</sub>) is given by a [[continuous function (topology)|continuous]] map ''f'' : ''X'' → ''Y'' with ''f''(''x''<sub>0</sub>) = ''y''<sub>0</sub>.
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| 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 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>. If ''f'' : ''X'' → ''Y'' 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 π(''X'', ''x''<sub>0</sub>) to π(''Y'', ''y''<sub>0</sub>). We thus obtain a functor from the category of pointed topological spaces to the [[category of groups]].
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| 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.
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| '''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 ''f'' : ''X'' → ''Y'' induces an [[algebra homomorphism]] C(''f'') : C(''Y'') → C(''X'') by the rule C(''f'')(φ) = φ o ''f'' for every φ in C(''Y'').
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| '''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.
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| 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.
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| '''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]] 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 ''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.
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| '''Lie algebras:''' Assigning to every real (complex) [[Lie group]] its real (complex) [[Lie algebra]] defines a functor.
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| '''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 ''C'' × ''C'' → ''C'' which is covariant in both arguments.<ref>{{citation|first1=Michiel|last1=Hazewinkel|authorlink1=Michiel Hazewinkel|first2=Nadezhda Mikhaĭlovna|last2=Gubareni|authorlink2=Nadezhda Mikhaĭlovna|first3=Nadiya|last3=Gubareni|authorlink3=Nadiya Gubareni|first4=Vladimir V.|last4=Kirichenko|authorlink4=Vladimir V. Kirichenko|title=Algebras, rings and modules|publisher=Springer|year=2004|isbn=978-1-4020-2690-4}}</ref>
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| '''Forgetful functors:''' The functor ''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.<ref>Jacobson (2009), p. 20, ex. 2.</ref> Functors like these, which "forget" some structure, are termed ''[[forgetful functor]]s''. Another example is the functor '''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).
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| '''Free functors:''' Going in the opposite direction of forgetful functors are free functors. The free functor ''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]].
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| '''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 '''Ab'''<sup>op</sup> × '''Ab''' → '''Ab''' (where '''Ab''' denotes the [[category of abelian groups]] with group homomorphisms). If ''f'' : ''A''<sub>1</sub> → ''A''<sub>2</sub> and ''g'' : ''B''<sub>1</sub> → ''B''<sub>2</sub> are morphisms in '''Ab''', then the group homomorphism Hom(''f'',''g'') : Hom(''A''<sub>2</sub>,''B''<sub>1</sub>) → Hom(''A''<sub>1</sub>,''B''<sub>2</sub>) is given by φ {{mapsto}} ''g'' ∘ φ ∘ ''f''. See [[Hom functor]].
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| '''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 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 ''C''<sup>op</sup> × ''C'' → '''Set'''. If ''f'' : ''X''<sub>1</sub> → ''X''<sub>2</sub> and ''g'' : ''Y''<sub>1</sub> → ''Y''<sub>2</sub> are morphisms in ''C'', then the group homomorphism Hom(''f'',''g'') : Hom(''X''<sub>2</sub>,''Y''<sub>1</sub>) → Hom(''X''<sub>1</sub>,''Y''<sub>2</sub>) is given by φ {{mapsto}} ''g'' ∘ φ ∘ ''f''.
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| Functors like these are called [[representable functor]]s. An important goal in many settings is to determine whether a given functor is representable.
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| ==Properties==
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| Two important consequences of the functor [[axiom]]s are:
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| * ''F'' transforms each [[commutative diagram]] in ''C'' into a commutative diagram in ''D'';
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| * if ''f'' is an [[isomorphism]] in ''C'', then ''F''(''f'') is an isomorphism in ''D''.
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| One can compose functors, i.e. if ''F'' is a functor from ''A'' to ''B'' and ''G'' is a functor from ''B'' to ''C'' then one can form the composite functor ''G∘F'' from ''A'' to ''C''. Composition of functors is associative where defined. Identity of composition of functors is identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the [[category of small categories]].
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| A small category with a single object is the same thing as a [[monoid]]: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid [[homomorphism]]s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
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| ==Relation to other categorical concepts==
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| Let ''C'' and ''D'' be categories. The collection of all functors ''C'' → ''D'' form the objects of a category: the [[functor category]]. Morphisms in this category are [[natural transformation]]s between functors.
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| Functors are often defined by [[universal property|universal properties]]; examples are the [[tensor product]], the [[direct sum of modules|direct sum]] and [[direct product]] of groups or vector spaces, construction of free groups and modules, [[direct limit|direct]] and [[inverse limit|inverse]] limits. The concepts of [[limit (category theory)|limit and colimit]] generalize several of the above.
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| Universal constructions often give rise to pairs of [[adjoint functors]].
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| ==Computer implementations==
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| Functors sometimes appear in [[functional programming]]. For instance, [[Haskell (programming language)|Haskell language]] has <code>Functor</code> [[class (computer programming)|class]] where [[Map (higher-order function)#Generalization|<code>fmap</code>]] is a [[polytypic]] function used to map [[function (computer programming)|functions]] (''morphisms'') on an existing class to functions on some new class.
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| ==See also==
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| {{Portal|Category theory}}
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| * [[Functor category]]
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| * [[Kan extension]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7}}.
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| ==External links==
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| {{Wiktionary|functor}}
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| * {{springer|title=Functor|id=p/f042140}}
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| * see {{nlab|id=functor}} and the variations discussed and linked to there.
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| * [[André Joyal]], [http://ncatlab.org/nlab CatLab], a wiki project dedicated to the exposition of categorical mathematics
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| * {{cite web | first = Chris | last = Hillman | title = A Categorical Primer | id = {{citeseerx|10.1.1.24.3264}} | postscript = : }} formal introduction to category theory.
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| * J. Adamek, H. Herrlich, G. Stecker, [http://katmat.math.uni-bremen.de/acc/acc.pdf Abstract and Concrete Categories-The Joy of Cats]
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| * [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/category-theory/ Category Theory]" — by Jean-Pierre Marquis. Extensive bibliography.
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| * [http://www.mta.ca/~cat-dist/ List of academic conferences on category theory]
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| * Baez, John, 1996,"[http://math.ucr.edu/home/baez/week73.html The Tale of ''n''-categories.]" An informal introduction to higher order categories.
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| * [http://wildcatsformma.wordpress.com WildCats] is a [[category theory]] package for [[Mathematica]]. Manipulation and visualization of objects, [[morphism]]s, categories, functors, [[natural transformation]]s, [[universal properties]].
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| * [http://www.youtube.com/user/TheCatsters The catsters], a YouTube channel about category theory.
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| *{{planetmath reference|id=5622|title=Category Theory}}
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| * [http://categorieslogicphysics.wikidot.com/events Video archive] of recorded talks relevant to categories, logic and the foundations of physics.
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| *[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page] which generates examples of categorical constructions in the category of finite sets.
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| {{Functors}}
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| [[Category:Functors| ]]
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| [[de:Kategorientheorie#Funktor]]
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