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In Category Theory , a functor is a special type of mapping between categories. Functors can be thought of as Morphism s in the Category Of Small Categories .

Functors were first considered in Algebraic Topology , where algebraic objects (like the Fundamental Group ) are associated to Topological Space s, and algebraic Homomorphism s are associated to Continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories.


DEFINITION


Let ''C'' and ''D'' be Categories . A functor ''F'' from ''C'' to ''D'' is a mapping that
  • associates to each object ''X'' in ''C'' an object ''F''(''X'') in ''D'',

  • associates to each morphism ''f'' : ''X'' → ''Y'' in ''C'' a morphism ''F''(''f'') : ''F''(''X'') → ''F''(''Y'') in ''D''

    • such that the following two properties hold:

  • F(id_{X}) = id_{F(X)} for every object X \in C

  • F(g \circ f) = F(g) \circ F(f) for all morphisms f:X ightarrow Y and g:Y ightarrow Z.


That is, functors must preserve identity morphisms and composition of morphisms.


Covariance and contravariance


There are many constructions in mathematics which 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
  • associates to each object X \in C an object F(X) \in D,

  • associates to each morphism f:X ightarrow Y \in C a morphism F(f):F(Y) ightarrow F(X) \in D such that

  • ---F(id_X)=id_{F(X)} for every object X \in C,

  • ---F(g \circ f) = F(f) \circ F(g) for all morphisms f:X ightarrow Y and g:Y ightarrow Z.


Note that contravariant functors reverse the direction of composition.

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 Dual Category C^{op}. Some authors prefer to write all expressions covariantly. That is, instead of saying F: C ightarrow D is a contravariant functor, they simply write F: C^{op} ightarrow D (or sometimes F:C ightarrow D^{op}) and call it a functor.

Contravariant functors are also occasionally called ''cofunctors''. This is a perhaps misleading usage of the prefix "co", which in a categorical context usually means "reverse all arrows". Recall that a functor ''F'' maps a morphism ''f'':''X''→''Y'' to a morphism ''F(f)'':''F(X)''→''F(Y)''. Reversing all the arrows would map morphisms ''f'':''X''←''Y'' to morphisms ''F(f)'':''F(X)''←''F(Y)'', but by transposing ''X'' and ''Y'' we see that this gives the same object as before. Consequently a cofunctor, under a strict interpretation, would be the same type of object as a functor. Nevertheless, this use of the prefix "co" is not without precedent. For example, some authors call elements of a Dual Space covectors.


EXAMPLES


Constant functor: A very boring functor ''C'' → ''D'' is one 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.

Diagonal functor: The Diagonal Functor is defined as the functor from ''D'' to the functor category ''D''''C'' which sends each object in ''D'' to the constant functor at that object.

Limit functor: For a fixed Index Category ''J'', if every functor ''J''→''C'' has a Limit (for instance if ''C'' is complete), then the limit functor ''C''''J''→''C'' assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the 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).

Power sets: The power set functor ''P'' : '''Set''' → '''Set''' maps each set to its in ''Y''.

Dual vector space: The map which assigns to every Vector Space its Dual Space and to every Linear Map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed Field to itself.

Fundamental group: Consider the category of Pointed Topological Space s, i.e. topological spaces with distinguished points. The objects are pairs (''X'', ''x''0), where ''X'' is a topological space and ''x''0 is a point in ''X''. A morphism from (''X'', ''x''0) to (''Y'', ''y''0) is given by a Continuous map ''f'' : ''X'' → ''Y'' with ''f''(''x''0) = ''y''0.

To every topological space ''X'' with distinguished point ''x''0, one can define the s, then every loop in ''X'' with base point ''x''0 can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''0. This operation is compatible with the homotopy Equivalence Relation and the composition of loops, and we get a Group Homomorphism from π(''X'', ''x''0) to π(''Y'', ''y''0). We thus obtain a functor from the category of pointed topological spaces to the Category Of Groups .

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 C(''f'') : C(''Y'') → C(''X'') by the rule C(''f'')(φ) = φ o ''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. Likewise, the map which sends every differentiable manifold to its Cotangent Bundle and every smooth map to its Pullback is a contravariant functor.

Doing these constructions pointwise gives covariant and contravariant functors from the category of pointed differentiable manifolds to the category of real vector spaces.

Group actions/representations: Every Group (or Groupoid ) ''G'' can be considered as a category with a single object. A functor from ''G'' to '''Set''' is 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'''''K'', 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''.

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 Maps as morphisms, then the Tensor Product V \otimes W defines a functor ''C'' × ''C'' → ''C'' which is covariant in both arguments.

Forgetful functors: The functor ''U'' : '''Grp''' → '''Set''' which maps a Group to its underlying set and a Group Homomorphism to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed '' Forgetful Functor s''. Another example is the functor '''Rng''' → '''Ab''' which maps a 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 ''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 .