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In [[category theory]], a '''functor''' is a map between categories satisfying certain relations.  Functors can be thought of as expressing relations between categories, and have a wide range of applications.  For example, given a problem about topological spaces, one might apply some functor (say, taking the homology groups), and obtain an equivalent problem about vector spaces.  Since problems about vector spaces are generally much easier to answer than those about general topological spaces, this may make it easier to understand the original problem.

In [[category theory]], a '''functor''' is a map between categories satisfying certain relations.  Functors can be thought of as expressing relations between categories, and have a wide range of applications.  For example, given a problem about topological spaces, one might apply some functor (say, taking the homology groups), and obtain an equivalent problem about vector spaces.  Since problems about vector spaces are generally much easier to answer than those about general topological spaces, this may make it easier to understand the original problem.

Functors, in a sense, provide for categories what group [[homomorphism|homomorphisms]] do for groups.  To be precise, a function <math>F : \mathcal C \to \mathcal D</math> between two categories associates to each object <math>X \in \textrm{Ob } \mathcal C</math> an object <math>F(X) \in \textrm{Ob } \mathcal D</math>, and to each morphism <math>f \in \mathcal C(X,Y)</math> a morphism <math>F(f) \in \mathcal D(FX,FY)</math> such that:

Functors, in a sense, provide for categories what group [[homomorphism]]s do for groups.  To be precise, a function <math>F : \mathcal C \to \mathcal D</math> between two categories associates to each object <math>X \in \textrm{Ob } \mathcal C</math> an object <math>F(X) \in \textrm{Ob } \mathcal D</math>, and to each morphism <math>f \in \mathcal C(X,Y)</math> a morphism <math>F(f) \in \mathcal D(FX,FY)</math> such that:

:<math>F(\textrm{id}_X) = id_{F(X)}\,</math>

:<math>F(\textrm{id}_X) = id_{F(X)}\,</math>

:<math>F(g \circ f) = F(g) \circ F(f)</math>.

:<math>F(g \circ f) = F(g) \circ F(f)</math>.

A simple example arises in the category of vector spaces. Fix a vector space <math>V</math>. We can define a functor <math>\textrm{Hom}(V,-) : \textrm{Vect}_k \to \textrm{Vect}_k</math> by

A simple example arises in the category of vector spaces. Fix a vector space <math>V</math>. We can define a functor <math>\textrm{Hom}(V,-) : \textrm{Vect}_k \to \textrm{Vect}_k</math> by

#An object <math>A</math> (that is a vector space) is sent to <math>\textrm{Hom}(V,A)</math>, the vector space of linear maps from <math>V</math> to <math>A</math>.  We can think of these linear maps as matrices.

#An object <math>A</math> (that is a vector space) is sent to <math>\textrm{Hom}(V,A)</math>, the vector space of linear maps from <math>V</math> to <math>A</math>.  We can think of these linear maps as matrices.

#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>.  I other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.

#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>).  In other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.

We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by

We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by