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add some examples. this article is still completely inaccessible to the uninitiated and probably requires the attentions of an expositor more gifted than i, but i'll look again later
Functors are the fundamental objects used to relate structures between different categories.
Functors are the fundamental objects used to relate structures between different categories.
The above is sometimes called a "covariant functor", to distinguish it from the less common "contravariant functor". In a contravariant functor, the composition order is reversed:
The traditional terminology for the above is a "covariant functor", to distinguish it from "contravariant functors", which reverse the direction of compositions, and instead satisfy
:<math>F(g \circ f) = F(f) \circ F(g)</math>.
:<math>F(g \circ f) = F(f) \circ F(g)</math>.
We will see an example of this below.
=== Examples ===
=== Examples ===
Algebraic topology was the first field in which the usefulness of the notion of a functor was recognized. A basic example is the [[fundamental group]] functor <math>\pi_1 : \textbf{Top}* \to \textbf{Grp}</math>. The action on objects is defined by sending a topological space to its fundamental group <math>\pi_1(X)</math>. Recall that a map between two topological spaces <math>f : X \to Y</math> induces a map <math>f_* : \pi_1(X) \to \pi(Y)</math> by <math>f_*([\gamma]) = [f \circ \gamma]</math>. Set <math>\pi_1(f) = f_*\,</math> so defined. The functoriality of <math>\pi_1</math> boils down to the fact that <math>F(\textrm{id}_X) = \textrm{id}_{\pi_1(X)}</math>, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that <math>F(g \circ f) = F(g) \circ F(f)</math>, explained at [[fundamental group]].
Algebraic topology was the first field in which the usefulness of the notion of a functor was recognized. A basic example is the [[fundamental group]] functor <math>\pi_1 : \textbf{Top}* \to \textbf{Grp}</math>. The action on objects is defined by sending a topological space to its fundamental group <math>\pi_1(X)</math>. Recall that a map between two topological spaces <math>f : X \to Y</math> induces a map <math>f_* : \pi_1(X) \to \pi(Y)</math> by <math>f_*([\gamma]) = [f \circ \gamma]</math>. Set <math>\pi_1(f) = f_*\,</math> so defined. The functoriality of <math>\pi_1</math> boils down to the fact that <math>F(\textrm{id}_X) = \textrm{id}_{\pi_1(X)}</math>, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that <math>F(g \circ f) = F(g) \circ F(f)</math>, explained at [[fundamental group]].
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.
#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>.
We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by
#An object <math>A</math> is sent to the vector space of linear maps <math>\textrm{Hom}(A,V)</math>.
#A map <math>\phi \in \textrm{Hom}(A,B)</math> is sent to a linear map <math>\textrm{Hom}(B,V) \to \textrm{Hom}(A,V)</math> by <math> \phi \mapsto (f \mapsto f \circ \phi)</math>. Given a linear map from B to V, we get one from A to V by composing with <math>\phi</math>. Note that morphisms now compose in the opposite direction: that is why this is a contravariant functor.
==In Computer Science==
==In Computer Science==