Functor

In category theory, a functor is a map between categories satisfying certain relations. Functors, in a sense, provide for categories what group 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:

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

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

Functors are the fundamental objects used to relate structures between different categories.

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.