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==In Pure Mathematics==

==In Pure Mathematics==

In [[category theory]], a '''functor''' is a map between categories satisfying certain relations.  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:

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:

:<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>.