In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term in a linguistic context;see function word. Let C and D be categories. A functor F from C to D is a mapping that That is, functors must preserve identity morphisms and composition of morphisms. There are many constructions in mathematics that 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