What is "functor"

n. 1 (context grammar English) a function word 2 (context computing English) a function object 3 (context category theory English) a structure-preserving mapping between categories: if ''F'' is a functor from category ''C'' to category ''D'', then ''F'' maps objects of ''C'' to objects of ''D'' and morphisms of ''C'' to morphisms of ''D'' such that any morphism ''f'':''X''→''Y'' of ''C'' is mapped to a morphism ''F''(''f''): ''F''(''X'') → ''F''(''Y'') of ''D'', such that if $h\; =\; g\; circ\; f$ then $F(h)\; =\; F(g)\; circ\; F(f)$, and such that identity morphisms (and only identity morphisms) are mapped to identity morphisms. Note: the functor just described is covariant.

In mathematics, a functor is a type of mapping between categories which is applied in category theory. Functors can be thought of as homomorphisms between categories. In the category of small categories, functors can be thought of more generally as morphisms.

Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areas within mathematics that category theory can make an abstraction of.

The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term in a linguistic context: see function word.

Functor may refer to:

• Functor in mathematics, a concept in the category theory
• Predicate functor in logic, a basic concept of predicate functor logic
• Function word in linguistics
• In computer programming:
  • function object used to pass function pointers along with its state
  • for use of the term in Prolog language, see Prolog syntax and semantics
  • In Haskell a Functor describes a generalisation of functions that perform mapping operations