Editing Functor (section)

{{short description|Mapping between categories}}
{{About|the mathematical concept}}
{{Redirect|Functoriality|the Langlands functoriality conjecture in number theory|Langlands program#Functoriality}}

In [[mathematics]], specifically [[category theory]], a '''functor''' is a [[Map (mathematics)|mapping]] between [[Category (mathematics)|categories]].  Functors were first considered in [[algebraic topology]], where algebraic objects (such as the [[fundamental group]]) are associated to [[topological space]]s, and maps between these algebraic objects are associated to [[continuous function|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 words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers [[Aristotle]] and [[Rudolf Carnap]], respectively.<ref>{{citation|first1=Saunders|last1=Mac Lane|author-link1=Saunders Mac Lane|title=Categories for the Working Mathematician|publisher=Springer-Verlag|location=New York|year=1971|isbn=978-3-540-90035-1|page=30}}</ref>  The latter used ''functor'' in a [[Linguistics|linguistic]] context;<ref>[[Rudolf Carnap|Carnap, Rudolf]] (1937). ''The Logical Syntax of Language'', Routledge & Kegan, pp.&nbsp;13–14.</ref>
see [[function word]].