This article is about the mathematical concept. For other uses, see Functor (disambiguation).
"Functoriality" redirects here. For the Langlands functoriality conjecture in number theory, see Langlands program § Functoriality.
In mathematics, specifically category theory, a functor is a mapping 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 words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively.[1] The latter used functor in a linguistic context;[2]
see function word.
^Mac Lane, Saunders (1971), Categories for the Working Mathematician, New York: Springer-Verlag, p. 30, ISBN 978-3-540-90035-1
^Carnap, Rudolf (1937). The Logical Syntax of Language, Routledge & Kegan, pp. 13–14.
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...
relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in...
between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category...
mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition...
contravariant functor acts as a covariant functor from the opposite category Cop to D. A natural transformation is a relation between two functors. Functors often...
calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes...
particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations...
mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure...
respect to some universe) and the morphisms functors. Fct(C, D), the functor category: the category of functors from a category C to a category D. Set, the...
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...
category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an...
Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's...
theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two...
properties. An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between...
geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each...
In some languages, particularly C++, function objects are often called functors (not related to the functional programming concept). A typical use of a...
composition are as in C. There is an obvious faithful functor I : S → C, called the inclusion functor which takes objects and morphisms to themselves. Let...
is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation...
a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to...
{\displaystyle C} and D {\displaystyle D} are preadditive categories, then a functor F : C → D {\displaystyle F:C\rightarrow D} is additive if it too is enriched...
Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal...
that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that...
of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category. A functor F : C → D...
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