functor

functor

English

Etymology

From function, modeled after factor.

Pronunciation

• (Received Pronunciation) IPA^(key): /ˈfʌŋktə/

Noun

functor (plural functors)

1. (grammar) A function word.
2. (object-oriented programming) A function object.
3. (category theory) A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows, in such a way as to preserve domains and codomains (of the arrows) as well as composition and identities.

   Hyponym: endofunctor

   In the category of categories, ${\\displaystyle \\mathbb {CAT} }$, the objects are categories and the morphisms are functors.

   • 1991, Natalie Wadhwa (translator), Yu. A. Brudnyǐ, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, Volume I, Elsevier (North-Holland), page 143,

     Choosing for ${\\displaystyle U}$ the operation of closure, regularization or relative completion, we obtain from a given functor ${\\displaystyle {\\mathcal {F}}\\in {\\mathcal {JF}}}$ the functors

     ${\\displaystyle {\\overline {F}}:{\\overrightarrow {X}}\\rightarrow {\\overline {F({\\overrightarrow {X}})}},F^{0}:{\\overrightarrow {X}}\\rightarrow F({\\overrightarrow {X}})^{0},F^{c}:{\\overrightarrow {X}}\\rightarrow F({\\overrightarrow {X}})^{c}}$.

   • 2004, William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, American Mathematical Society, page 165,

     Given a homotopical category ${\\displaystyle X}$ and a functor ${\\displaystyle u:A\\rightarrow B}$, a homotopical ${\\displaystyle u}$-colimit (resp. ${\\displaystyle u}$-limit) functor on ${\\displaystyle X}$ will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor ${\\displaystyle X^{u}:X^{B}\\rightarrow X^{A}}$ (47.1).

   • 2009, Benoit Fresse, Modules Over Operads and Functors, Springer, Lecture Notes in Mathematics: 1967, page 35,

     In this chapter, we recall the definition of the category of ${\\displaystyle \\Sigma _{*}}$-objects and we review the relationship between ${\\displaystyle \\Sigma _{*}}$-objects and functors. In short, a ${\\displaystyle \\Sigma _{*}}$-object (in English words, a symmetric sequence of objects, or simply a symmetric object) is the coefficient sequence of a generalized symmetric functor ${\\displaystyle S(M):X\\rightarrow S(M,X)}$, defined by a formula of the form

     ${\\displaystyle S(M,X)=\\bigoplus _{r=0}^{\\infty }\\left(M(r)\\otimes X^{\\otimes r}\\right)_{\\Sigma _{r}}}$.

4. (functional programming) A structure allowing a function to apply within a generic type, in a way that is conceptually similar to a functor in category theory.

Derived terms

• adjoint functor
• contravariant functor
• faithful functor
• forgetful functor
• full functor
• functor category
• group functor
• identity functor
• representable functor
• underlying functor
• Yoneda functor

 

Translations

grammar: function word — see function word

object-oriented programming: function object

Finnish: funktio-olio French: foncteur (fr) m

category theory: category mapping

Bashkir: функтор (funktor) Chinese: Mandarin: 函子 (hánzǐ) Finnish: funktori French: foncteur (fr) m German: Funktor m Hungarian: funktor Icelandic: varpi m Irish: feidhmeoir Italian: funtore m Japanese: 関手 (かんしゅ, kanshu) Kazakh: функтор (funktor) Kyrgyz: функтор (funktor) Polish: funktor m Portuguese: functor m, funtor m Russian: функтор (ru) (funktor) Spanish: funtor m Swedish: funktor (sv) c

Further reading

• function word on Wikipedia.Wikipedia
• functor on Wikipedia.Wikipedia
• functor (functional programming) on Wikipedia.Wikipedia
• “functor”, in Lexico, Dictionary.com; Oxford University Press, 2019–2022.

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Portuguese

Alternative forms

• funtor

Noun

functor m (plural functores)

1. (category theory) functor (a mapping between categories)

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Romanian

Etymology

From French functeur.

Noun

functor m (plural functori)

1. functor

Declension

Declension of functor

	singular		plural
	indefinite articulation	definite articulation	indefinite articulation	definite articulation
nominative/accusative	(un) functor	functorul	(niște) functori	functorii
genitive/dative	(unui) functor	functorului	(unor) functori	functorilor
vocative	functorule		functorilor