functor

English

Etymology

From function, modeled after factor.

Pronunciation

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

• Audio (US):

  (file)

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 (either covariantly or contravariantly), in such a way as to preserve morphism composition and identities.

   Hypernym: morphism

   Hyponym: endofunctor

   In the category of categories $𝐂𝐚𝐭$ 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 $U$ the operation of closure, regularization or relative completion, we obtain from a given functor $ℱ\in 𝒥ℱ$ the functors

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

   • 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 $X$ and a functor $u:A\to B$, a homotopical $u$-colimit (resp. $u$-limit) functor on $X$ will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor $X^u:X^B\to X^A$ (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 ${\mathrm{\Sigma }}_{\ast }$-objects and we review the relationship between ${\mathrm{\Sigma }}_{\ast }$-objects and functors. In short, a ${\mathrm{\Sigma }}_{\ast }$-object (in English words, a symmetric sequence of objects, or simply a symmetric object) is the coefficient sequence of a generalized symmetric functor $S(M):X\to S(M,X)$, defined by a formula of the form

     $S(M,X)=\underset{r=0}{\overset{\mathrm{\infty }}{⨁}}{(M(r)\otimes X^{\otimes r})}_{{\mathrm{\Sigma }}_r}$.

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
• anafunctor
• bifunctor
• contravariant functor
• faithful functor
• forgetful functor
• full functor
• functor category
• functorial
• group functor
• identity functor
• overfunctor
• profunctor
• pseudofunctor
• representable functor
• subfunctor
• underlying functor
• Yoneda functor

Translations

grammar: function word — see function word

object-oriented programming: function object

• Finnish: funktio-olio
• French: foncteur (fr) m
• Polish: funktor m inan

category theory: category mapping

• Bashkir: функтор (funktor)
• Chinese:

  Mandarin: 函子 (hánzǐ)

• Finnish: funktori (fi), ylinuoli
• 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 (fuwnktor)
• Kyrgyz: функтор (funktor)
• Polish: funktor m inan
• Portuguese: functor m, funtor m
• Russian: функтор (ru) (funktor)
• Spanish: funtor m, functor 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.

Portuguese

Alternative forms

• funtor

Noun

functor m (plural functores)

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

Further reading

• “functor”, in Dicionário Priberam da Língua Portuguesa (in Portuguese), Lisbon: Priberam, 2008–2026

Romanian

Etymology

Borrowed from French functeur.

Noun

functor m (plural functori)

1. functor

Declension

	singular		plural
	indefinite	definite	indefinite	definite
nominative-accusative	functor	functorul	functori	functorii
genitive-dative	functor	functorului	functori	functorilor
vocative	functorule		functorilor

Spanish

Noun

functor m (plural functores)

1. alternative form of funtor