functor

From function, modeled after factor.

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

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, ${\displaystyle \mathbb {CAT} }$, 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.

• 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.

• funtor

functor m (plural functores)

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

Borrowed from French functeur.

functor m (plural functori)

1. functor