# functor

## English

### Etymology

From function, modeled after factor.

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