FUNCTOR

Etymology

Noun

functor (plural functors)

(grammar) A function word.

(object-oriented programming) A function object.

(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, $$\mathbb{CAT}$$ , the objects are categories and the morphisms are functors.

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

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

Source: Wiktionary