Definition of functor

The term "functor" in mathematics, and specifically in category theory, was coined by the American mathematician Samuel Eilenberg and his collaborator Saunders Mac Lane in the late 1940s. They introduced the concept as a way to abstract and generalize certain algebraic structures, such as groups or rings, to more complex settings. The word "functor" arises from the idea that such mathematical structures can be mapped between categories (sets of objects and arrows between them that satisfy certain rules) in a functory way. A functor is essentially a function that preserves the structure of the category on which it operates, allowing for new categories to be constructed from existing ones. The name "functor" was originally suggested by Mac Lane, who adapted it from the term "operator" used in functional analysis, a branch of mathematics that deals with functions between spaces. In category theory, a functor is viewed as an operator that takes objects and morphisms (maps between objects) from one category and maps them into objects and morphisms in another category while preserving certain properties. In short, the term "functor" has become a standard term in mathematics due to its descriptive and intuitive meaning, reflecting the way in which mathematical structures are transformed and transported between different categories.