Functor

A functor maps objects and morphisms from one category to another while preserving identity morphisms and composition.

Category: A structure consisting of objects and morphisms, with identity morphisms and associative composition. Functors act between categories and preserve their identities and composition.
Object (category theory): An element of a category connected to other objects by morphisms. A functor assigns each source object a target object.
Natural transformation: A family of morphisms relating two functors while commuting with every source morphism. It relates functors rather than mapping objects and morphisms between categories.
Fundamental group: A group encoding the loop structure of a pointed topological space up to continuous deformation. The fundamental group construction maps continuous maps to group homomorphisms.
Samuel Eilenberg: A Polish-American mathematician who helped found category theory and homological algebra. Eilenberg introduced functors with Saunders Mac Lane in their 1945 work.
Identity morphism: A morphism from an object to itself that acts as an identity under composition. A functor must map each identity morphism to the identity of its image object.
Morphism: A structure-preserving map between objects in a mathematical setting. A functor assigns each source morphism a target morphism.
Contravariant functor: A map that reverses morphism direction while preserving identities and reversing composition order. It reverses arrows, unlike the ordinary direction-preserving functor defined here.
Homology (mathematics): A family of algebraic invariants measuring holes in topological spaces or chain complexes. Homology assigns algebraic maps to continuous maps in a functorial way.
Saunders Mac Lane: An American mathematician who helped develop category theory and modern algebra. Mac Lane co-introduced functors as part of a framework for structural mathematical relations.

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