Category

A category consists of objects and morphisms between them, with an identity morphism for each object and associative composition of composable morphisms.

Morphism: A structure-preserving map between objects in a category, composed according to that category’s rules. Morphisms provide the arrows whose identities and composition complete the category axioms.
Category of sets: The category whose objects are sets and whose morphisms are functions. It is the basic example in which objects and arrows have familiar meanings.
Functor: A map between categories that sends objects and morphisms to objects and morphisms while preserving identities and composition. Functors express structure-preserving relationships between categories.
Samuel Eilenberg: A Polish-American mathematician who helped found category theory and algebraic topology. Eilenberg introduced category theory with Saunders Mac Lane in the 1940s.
Groupoid: A category in which every morphism is invertible. Groupoids add inverses to all arrows, unlike general categories.
Identity morphism: A morphism from an object to itself that leaves every composable morphism unchanged under composition. Each object must have an identity morphism for the category axioms to hold.
Category of groups: The category whose objects are groups and whose morphisms are group homomorphisms. It illustrates how categories encode algebraic structures and structure-preserving maps.
Natural transformation: A family of morphisms between two functors that is compatible with every morphism in their source category. Natural transformations compare functors while respecting the categories’ morphisms.
Saunders Mac Lane: An American mathematician who helped found category theory and shaped its early development. Mac Lane developed the theory’s foundational language and exposition.
Preadditive category: A category whose hom-sets are abelian groups and whose composition is bilinear. Preadditive categories add algebraic structure to morphisms beyond the basic axioms.

Linked from 10 pages