Category theory

A branch of mathematics that studies objects and structure-preserving morphisms, emphasizing how morphisms compose and how mathematical structures relate.

Category: A structure consisting of objects, morphisms between objects, identity morphisms, and associative composition. This is the basic structure category theory studies.
Composition of morphisms: The operation that combines consecutive morphisms into a single morphism. Associative composition is one of the defining laws of a category.
Homological algebra: The study of algebraic structures and invariants using chain complexes and homology. Categories and functors clarify how homological constructions relate across settings.
Samuel Eilenberg: A Polish-American mathematician who helped establish category theory and algebraic topology. Eilenberg co-introduced categories and functors to formalize constructions in topology.
Type theory: A formal system that organizes mathematical and computational expressions by types and rules of construction. Type theory offers a distinct foundation that can also express categorical structures.
Morphism: A structure-preserving map between objects in a mathematical category. Morphisms encode the relationships whose composition organizes the theory.
Identity morphism: A morphism from an object to itself that leaves every composable morphism unchanged. Identity morphisms complete the composition laws for each object.
Algebraic topology: The study of topological spaces through algebraic invariants such as homology and homotopy groups. Functoriality explains how continuous maps induce maps on algebraic invariants.
Saunders Mac Lane: An American mathematician who co-founded category theory and shaped its development. Mac Lane co-created the framework and later systematized its basic language.
Category of sets: The category whose objects are sets and whose morphisms are functions. It is a familiar concrete example, rather than an alternative foundation for every category.

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