Category Theory

The study of structure-preserving maps between mathematical objects, expressed through objects, arrows (morphisms), and functors.

• A category consists of objects and composable morphisms satisfying identity and associativity
• functors map between categories, preserving composition and identity
• natural transformations map between functors, forming morphisms of morphisms
• isomorphism identifies structurally identical objects regardless of representation
• adjunctions capture universal construction patterns across mathematics
• The Yoneda lemma states that an object is fully determined by its relationships to all other objects
• Unifying language for set theory, algebra, topology, logic, and type theory
• Related: graph theory, linear algebra, information theory