Category theory

Definition

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Related concepts

2-category 2-functor 2-group 2-ring 3-category ACM Computing Classification System Abelian category Abstract algebra Additive category Adjoint functors Algebra Algebraic category Algebraic geometry Algebraic number theory Algebraic structure Algebraic topology Algorithm Algorithm design Algorithmic efficiency Analysis of algorithms Analytic geometry Analytic number theory Application security Applied category theory Applied mathematics Arithmetic Arithmetic geometry Artificial intelligence Associativity Augmented reality Automata theory Automated planning and scheduling Automorphism Axiom of reducibility Axiomatic set theory Axiomatic system Benjamin C. Pierce Bicategory Binary operation Calculus Cambridge University Press Carlos Simpson Cartesian closed category Categorical abstract machine Categorical logic Categorical theory Categories for the Working Mathematician Categorification Category (mathematics)Category of abelian groups Category of groups Category of magmas Category of modules Category of relations Category of rings Category of sets Charles Wells (mathematician)Class (set theory)Coequalizer Cokernel Colimit Combinatorics Comma category Communication protocol Commutative algebra Commutative diagram Compiler construction Complete category Complex analysis Complex geometry Composition of functions Computability theory Computational biology Computational chemistry Computational complexity Computational complexity theory Computational engineering Computational geometry Computational intelligence Computational mathematics Computational physics Computational problem Computational social science Computer accessibility Computer algebra Computer animation Computer architecture Computer data storage Computer graphics Computer hardware Computer network Computer science Computer security Computer vision Computing Computing platform Concrete category Concurrency (computer science)Concurrent computing Cone (category theory)Conservative functor Constructive set theory Constructivism (mathematics)Contravariant functor Control flow Control theory Coproduct Cross-validation (statistics)Cryptography Cyber-physical system Cyberwarfare Data mining Database Decision support system Dependability Dependent type Derived functor Descriptive set theory Determinacy Diagonal functor Diagram (category theory)Differential equation Differential geometry Differential topology Digital art Digital library Digital marketing Diophantine geometry Direct limit Direct product Discrete geometry Discrete mathematics Distributed artificial intelligence Distributed computing Document management system Domain-specific language Domain theory Dual (category theory)Duality (mathematics)E-commerce Educational technology Electronic design automation Electronic publishing Electronic voting Elementary algebra Embedded system Emmy Noether Empty set En-ring End (category theory)Endomorphism Engineering mathematics Enriched category Enterprise information system Enterprise software Epimorphism Equaliser (mathematics)Equivalence of categories Essentially surjective functor Euclidean geometry Exact functor Exponential object Fault tolerance Feynman diagrams Finite geometry Forgetful functor Form factor (design)Formal language Formal methods Formal system Foundations of mathematics Free category Full and faithful functors Function (mathematics)Function composition Functional analysis Functional programming Functor Functor category Future of mathematics Game theory Garrett Birkhoff General topology Geographic information system Geometric topology Geometry Girard's paradox Glossary of category theory Glossary of mathematical symbols Graph theory Graphics processing unit Green computing Group theory Guerino Mazzola Hardware acceleration Hardware security Harmonic analysis Health informatics Higher-dimensional algebra Higher Topos Theory Higher category theory Hilbert system History of mathematics History of type theory Homological algebra Homology (mathematics)Homomorphism Homotopy hypothesis Homotopy theory Homotopy type theory Horst Herrlich Human-centered computing Human–computer interaction Hypercomplex analysis Identity (mathematics)Identity element Identity function Identity morphism Image compression Industrial process control Informal mathematics Information retrieval Information security Information system Information theory Initial and terminal objects Integrated circuit Integrated development environment Interaction design Interpreter (computing)Intrusion detection system Intuitionistic logic Intuitionistic type theory Inverse limit Isomorphism Isomorphism of categories Jacob Lurie John Baez Joseph Goguen Kan extension Kernel (category theory)Kleisli category Knot theory Knowledge representation and reasoning Lambda calculus Large numbers Library (computing)Limit (category theory)Linear algebra List of category theory topics List of computer size categories List of films about mathematicians List of mathematical logic topics List of mathematicians List of publications in mathematics List of set theory topics Lists of mathematics topics Localization of a category Logic in computer science M. Cristina Pedicchio Machine learning Manchester University Map (mathematics)Masaki Kashiwara Mathematica Mathematical Logic Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical constructivism Mathematical economics Mathematical finance Mathematical geology Mathematical induction Mathematical linguistics Mathematical logic Mathematical object Mathematical optimization Mathematical physics Mathematical proof Mathematical psychology Mathematical sociology Mathematical software Mathematical statistics Mathematical structure Mathematicians Mathematics Mathematics and art Mathematics education Measure (mathematics)Michael Barr (mathematician)Middleware Mobile computing Modal logic Model category Model of computation Model theory Modeling language Monoid Monoidal category Monoidal functor Monomorphism Morphism Multi-task learning Multilinear algebra Multimedia database Multiprocessing Multithreading (computer architecture)N-category N-group (category theory)N-monoid Naive set theory Natural deduction Natural language processing Natural number Natural transformation Network architecture Network performance Network scheduler Network security Network service Networking hardware Number Number theory Numerical analysis Object (category theory)Open source Operating system Operations research Opposite category Order theory Ordinal number Outline of category theory Outline of computer science Overcategory Parallel computing Peano axioms Peripheral Peter J. Freyd Philosophy of artificial intelligence Philosophy of mathematics Photograph manipulation Physics Pierre Schapira (mathematician)PlanetMath Pointless topology Pre-abelian category Preadditive category Printed circuit board Probability Probability theory Processor (computing)Product (category theory)Product category Product topology Programming language Programming language theory Programming paradigm Programming team Programming tool Pseudo-functor Pullback (category theory)Pushout (category theory)Quantum computing Quasi-category Quotient category Quotient space (disambiguation)Randomized algorithm Real-time computing Real analysis Recreational mathematics Reinforcement learning Rendering (computer graphics)Representable functor Requirements analysis Retract (category theory)Robert Goldblatt Ronald Brown (mathematician)Russell's paradox Samuel Eilenberg Saunders Mac Lane Scheme theory Section (category theory)Security hacker Security service (telecommunication)Semantics (computer science)Set (mathematics)Set theory Sheaf (mathematics)Simplex category Smooth functor Social computing Social software Software configuration management Software construction Software deployment Software design Software development Software development process Software engineering Software framework Software maintenance Software quality Software repository Solid modeling Stanford Encyclopedia of Philosophy Stanford University Stanisław Ulam Statistics Stephen Schanuel Stochastic computing Strict 2-category Strict n-category String diagram Structuralism (philosophy of mathematics)Subcategory Supervised learning Symmetric monoidal category System on a chip Systems science Tetracategory The Universal Book of Mathematics Theoretical computer science Theory of computation Timeline of category theory and related mathematics Timeline of mathematics Tom Leinster Topological invariant Topology Topos Traced monoidal category Tricategory Type theory Ubiquitous computing Univalent foundations Universal algebra Universal construction Universal properties Universal property Unsupervised learning Very-large-scale integration Video game Virtual machine Virtual reality Visualization (graphics)Weak n-category William Lawvere Wireless sensor network Word processor World Wide Web Yoneda lemma Zermelo set theory Zermelo–Fraenkel set theory Zero morphism Ω (ordinal number)∞-groupoid ∞-topos

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