Category (mathematics)

Definition

In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

Related concepts

2-functor 2-group 2-ring 3-category Abelian category Abelian group Additive category Adjoint functors Algebraic category Algebraic structure Allegory (category theory)Antisymmetric relation Associative Associative property Associativity Automorphism Automorphism group Bicategory Bilinear form Bimorphism Binary operation Binary relation Bundle map Cartesian closed category Categories for the Working Mathematician Categorification Category (disambiguation)Category of Markov kernels Category of abelian groups Category of fields Category of graphs Category of groups Category of magmas Category of manifolds Category of measurable spaces Category of metric spaces Category of modules Category of monoids Category of preordered sets Category of relations Category of representations Category of rings Category of schemes Category of sets Category of small categories Category of topological spaces Category theory Charles Wells (mathematician)Class (set theory)Closure (mathematics)Coequalizer Cokernel Colimit Comma category Commutative diagram Complete category Complete partial order Computer science Concrete category Cone (category theory)Conservative functor Continuous function (topology)Continuous map Continuously differentiable Coproduct Derived functor Diagonal functor Diagram (category theory)Direct limit Directed graph Discrete category Duality (mathematics)En-ring Encyclopedia of Mathematics End (category theory)Endomorphism Enriched category Epimorphism Equaliser (mathematics)Equivalence of categories Equivalence relation Equivariant map Essentially surjective functor European Mathematical Society Exact functor Exponential object Fiber bundle Field (mathematics)Field homomorphism Forgetful functor Free category Full and faithful functors Full subcategory Function (mathematics)Function composition Functor Functor category Fundamental group Fundamental groupoid GNU Free Documentation License Generating set Glossary of category theory Graph (discrete mathematics)Graph homomorphism Graph theory Group (mathematics)Group action (mathematics)Group homomorphism Group representations Groupoid Higher-dimensional algebra Higher category theory Homotopy Homotopy hypothesis Identity element Identity function Initial and terminal objects Inverse limit Isomorphism Isomorphism of categories K-Vect Kan extension Kernel (category theory)Kleisli category Limit (category theory)Linear map Localization of a category Loop (algebra)Loop (graph theory)Magma (algebra)Markov kernel Mathematical object Mathematics Measurable function Measure space Metric space Michael Barr (mathematician)Model category Module (mathematics)Module homomorphism Monoid Monoidal category Monoidal functor Monoids Monomorphism Morphism Morphism of schemes N-group (category theory)N-monoid N-tuple NLab Nathan Jacobson Natural transformation Natural transformations Opposite category Ordinal number Outline of category theory Overcategory Partial groupoid Partially ordered set Pre-abelian category Preadditive category Preorder Product (category theory)Product category Proper class Pseudo-functor Pullback (category theory)Pushout (category theory)Quantaloid Quasi-category Quasigroup Quiver (mathematics)Quotient category Reflexive relation Relation (mathematics)Representable functor Retract (category theory)Ring (mathematics)Ring homomorphism Saunders Mac Lane Scheme (mathematics)Scott continuity Section (category theory)Semantics of programming languages Semigroup Semigroupoid Set (mathematics)Set theory Short map Simplex category Smooth functor Smooth manifold Space (mathematics)Stanford Encyclopedia of Philosophy Strict 2-category Strict n-category String diagram Structure (mathematics)Subcategory Symmetric monoidal category Table of mathematical symbols Tetracategory Thin category Topological space Topos Total function Total order Traced monoidal category Transitive relation Tricategory Unital magma Universal construction Universal property Vector space Vertex (graph theory)Weak n-category William Lawvere Yoneda lemma Zero morphism ∞-groupoid ∞-topos

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