Functor

Definition

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in every area of mathematics where category theory is applied.

Related concepts

1-forms 2-functor 2-group 2-ring 3-category Abelian category Abelian group Additive category Adjoint functors Algebra homomorphism Algebraic category Algebraic function Algebraic topology Anafunctor Analytic function André Joyal Aristotle Associative algebra Axiom Axiom of choice Bicategory Bijection Binary relation Boolean-valued function Boolean function Cartesian closed category 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 Category of small categories Category of vector spaces Category theory Coequalizer Cokernel Colimit Comma category Commutative diagram Complete category Complex-valued function Concrete category Cone (category theory)Conservative functor Constant function Continuous function Continuous function (topology)Coproduct Cotangent bundle Cotangent space Covariance and contravariance (computer science)Covariance and contravariance of vectors Derivative Derived functor Diagonal functor Diagram (category theory)Differentiable manifold Direct limit Direct product Direct sum of modules Dual space Einstein summation En-ring Encyclopedia of Mathematics End (category theory)Enriched category Epimorphism Equaliser (mathematics)Equivalence of categories Equivalence relation Essentially surjective functor European Mathematical Society Exact functor Exponential object Field (mathematics)Forgetful functor Free category Free group Free object Freyd adjoint functor theorem Full and faithful functors Function (computer programming)Function (mathematics)Function composition Function of a complex variable Function of a real variable Function of several complex variables Function of several real variables Function space Function word Functional programming Functor (disambiguation)Functor (functional programming)Functor category Fundamental group Glossary of category theory Group (mathematics)Group action (mathematics)Group homomorphism Groupoid Haskell (programming language)Higher-dimensional algebra Higher-order function Higher category theory History of the function concept Hom functor Homomorphism Homotopy Homotopy hypothesis Identity function Ieke Moerdijk Implicit function Index category Initial and terminal objects Injective function Integer-valued function Inverse function Inverse image Inverse limit Isomorphism Isomorphism of categories Kan extension Kernel (category theory)Kleisli category Lambda calculus Langlands program Lie algebra Lie group Limit (category theory)Linear map Linear operator Linear representation Linguistics List of mathematical functions Localization of a category Map (higher-order function)Map (mathematics)Mathematica Mathematical object Mathematics Measurable function Michiel Hazewinkel Model category Monoid Monoidal category Monoidal functor Monomorphism Morphism Multivalued function N-group (category theory)N-monoid NLab Nadezhda Mikhaĭlovna Nadiya Gubareni Nathan Jacobson Natural transformation Open set Opposite category Ordered pair Outline of category theory Overcategory Partial function Partially ordered set Pointed space Pointed topological space Polynomial Polynomial functor Polytypic function Power set Pre-abelian category Preadditive category Presheaf Presheaf (category theory)Product (category theory)Product category Profunctor Pseudo-functor Pseudofunctor Pullback (category theory)Pushout (category theory)Quasi-category Quotient category Rational function Real-valued function Relation (mathematics)Representable functor Restriction (mathematics)Ring (algebra)Ring homomorphism Rudolf Carnap Saunders Mac Lane Sequence Set-valued function Simplex category Smooth function Smooth functor Smooth map Stanford Encyclopedia of Philosophy Strict 2-category Strict n-category String diagram Subcategory Surjective function Symmetric monoidal category Tangent bundle Tangent space Tensor product Tetracategory Topological space Topology Topos Traced monoidal category Tricategory Trivial topology Type class Universal construction Universal properties Universal property Vector bundle Vector field Vector space Vladimir V. Kirichenko Weak n-category Yoneda lemma Zero morphism ∞-groupoid ∞-topos

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