Isomorphism

Definition

In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as ⁠⁠. The word is derived from Ancient Greek ἴσος 'equal' and μορφή 'form, shape'.

Related concepts

2-functor 2-group 2-ring 3-category ABO blood group system Abelian category Abelian group Additive category Adjoint functors Affine transformation Algebra Algebraic category Algebraic structure Ancient Greek language Antisymmetric relation Asymmetric relation Automorphism Balanced category Barry Mazur Bertrand Russell Bicategory Bijection Bijective Binary relation Bisimulation Canonical map Cartesian closed category 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 topological spaces Category theory Chinese remainder theorem Classification of finite simple groups Closed surface Coequalizer Coherent isomorphism Cokernel Colimit Comma category Commutative diagram Complete category Computer language Concrete category Cone (category theory)Conjugacy class Connected relation Connected space Conservative functor Coprime Coproduct Covering space Cybernetics Derived functor Diagonal functor Diagram (category theory)Diffeomorphism Differentiable manifold Differential equations Differential structure Direct limit En-ring Encyclopedia of Mathematics End (category theory)Enriched category Epimorphism Equaliser (mathematics)Equality (mathematics)Equivalence class Equivalence of categories Equivalence relation Eric W. Weisstein Essentially surjective functor European Mathematical Society Even number Exact functor Exponential function Exponential object Extensional definition Field (mathematics)Field automorphism Finite-dimensional vector space Finite group Finite simple groups Forgetful functor Free category Full and faithful functors Functor Functor category Fundamental group Galois theory Geometry Glossary of category theory Good regulator theorem Graph isomorphism Graph theory Group (mathematics)Group homomorphism Group isomorphism Heap (mathematics)Higher-dimensional algebra Higher category theory Hilbert space Homeomorphism Homogeneous relation Homomorphism Homotopy hypothesis Homotopy theory If and only if Initial and terminal objects Integer Intensional definition Introduction to Mathematical Philosophy Inverse function Inverse limit Invertible matrices Irreflexive relation Isometry Isomorphism (disambiguation)Isomorphism class Isomorphism of categories Isomorphism theorem Isomorphism theorems Kan extension Kernel (category theory)Kleisli category Laplace transform Limit (category theory)Linear isomorphism Linear subspace Localization of a category Logarithm function Logical atomism Loop (topology)Ludwig Wittgenstein Map (mathematics)MathWorld Mathematical analysis Mathematical jargon Mathematical object Mathematical structure Mathematics Metric space Model category Modular arithmetic Monoidal category Monoidal functor Monomorphism Morphism Multiplicative group N-group (category theory)N-monoid Natural transformation Opposite category Order isomorphism Order theory Ordered set Ordinal number Outline of category theory Overcategory Partial order Permutation Positive real numbers Pre-abelian category Preadditive category Prime number Product (category theory)Product category Projective transformation Proper subset Pseudo-functor Pullback (category theory)Pushout (category theory)Quasi-category Quotient category Rational number Real number Reflexive relation Representable functor Rigid transformation Ring (mathematics)Ring isomorphism Roots of unity Ruler Set (mathematics)Set builder notation Simplex category Slide rule Smooth functor Steve Awodey Strict 2-category Strict n-category Strict weak order String diagram Subalgebra Subcategory Subgroup Substructure (mathematics)Surface (topology)Symmetric monoidal category Symmetric relation Symplectic manifold Symplectomorphism Table of logarithms Tetracategory Topological space Topos Total order Traced monoidal category Transformation (function)Transitive relation Tricategory Type conversion Type theory Underlying set Universal construction Universal properties Universal property Up to Variety (universal algebra)Vector space Vertex (graph theory)Weak n-category Well-order Yoneda lemma Zero morphism ∞-groupoid ∞-topos

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