TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an isomorphism is a structure-preserving
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
between two
structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ...
of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the
Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An
automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a
canonical map In mathematics, a canonical map, also called a natural map, is a Function (mathematics), map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the wi ...
that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a
universal property In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, all
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
with elements are canonically isomorphic, with a unique isomorphism. The
isomorphism theorems In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
provide canonical isomorphisms that are not unique. The term is mainly used for
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. In this case, mappings are called
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
s, and a homomorphism is an isomorphism
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example: * An
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
is an isomorphism of
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s. * A
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
is an isomorphism of
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s. * A
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is an isomorphism of spaces equipped with a
differential structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, typically
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s. * A
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

is an automorphism of a set. * In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, isomorphisms and automorphisms are often called
transformations Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Transf ...
, for example
rigid transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s,
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
s,
projective transformation In projective geometry, a homography is an isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
s.
Category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.

Examples

Logarithm and exponential

Let $\R^+$ be the
multiplicative group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of
positive real numbersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, and let $\R$ be the additive group of real numbers. The
logarithm function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$\log : \R^+ \to \R$ satisfies $\log\left(xy\right) = \log x + \log y$ for all $x, y \in \R^+,$ so it is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

. The
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

$\exp : \R \to \R^+$ satisfies $\exp\left(x+y\right) = \left(\exp x\right)\left(\exp y\right)$ for all $x, y \in \R,$ so it too is a homomorphism. The identities $\log \exp x = x$ and $\exp \log y = y$ show that $\log$ and $\exp$ are inverses of each other. Since $\log$ is a homomorphism that has an inverse that is also a homomorphism, $\log$ is an isomorphism of groups. The $\log$ function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a
ruler A ruler, sometimes called a rule or line gauge, is a device used in geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of spac ...

and a table of logarithms, or using a
slide rule The slide rule is a mechanical . The slide rule is used primarily for and and for functions such as , , s, and . They are not designed for addition or subtraction which was usually performed manually, with used to keep track of the magnitude ...

with a logarithmic scale.

Integers modulo 6

Consider the group $\left(\Z_6, +\right),$ the integers from 0 to 5 with addition modulo 6. Also consider the group $\left\left(\Z_2 \times \Z_3, +\right\right),$ the ordered pairs where the ''x'' coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the ''x''-coordinate is modulo 2 and addition in the ''y''-coordinate is modulo 3. These structures are isomorphic under addition, under the following scheme: $\begin (0, 0) &\mapsto 0 \\ (1, 1) &\mapsto 1 \\ (0, 2) &\mapsto 2 \\ (1, 0) &\mapsto 3 \\ (0, 1) &\mapsto 4 \\ (1, 2) &\mapsto 5 \\ \end$ or in general $\left(a, b\right) \mapsto \left(3 a + 4 b\right) \mod 6.$ For example, $\left(1, 1\right) + \left(1, 0\right) = \left(0, 1\right),$ which translates in the other system as $1 + 3 = 4.$ Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same. More generally, the
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of two
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

s $\Z_m$ and $\Z_n$ is isomorphic to $\left(\Z_, +\right)$ if and only if ''m'' and ''n'' are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, per the
Chinese remainder theorem In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
.

Relation-preserving isomorphism

If one object consists of a set ''X'' with a
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
R and the other object consists of a set ''Y'' with a binary relation S then an isomorphism from ''X'' to ''Y'' is a bijective function $f : X \to Y$ such that: $\operatorname(f(u),f(v)) \quad \text \quad \operatorname(u,v)$ S is reflexive,
irreflexive In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...
,
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...
, antisymmetric, asymmetric,
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
, total, trichotomous, a
partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ...
,
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a \ ...
,
well-order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
,
strict weak order The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy. In mathematics Mathematics (from Ancient Greek, Gre ...
,
total preorder The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy. In mathematics Mathematics (from Ancient Greek, Gre ...
(weak order), an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, or a relation with any other special properties, if and only if R is. For example, R is an
ordering Order or ORDER or Orders may refer to: * Orderliness, a desire for organization * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements hav ...
≤ and S an ordering $\scriptstyle \sqsubseteq,$ then an isomorphism from ''X'' to ''Y'' is a bijective function $f : X \to Y$ such that $f(u) \sqsubseteq f(v) \quad \text \quad u \leq v.$ Such an isomorphism is called an or (less commonly) an . If $X = Y,$ then this is a relation-preserving
automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.

Applications

In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, isomorphisms are defined for all
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. Some are more specifically studied; for example: *
Linear isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s between
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s; they are specified by
invertible matrices In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
. *
Group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two gr ...
s between
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
; the classification of
isomorphism class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
es of
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s is an open problem. *
Ring isomorphism In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
between
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
. * Field isomorphisms are the same as ring isomorphism between
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
; their study, and more specifically the study of
field automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s is an important part of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
. Just as the
automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
form a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, the isomorphisms between two algebras sharing a common structure form a
heap Heap or HEAP may refer to: Computing and mathematics * Heap (data structure), a data structure commonly used to implement a priority queue * Heap (mathematics), a generalization of a group * Heap (programming) (or free store), an area of memory for ...
. Letting a particular isomorphism identify the two structures turns this heap into a group. In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the
Laplace transform In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable t (often time in physics, time) to a function of a complex analysis, complex variable s (co ...
is an isomorphism mapping hard
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
into easier
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

ic equations. In
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, an isomorphism between two graphs ''G'' and ''H'' is a
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
map ''f'' from the vertices of ''G'' to the vertices of ''H'' that preserves the "edge structure" in the sense that there is an edge from
vertex Vertex (Latin: peak; plural vertices or vertexes) means the "top", or the highest geometric point of something, usually a curved surface or line, or a point where any two geometric sides or edges meet regardless of elevation; as opposed to an Apex ( ...
''u'' to vertex ''v'' in ''G'' if and only if there is an edge from $f\left(u\right)$ to $f\left(v\right)$ in ''H''. See
graph isomorphism In graph theory, an isomorphism of Graph (discrete mathematics), graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are Adjacent (graph theo ...

. In mathematical analysis, an isomorphism between two
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s is a bijection preserving addition, scalar multiplication, and inner product. In early theories of
logical atomism Logical atomism is a philosophical view that originated in the early 20th century with the development of analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fu ...
, the formal relationship between facts and true propositions was theorized by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
and
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationali ...

to be isomorphic. An example of this line of thinking can be found in Russell's ''
Introduction to Mathematical Philosophy ''Introduction to Mathematical Philosophy'' is a book by philosopher Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath, philosopher, Mathematical logic, logician, mathema ...
''. In
cybernetics Cybernetics is a wide-ranging field concerned with regulatory and purposive systems A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influen ...

, the good regulator or Conant–Ashby theorem is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system.

Category theoretic view

In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, given a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
''C'', an isomorphism is a morphism $f : a \to b$ that has an inverse morphism $g : b \to a,$ that is, $f g = 1_b$ and $g f = 1_a.$ For example, a bijective
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is an isomorphism between
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, and a bijective
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
whose inverse is also continuous is an isomorphism between
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s, called a
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
. Two categories and are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
if there exist
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s $F : C \to D$ and $G : D \to C$ which are mutually inverse to each other, that is, $FG = 1_D$ (the identity functor on ) and $GF = 1_C$ (the identity functor on ).

Isomorphism vs. bijective morphism

In a
concrete category In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
(roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the
category of topological spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
or categories of algebraic objects (like the
category of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
, and the
category of modulesIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
), an isomorphism must be bijective on the
underlying set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).

Relation with equality

In certain areas of mathematics, notably category theory, it is valuable to distinguish between on the one hand and on the other. Equality is when two objects are exactly the same, and everything that is true about one object is true about the other, while an isomorphism implies everything that is true about a designated part of one object's structure is true about the other's. For example, the sets $A = \left\ \quad \text \quad B = \$ are ; they are merely different representations—the first an intensional one (in
set builder notation Set, The Set, or SET may refer to: Science, technology, and mathematics Mathematics *Set (mathematics) In mathematics, a set is a collection of Distinct (mathematics), distinct Element (mathematics), elements. The elements that make up a set ...
), and the second
extensionalIn philosophy of language In analytic philosophy, philosophy of language investigates the nature of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, ...
(by explicit enumeration)—of the same subset of the integers. By contrast, the sets $\$ and $\$ are not —the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined
up to isomorphism Two mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...
by their
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is :$\text \mapsto 1, \text \mapsto 2, \text \mapsto 3,$ while another is $\text \mapsto 3, \text \mapsto 2, \text \mapsto 1,$ and no one isomorphism is intrinsically better than any other.$A, B, C$ have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus one particular isomorphism is "natural", namely :$\text \mapsto 1, \text \mapsto 2, \text \mapsto 3.$ More formally, as these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as they are naturally isomorphic (there is a unique isomorphism, given above), since finite total orders are uniquely determined up to unique isomorphism by
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. This intuition can be formalized by saying that any two finite totally ordered sets of the same cardinality have a natural isomorphism, the one that sends the least element of the first to the least element of the second, the least element of what remains in the first to the least element of what remains in the second, and so forth, but in general, pairs of sets of a given finite cardinality are not naturally isomorphic because there is more than one choice of map—except if the cardinality is 0 or 1, where there is a unique choice.
In fact, there are precisely $3! = 6$ different isomorphisms between two sets with three elements. This is equal to the number of
automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of a given three-element set (which in turn is equal to the order of the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted $\operatorname\left(A,B\right),$ is a torsor for the automorphism group of ''A,'' $\operatorname\left(A\right)$ and also a torsor for the automorphism group of ''B.'' In fact, automorphisms of an object are a key reason to be concerned with the distinction between isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or with its double dual, as elaborated in the sequel.
On this view and in this sense, these two sets are not equal because one cannot consider them : one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism. Sometimes the isomorphisms can seem obvious and compelling, but are still not equalities. As a simple example, the genealogy, genealogical relationships among Joseph Kennedy, Joe, John F. Kennedy, John, and Robert F. Kennedy, Bobby Kennedy are, in a real sense, the same as those among the American football quarterbacks in the Manning family: Archie Manning, Archie, Peyton Manning, Peyton, and Eli Manning, Eli. The father-son pairings and the elder-brother-younger-brother pairings correspond perfectly. That similarity between the two family structures illustrates the origin of the word (Greek ''iso''-, "same", and -''morph'', "form" or "shape"). But because the Kennedys are not the same people as the Mannings, the two genealogical structures are merely isomorphic and not equal. Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a finite-dimensional vector space ''V'' and its dual space $V^* = \left\$ of linear maps from ''V'' to its field of scalars $\mathbf.$ These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism $\scriptstyle V \mathrel V^*.$ If one chooses a basis for ''V'', then this yields an isomorphism: For all $u, v \in V,$ $v \mathrel \phi_v \in V^* \quad \text \quad \phi_v(u) = v^\mathrm u.$ This corresponds to transforming a column vector (element of ''V'') to a row vector (element of ''V''*) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis". More subtly, there a map from a vector space ''V'' to its double dual $V^ = \left\$ that does not depend on the choice of basis: For all $v \in V \text \varphi \in V^*,$ $v \mathrel x_v \in V^ \quad \text \quad x_v(\phi) = \phi(v).$ This leads to a third notion, that of a natural isomorphism: while $V$ and $V^$ are different sets, there is a "natural" choice of isomorphism between them. This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a natural transformation; briefly, that one may identify, or more generally map from, a finite-dimensional vector space to its double dual, $\scriptstyle V \mathrel V^,$ for vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory. However, there is a case where the distinction between natural isomorphism and equality is usually not made. That is for the objects that may be characterized by a
universal property In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
. In fact, there is a unique isomorphism, necessarily natural, between two objects sharing the same universal property. A typical example is the set of real numbers, which may be defined through infinite decimal expansion, infinite binary expansion, Cauchy sequences, Dedekind cuts and many other ways. Formally, these constructions define different objects which are all solutions with the same universal property. As these objects have exactly the same properties, one may forget the method of construction and consider them as equal. This is what everybody does when referring to " set of the real numbers". The same occurs with quotient space (topology), quotient spaces: they are commonly constructed as sets of equivalence classes. However, referring to a set of sets may be counterintuitive, and so quotient spaces are commonly considered as a pair of a set of undetermined objects, often called "points", and a surjective map onto this set. If one wishes to distinguish between an arbitrary isomorphism (one that depends on a choice) and a natural isomorphism (one that can be done consistently), one may write $\, \approx \,$ for an unnatural isomorphism and for a natural isomorphism, as in $V \approx V^*$ and $V \cong V^.$ This convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction. Generally, saying that two objects are is reserved for when there is a notion of a larger (ambient) space that these objects live in. Most often, one speaks of equality of two subsets of a given set (as in the integer set example above), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensional space $S^2 := \left\$ and the Riemann sphere $\widehat$ which can be presented as the one-point compactification of the complex plane $\Complex \cup \$ as the complex projective line (a quotient space) $\mathbf_^1 := \left(\Complex^2\setminus \\right) / \left(\Complex^*\right)$ are three different descriptions for a mathematical object, all of which are isomorphic, but not because they are not all subsets of a single space: the first is a subset of $\R^3,$ the second is $\Complex \cong \R^2$Being precise, the identification of the complex numbers with the real plane, $\C \cong \R \cdot 1 \oplus \R \cdot i = \R^2$ depends on a choice of $i;$ one can just as easily choose $\left(-i\right),$ which yields a different identification—formally, complex conjugation is an automorphism—but in practice one often assumes that one has made such an identification. plus an additional point, and the third is a subquotient of $\Complex^2.$ In the context of category theory, objects are usually at most isomorphic—indeed, a motivation for the development of category theory was showing that different constructions in homology theory yielded equivalent (isomorphic) groups. Given maps between two objects ''X'' and ''Y'', however, one asks if they are equal or not (they are both elements of the set $\hom\left(X, Y\right),$ hence equality is the proper relationship), particularly in commutative diagrams.

*Bisimulation *Equivalence relation *Heap (mathematics) *Isometry *Isomorphism class *Isomorphism theorem *Universal property *Coherent isomorphism