TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an isomorphism is a structure-preserving mapping 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. The word isomorphism is derived from the
Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
: 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 is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map 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 mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
), 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 Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
with elements are canonically isomorphic, with a unique isomorphism. The
isomorphism theorems In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
provide canonical isomorphisms that are not unique. The term is mainly used for algebraic structures. In this case, mappings are called homomorphisms, and a homomorphism is an isomorphism
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
. In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example: * An
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
is an isomorphism of
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. * A
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
is an isomorphism of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s. * A
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
is an isomorphism of spaces equipped with a
differential structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for d ...
, typically
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s. * A symplectomorphism is an isomorphism of symplectic manifolds. * A
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
is an automorphism of a set. * In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, isomorphisms and automorphisms are often called transformations, for example rigid transformations,
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
s,
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In gener ...
s.
Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, 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 of positive real numbers, and let $\R$ be the additive group of real numbers. The
logarithm function In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
$\log : \R^+ \to \R$ satisfies $\log\left(xy\right) = \log x + \log y$ for all $x, y \in \R^+,$ so it is a group homomorphism. The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
$\exp : \R \to \R^+$ 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 and a table of logarithms, or using a slide rule 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 product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of two cyclic groups $\Z_m$ and $\Z_n$ is isomorphic to $\left(\Z_, +\right)$ if and only if ''m'' and ''n'' are coprime, per the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
.

## Relation-preserving isomorphism

If one object consists of a set ''X'' with a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
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,
symmetric Symmetry (from grc, συμμετρία "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 precise definiti ...
, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order,
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 X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
, well-order, strict weak order,
total preorder In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered se ...
(weak order), an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
, or a relation with any other special properties, if and only if R is. For example, R is an ordering ≤ and S an ordering $\sqsubseteq,$ 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.

# Applications

In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example: *
Linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
s between
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s; they are specified by
invertible matrices In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicat ...
. * Group isomorphisms between groups; the classification of
isomorphism class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the st ...
es of finite groups is an open problem. *
Ring isomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
between rings. * Field isomorphisms are the same as ring isomorphism between
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
; their study, and more specifically the study of
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorph ...
s is an important part of Galois theory. Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing a common structure form a heap. 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 continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied i ...
, the Laplace transform is an isomorphism mapping hard differential equations into easier
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic equations. In
graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' ( ...
, an isomorphism between two graphs ''G'' and ''H'' is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 ''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 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 in ''G'' if and only if f(u) and f(v) ...
. In mathematical analysis, an isomorphism between two
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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. Its principal exponent was the British philosopher Bertrand Russell. It is also widely held that the early works of his ...
, 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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
and
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian- British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is con ...
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 (1919 first edition) by philosopher Bertrand Russell, in which the author seeks to create an accessible introduction to various topics within the foundations of mathematics. According to the pr ...
''. In cybernetics, 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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, given a category ''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, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
is an isomorphism between
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
s, and a bijective
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
whose inverse is also continuous is an isomorphism between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s, called a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
. Two categories and are isomorphic if there exist functors $F : C \to D$ 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, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, ''see Relative concreteness below''). This functor makes it possible to think of the objects of ...
(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, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again conti ...
or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules), an isomorphism must be bijective on the underlying sets. 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), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets $\$ and $\$ are not —the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is :$\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, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. This intuition can be formalized by saying that any two finite
totally ordered set 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 X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
s of the same cardinality have a natural isomorphism, the one that sends the
least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
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 automorphisms of a given three-element set (which in turn is equal to the order of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite 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. Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a
finite-dimensional vector space In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
''V'' and its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
$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 $V \mathrel V^*.$ 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 In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, ...
(element of ''V'') to a
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, ...
(element of ''V''*) by
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
, 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 In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
; briefly, that one may identify, or more generally map from, a finite-dimensional vector space to its double dual, $V \mathrel V^,$ 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 mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
. 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, which may be defined through infinite decimal expansion, infinite binary expansion,
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s,
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the ra ...
s 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 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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
$\widehat$ which can be presented as the one-point compactification of the complex plane $\Complex \cup \$ as the complex
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
(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, $\Complex \cong \R \cdot 1 \oplus \R \cdot i = \R^2$ depends on a choice of $i;$ one can just as easily choose $\left(-i\right),$ which yields a different identification—formally,
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a ...
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 diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s. See also: homotopy type theory, in which isomorphisms can be treated as kinds of equality.

* Bisimulation *
Equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
* Heap (mathematics) *
Isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
*
Isomorphism class In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the st ...
* Isomorphism theorem *
Universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
* Coherent isomorphism