Khanh: /* Relation with equality */

2021-12-18T17:35:51Z

Relation with equality

Khanh: /* Category theoretic view */

2021-12-18T17:35:11Z

Category theoretic view

Khanh at 17:32, 18 December 2021

2021-12-18T17:32:51Z

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In [[mathematics]], an '''isomorphism''' is a structure-preserving [[Map (mathematics)|mapping]] between two [[Mathematical structure|structures]] of the same type that can be reversed by an [[inverse function|inverse mapping]]. Two mathematical structures are '''isomorphic''' if an isomorphism exists between them. The word isomorphism is derived from the [[Ancient Greek]]: [[wikt:ἴσος|ἴσος]] ''isos'' "equal", and [[wikt:μορφή|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 {{em|the same [[up to]] an isomorphism}}.

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]]), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every [[prime number]] {{mvar|p}}, all [[Field (mathematics)|fields]] with {{mvar|p}} elements are canonically isomorphic, with a unique isomorphism. The [[isomorphism theorems]] provide canonical isomorphisms that are not unique.

The term {{em|isomorphism}} is mainly used for [[algebraic structure]]s. In this case, mappings are called [[homomorphism]]s, and a homomorphism is an isomorphism [[if and only if]] it is [[bijective]].

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:
* An [[isometry]] is an isomorphism of [[metric space]]s.
* A [[homeomorphism]] is an isomorphism of [[topological space]]s.
* A [[diffeomorphism]] is an isomorphism of spaces equipped with a [[differential structure]], typically [[differentiable manifold]]s.
* A [[permutation]] is an automorphism of a [[set (mathematics)|set]].
* In [[geometry]], isomorphisms and automorphisms are often called [[transformation (function)|transformations]], for example [[rigid transformation]]s, [[affine transformation]]s, [[projective transformation]]s.

[[Category theory]], 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 <math>\R^+</math> be the [[multiplicative group]] of [[positive real numbers]], and let <math>\R</math> be the additive group of real numbers.

The [[logarithm function]] <math>\log : \R^+ \to \R</math> satisfies <math>\log(xy) = \log x + \log y</math> for all <math>x, y \in \R^+,</math> so it is a [[group homomorphism]]. The [[exponential function]] <math>\exp : \R \to \R^+</math> satisfies <math>\exp(x+y) = (\exp x)(\exp y)</math> for all <math>x, y \in \R,</math> so it too is a homomorphism.

The identities <math>\log \exp x = x</math> and <math>\exp \log y = y</math> show that <math>\log</math> and <math>\exp</math> are [[inverse function|inverses]] of each other. Since <math>\log</math> is a homomorphism that has an inverse that is also a homomorphism, <math>\log</math> is an isomorphism of groups.

The <math>\log</math> 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 <math>(\Z_6, +),</math> the integers from 0 to 5 with addition [[Modular arithmetic|modulo]] 6. Also consider the group <math>\left(\Z_2 \times \Z_3, +\right),</math> 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:
<math display="block">\begin{alignat}{4}
(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{alignat}</math>
or in general <math>(a, b) \mapsto (3 a + 4 b) \mod 6.</math>

For example, <math>(1, 1) + (1, 0) = (0, 1),</math> which translates in the other system as <math>1 + 3 = 4.</math>

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 of groups|direct product]] of two [[cyclic group]]s <math>\Z_m</math> and <math>\Z_n</math> is isomorphic to <math>(\Z_{mn}, +)</math> if and only if ''m'' and ''n'' are [[coprime]], per the [[Chinese remainder theorem]].

===Relation-preserving isomorphism===
If one object consists of a set ''X'' with a [[binary relation]] 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 <math>f : X \to Y</math> such that:
<math display="block">\operatorname{S}(f(u),f(v)) \quad \text{ if and only if } \quad \operatorname{R}(u,v) </math>

S is [[Reflexive relation|reflexive]], [[Irreflexive relation|irreflexive]], [[Symmetric relation|symmetric]], [[Antisymmetric relation|antisymmetric]], [[Asymmetric relation|asymmetric]], [[Transitive relation|transitive]], [[Connected relation|total]], [[Binary relation#Relations over a set|trichotomous]], a [[partial order]], [[total order]], [[well-order]], [[strict weak order]], [[Strict weak order#Total preorders|total preorder]] (weak order), an [[equivalence relation]], or a relation with any other special properties, if and only if R is.

For example, R is an [[Order theory|ordering]] ≤ and S an ordering <math>\scriptstyle \sqsubseteq,</math> then an isomorphism from ''X'' to ''Y'' is a bijective function <math>f : X \to Y</math> such that
<math display="block">f(u) \sqsubseteq f(v) \quad \text{ if and only if } \quad u \leq v.</math>
Such an isomorphism is called an {{em|[[order isomorphism]]}} or (less commonly) an {{em|isotone isomorphism}}.

If <math>X = Y,</math> then this is a relation-preserving [[automorphism]].

==Applications==
In [[algebra]], isomorphisms are defined for all [[algebraic structure]]s. Some are more specifically studied; for example:
* [[Linear isomorphism]]s between [[vector space]]s; they are specified by [[invertible matrices]].
* [[Group isomorphism]]s between [[group (mathematics)|groups]]; the classification of [[isomorphism class]]es of [[finite group]]s is an open problem.
* [[Ring isomorphism]] between [[ring (mathematics)|rings]].
* Field isomorphisms are the same as ring isomorphism between [[field (mathematics)|fields]]; their study, and more specifically the study of [[field automorphism]]s is an important part of [[Galois theory]].

Just as the [[automorphism]]s of an [[algebraic structure]] form a [[group (mathematics)|group]], the isomorphisms between two algebras sharing a common structure form a [[heap (mathematics)|heap]]. Letting a particular isomorphism identify the two structures turns this heap into a group.

In [[mathematical analysis]], the [[Laplace transform]] is an isomorphism mapping hard [[differential equations]] into easier [[algebra]]ic equations.

In [[graph theory]], an isomorphism between two graphs ''G'' and ''H'' is a [[bijective]] 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 (graph theory)|vertex]] ''u'' to vertex ''v'' in ''G'' if and only if there is an edge from <math>f(u)</math> to <math>f(v)</math> in ''H''. See [[graph isomorphism]].

In mathematical analysis, an isomorphism between two [[Hilbert space]]s is a bijection preserving addition, scalar multiplication, and inner product.

In early theories of [[logical atomism]], the formal relationship between facts and true propositions was theorized by [[Bertrand Russell]] and [[Ludwig Wittgenstein]] to be isomorphic. An example of this line of thinking can be found in Russell's ''[[Introduction to Mathematical Philosophy]]''.

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]], given a [[category (mathematics)|category]] ''C'', an isomorphism is a morphism <math>f : a \to b</math> that has an inverse morphism <math>g : b \to a,</math> that is, <math>f g = 1_b</math> and <math>g f = 1_a.</math> For example, a bijective [[linear map]] is an isomorphism between [[vector space]]s, and a bijective [[continuous function]] whose inverse is also continuous is an isomorphism between [[topological space]]s, called a [[homeomorphism]].

Two categories {{mvar|C}} and {{mvar|D}} are [[Isomorphism of categories|isomorphic]] if there exist [[functor]]s <math>F : C \to D</math> and <math>G : D \to C</math> which are mutually inverse to each other, that is, <math>FG = 1_D</math> (the identity functor on {{mvar|D}}) and <math>GF = 1_C</math> (the identity functor on {{mvar|C}}).

===Isomorphism vs. bijective morphism===
In a [[concrete category]] (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]] 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 set]]s. In algebraic categories (specifically, categories of [[variety (universal algebra)|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 {{em|[[Equality (mathematics)|equality]]}} on the one hand and {{em|isomorphism}} 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
<math display="block">A = \left\{ x \in \Z \mid x^2 < 2\right\} \quad \text{ and } \quad B = \{-1, 0, 1\}</math>
are {{em|equal}}; they are merely different representations—the first an [[intensional definition|intensional]] one (in [[set builder notation]]), and the second [[extensional definition|extensional]] (by explicit enumeration)—of the same subset of the integers. By contrast, the sets <math>\{A, B, C\}</math> and <math>\{1, 2, 3\}</math> are not {{em|equal}}—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]] (number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is
:<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3,</math> while another is <math>\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,</math>
and no one isomorphism is intrinsically better than any other. On this view and in this sense, these two sets are not equal because one cannot consider them {{em|identical}}: 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 {{em|isomorphism}} (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]] <math>V^* = \left\{ \varphi : V \to \mathbf{K} \right\}</math> of linear maps from ''V'' to its field of scalars <math>\mathbf{K}.</math>
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 <math>\scriptstyle V \mathrel{\overset{\sim}{\to}} V^*.</math>
If one chooses a basis for ''V'', then this yields an isomorphism: For all <math>u, v \in V,</math>
<math display="block">v \mathrel{\overset{\sim}{\mapsto}} \phi_v \in V^* \quad \text{ such that } \quad \phi_v(u) = v^\mathrm{T} u.</math>

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 {{em|is}} a map from a vector space ''V'' to its [[double dual]] <math>V^{**} = \left\{ x : V^* \to \mathbf{K} \right\}</math> that does not depend on the choice of basis: For all <math>v \in V \text{ and } \varphi \in V^*,</math>
<math display="block">v \mathrel{\overset{\sim}{\mapsto}} x_v \in V^{**} \quad \text{ such that } \quad x_v(\phi) = \phi(v).</math>

This leads to a third notion, that of a [[natural isomorphism]]: while <math>V</math> and <math>V^{**}</math> 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 {{em|consistently}} identify, or more generally map from, a finite-dimensional vector space to its double dual, <math>\scriptstyle V \mathrel{\overset{\sim}{\to}} V^{**},</math> for {{em|any}} 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 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]]s, which may be defined through infinite decimal expansion, infinite binary expansion, [[Cauchy sequence]]s, [[Dedekind cut]]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 "{{em|the}} set of the real numbers". The same occurs with [[quotient space (topology)|quotient space]]s: they are commonly constructed as sets of [[equivalence class]]es. 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 <math>\, \approx \,</math> for an [[unnatural isomorphism]] and {{math|<big>≅</big>}} for a natural isomorphism, as in <math>V \approx V^*</math> and <math>V \cong V^{**}.</math>
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 {{em|equal}} 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
<math display="block">S^2 := \left\{ (x,y,z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}</math> and the [[Riemann sphere]] <math>\widehat{\Complex}</math>
which can be presented as the [[one-point compactification]] of the complex plane <math>\Complex \cup \{ \infty \}</math> {{em|or}} as the complex [[projective line]] (a quotient space)
<math display="block">\mathbf{P}_{\Complex}^1 := \left(\Complex^2\setminus \{(0,0)\}\right) / \left(\Complex^*\right)</math>
are three different descriptions for a mathematical object, all of which are isomorphic, but not {{em|equal}} because they are not all subsets of a single space: the first is a subset of <math>\R^3,</math> the second is <math>\Complex \cong \R^2</math> plus an additional point, and the third is a [[subquotient]] of <math>\Complex^2.</math>

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 <math>\hom(X, Y),</math> hence equality is the proper relationship), particularly in [[commutative diagram]]s.

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Isomorphism Isomorphism, Wikipedia] under a CC BY-SA license

@@ Line 113: / Line 113: @@
 Generally, saying that two objects are ''equal'' 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
 <math display="block">S^2 := \left\{ (x,y,z) \in \R^3 \mid x^2 + y^2 + z^2 = 1\right\}</math> and the Riemann sphere <math>\widehat{\Complex}</math>
-which can be presented as the one-point compactification of the complex plane <math>\Complex \cup \{ \infty \}</math> {{em|or}} as the complex projective line (a quotient space)
+which can be presented as the one-point compactification of the complex plane <math>\Complex \cup \{ \infty \}</math> ''or'' as the complex projective line (a quotient space)
 <math display="block">\mathbf{P}_{\Complex}^1 := \left(\Complex^2\setminus \{(0,0)\}\right) / \left(\Complex^*\right)</math>
 are three different descriptions for a mathematical object, all of which are isomorphic, but not ''equal'' because they are not all subsets of a single space: the first is a subset of <math>\R^3,</math> the second is <math>\Complex \cong \R^2</math> plus an additional point, and the third is a subquotient of <math>\Complex^2.</math>

@@ Line 79: / Line 79: @@
 In category theory, given a category ''C'', an isomorphism is a morphism <math>f : a \to b</math> that has an inverse morphism <math>g : b \to a,</math> that is, <math>f g = 1_b</math> and <math>g f = 1_a.</math> For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.
-Two categories {{mvar|C}} and {{mvar|D}} are [[Isomorphism of categories|isomorphic]] if there exist [[functor]]s <math>F : C \to D</math> and <math>G : D \to C</math> which are mutually inverse to each other, that is, <math>FG = 1_D</math> (the identity functor on {{mvar|D}}) and <math>GF = 1_C</math> (the identity functor on {{mvar|C}}).
+Two categories {{mvar|C}} and {{mvar|D}} are isomorphic if there exist functors <math>F : C \to D</math> and <math>G : D \to C</math> which are mutually inverse to each other, that is, <math>FG = 1_D</math> (the identity functor on {{mvar|D}}) and <math>GF = 1_C</math> (the identity functor on {{mvar|C}}).
 ===Isomorphism vs. bijective morphism===

Isomorphisms - Revision history

Khanh: /* Relation with equality */

Khanh: /* Category theoretic view */

Khanh at 17:32, 18 December 2021

Khanh: Created page with "In mathematics, an '''isomorphism''' is a structure-preserving mapping between two structures of the same type that can be..."