Category theory abFrom Wikipedia, the free encyclopedia

Contents

1 Automorphism 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Automorphism group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Inner and outer automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Category theory 42.1 An abstraction of other mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Categories, objects, and morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Other concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6.1 Universal constructions, limits, and colimits . . . . . . . . . . . . . . . . . . . . . . . . . 72.6.2 Equivalent categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6.3 Further concepts and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6.4 Higher-dimensional categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.7 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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3 Endomorphism 113.1 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Endomorphism ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Endofunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Epimorphism 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Equivalence of categories 165.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Identity function 196.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Isomorphism 217.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.1.1 Logarithm and exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.1.2 Integers modulo 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.1.3 Relation-preserving isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.2 Isomorphism vs. bijective morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Relation with equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Isomorphism of categories 268.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Mathematical structure 279.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10 Monomorphism 2910.1 Relation to invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

11 Section (category theory) 3111.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 32

11.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 1

Automorphism

In mathematics, an automorphism is an isomorphismfrom a mathematical object to itself. It is, in some sense,a symmetry of the object, and a way of mapping the ob-ject to itself while preserving all of its structure. The setof all automorphisms of an object forms a group, calledthe automorphism group. It is, loosely speaking, thesymmetry group of the object.

1.1 Denition

The exact denition of an automorphism depends on thetype of mathematical object in question and what, pre-cisely, constitutes an isomorphism of that object. Themost general setting in which these words have mean-ing is an abstract branch of mathematics called categorytheory. Category theory deals with abstract objects andmorphisms between those objects.In category theory, an automorphism is an endomorphism(i.e. a morphism from an object to itself) which is also anisomorphism (in the categorical sense of the word).This is a very abstract denition since, in category theory,morphisms aren't necessarily functions and objects aren'tnecessarily sets. In most concrete settings, however, theobjects will be sets with some additional structure and themorphisms will be functions preserving that structure.In the context of abstract algebra, for example, a mathe-matical object is an algebraic structure such as a group,ring, or vector space. An isomorphism is simply abijective homomorphism. (The denition of a homomor-phism depends on the type of algebraic structure; see, forexample: group homomorphism, ring homomorphism,and linear operator).The identity morphism (identity mapping) is called thetrivial automorphism in some contexts. Respectively,other (non-identity) automorphisms are called nontrivialautomorphisms.

1.2 Automorphism groupIf the automorphisms of an objectX form a set (instead ofa proper class), then they form a group under compositionof morphisms. This group is called the automorphismgroup of X. That this is indeed a group is simple to see:

Closure: composition of two endomorphisms is an-other endomorphism.

Associativity: composition of morphisms is alwaysassociative.

Identity: the identity is the identity morphism froman object to itself, which exists by denition.

Inverses: by denition every isomorphism has an in-verse which is also an isomorphism, and since theinverse is also an endomorphism of the same objectit is an automorphism.

The automorphism group of an object X in a category Cis denoted AutC(X), or simply Aut(X) if the category isclear from context.

1.3 Examples In set theory, an arbitrary permutation of the ele-

ments of a set X is an automorphism. The automor-phism group of X is also called the symmetric groupon X.

In elementary arithmetic, the set of integers, Z, con-sidered as a group under addition, has a unique non-trivial automorphism: negation. Considered as aring, however, it has only the trivial automorphism.Generally speaking, negation is an automorphism ofany abelian group, but not of a ring or eld.

A group automorphism is a group isomorphismfrom a group to itself. Informally, it is a permu-tation of the group elements such that the structureremains unchanged. For every group G there is anatural group homomorphism G Aut(G) whoseimage is the group Inn(G) of inner automorphisms

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and whose kernel is the center of G. Thus, if G hastrivial center it can be embedded into its own auto-morphism group.[1]

In linear algebra, an endomorphism of a vector spaceV is a linear operator V V. An automorphism isan invertible linear operator on V. When the vectorspace is nite-dimensional, the automorphism groupof V is the same as the general linear group, GL(V).

A eld automorphism is a bijective ring homomor-phism from a eld to itself. In the cases of therational numbers (Q) and the real numbers (R)there are no nontrivial eld automorphisms. Somesubelds of R have nontrivial eld automorphisms,which however do not extend to all of R (becausethey cannot preserve the property of a number hav-ing a square root in R). In the case of the complexnumbers, C, there is a unique nontrivial automor-phism that sendsR intoR: complex conjugation, butthere are innitely (uncountably) many wild au-tomorphisms (assuming the axiom of choice).[2][3]Field automorphisms are important to the theory ofeld extensions, in particular Galois extensions. Inthe case of a Galois extension L/K the subgroup ofall automorphisms of L xing K pointwise is calledthe Galois group of the extension.

The eld Qp of p-adic numbers has no nontrivialautomorphisms.

In graph theory an automorphism of a graph is a per-mutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by anedge, so are their images under the permutation.

For relations, see relation-preserving automorphism. In order theory, see order automorphism.

In geometry, an automorphism may be called amotion of the space. Specialized terminology is alsoused:

In metric geometry an automorphism is a self-isometry. The automorphism group is alsocalled the isometry group.

In the category of Riemann surfaces, an au-tomorphism is a bijective biholomorphic map(also called a conformal map), from a surfaceto itself. For example, the automorphisms ofthe Riemann sphere are Mbius transforma-tions.

An automorphism of a dierentiable manifoldM is a dieomorphism from M to itself. Theautomorphism group is sometimes denotedDi(M).

In topology, morphisms between topologi-cal spaces are called continuous maps, andan automorphism of a topological space is

a homeomorphism of the space to itself, orself-homeomorphism (see homeomorphismgroup). In this example it is not sucient fora morphism to be bijective to be an isomor-phism.

1.4 HistoryOne of the earliest group automorphisms (automorphismof a group, not simply a group of automorphisms ofpoints) was given by the Irish mathematician WilliamRowan Hamilton in 1856, in his icosian calculus, wherehe discovered an order two automorphism,[4] writing:

so that is a new fth root of unity, con-nected with the former fth root by relationsof perfect reciprocity.

1.5 Inner and outer automor-phisms

In some categoriesnotably groups, rings, and Lie alge-brasit is possible to separate automorphisms into twotypes, called inner and outer automorphisms.In the case of groups, the inner automorphisms are theconjugations by the elements of the group itself. For eachelement a of a group G, conjugation by a is the operationa : G G given by a(g) = aga1 (or a1ga; usagevaries). One can easily check that conjugation by a is agroup automorphism. The inner automorphisms form anormal subgroup of Aut(G), denoted by Inn(G); this iscalled Goursats lemma.The other automorphisms are called outer automor-phisms. The quotient group Aut(G) / Inn(G) is usuallydenoted by Out(G); the non-trivial elements are the cosetsthat contain the outer automorphisms.The same denition holds in any unital ring or algebrawhere a is any invertible element. For Lie algebras thedenition is slightly dierent.

1.6 See also Endomorphism ring

Antiautomorphism

Frobenius automorphism

Morphism

Characteristic subgroup

1.8. EXTERNAL LINKS 3

1.7 References[1] PJ Pahl, R Damrath (2001). "7.5.5 Automorphisms.

Mathematical foundations of computational engineering(Felix Pahl translation ed.). Springer. p. 376. ISBN 3-540-67995-2.

[2] Yale, Paul B. (May 1966). Automorphisms of the Com-plex Numbers (PDF). Mathematics Magazine 39 (3):135141. doi:10.2307/2689301. JSTOR 2689301.

[3] Lounesto, Pertti (2001), Cliord Algebras and Spinors(2nd ed.), Cambridge University Press, pp. 2223, ISBN0-521-00551-5

[4] Sir William Rowan Hamilton (1856). Memorandumrespecting a new System of Roots of Unity (PDF).Philosophical Magazine 12: 446.

1.8 External links Automorphism at Encyclopaedia of Mathematics Weisstein, Eric W., Automorphism, MathWorld.

Chapter 2

Category theory

Schematic representation of a category with objects X, Y, Z andmorphisms f, g, g f. (The categorys three identity morphisms1X, 1Y and 1Z, if explicitly represented, would appear as threearrows, next to the letters X, Y, and Z, respectively, each havingas its shaft a circular arc measuring almost 360 degrees.)

Category theory[1] formalizes mathematical structureand its concepts in terms of a collection of objects andof arrows (also called morphisms). A category hastwo basic properties: the ability to compose the arrowsassociatively and the existence of an identity arrow foreach object. Category theory can be used to formal-ize concepts of other high-level abstractions such as sets,rings, and groups.Several terms used in category theory, including the termmorphism, are used dierently from their uses in therest of mathematics. In category theory, a morphismobeys a set of conditions specic to category theory it-self. Thus, care must be taken to understand the contextin which statements are made.

2.1 An abstraction of other mathe-matical concepts

Many signicant areas of mathematics can be formalisedby category theory as categories. Category theory is anabstraction of mathematics itself that allows many intri-cate and subtle mathematical results in these elds to bestated, and proved, in a much simpler way than withoutthe use of categories.[2]

The most accessible example of a category is the categoryof sets, where the objects are sets and the arrows are func-tions from one set to another. However, the objects ofa category need not be sets, and the arrows need not befunctions; any way of formalising a mathematical conceptsuch that it meets the basic conditions on the behaviour ofobjects and arrows is a valid category, and all the resultsof category theory will apply to it.The arrows of category theory are often said to repre-sent a process connecting two objects, or in many cases astructure-preserving transformation connecting two ob-jects. There are however many applications where muchmore abstract concepts are represented by objects andmorphisms. The most important property of the arrowsis that they can be composed, in other words, arrangedin a sequence to form a new arrow.Categories now appear in most branches of mathematics,some areas of theoretical computer science where theycan correspond to types, and mathematical physics wherethey can be used to describe vector spaces. Categorieswere rst introduced by Samuel Eilenberg and SaundersMac Lane in 194245, in connection with algebraictopology.Category theory has several faces known not just to spe-cialists, but to other mathematicians. A term dating fromthe 1940s, "general abstract nonsense", refers to its highlevel of abstraction, compared to more classical branchesof mathematics. Homological algebra is category theoryin its aspect of organising and suggesting manipulationsin abstract algebra.

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2.3. CATEGORIES, OBJECTS, AND MORPHISMS 5

2.2 Utility

2.2.1 Categories, objects, and morphisms

The study of categories is an attempt to axiomatically cap-ture what is commonly found in various classes of relatedmathematical structures by relating them to the structure-preserving functions between them. A systematic studyof category theory then allows us to prove general resultsabout any of these types of mathematical structures fromthe axioms of a category.Consider the following example. The classGrp of groupsconsists of all objects having a group structure. Onecan proceed to prove theorems about groups by makinglogical deductions from the set of axioms. For example,it is immediately proven from the axioms that the identityelement of a group is unique.Instead of focusing merely on the individual objects (e.g.,groups) possessing a given structure, category theory em-phasizes the morphisms the structure-preserving map-pings between these objects; by studying these mor-phisms, we are able to learn more about the structure ofthe objects. In the case of groups, the morphisms arethe group homomorphisms. A group homomorphism be-tween two groups preserves the group structure in a pre-cise sense it is a process taking one group to another,in a way that carries along information about the struc-ture of the rst group into the second group. The studyof group homomorphisms then provides a tool for study-ing general properties of groups and consequences of thegroup axioms.A similar type of investigation occurs in many mathemat-ical theories, such as the study of continuous maps (mor-phisms) between topological spaces in topology (the as-sociated category is called Top), and the study of smoothfunctions (morphisms) in manifold theory.Not all categories arise as structure preserving (set)functions, however; the standard example is the categoryof homotopies between pointed topological spaces.If one axiomatizes relations instead of functions, one ob-tains the theory of allegories.

2.2.2 Functors

Main article: FunctorSee also: Adjoint functors Motivation

A category is itself a type of mathematical structure, sowe can look for processes which preserve this structurein some sense; such a process is called a functor.Diagram chasing is a visual method of arguing with ab-stract arrows joined in diagrams. Functors are rep-resented by arrows between categories, subject to spe-cic dening commutativity conditions. Functors can de-

ne (construct) categorical diagrams and sequences (viz.Mitchell, 1965). A functor associates to every object ofone category an object of another category, and to everymorphism in the rst category a morphism in the second.In fact, what we have done is dene a category of cate-gories and functors the objects are categories, and themorphisms (between categories) are functors.By studying categories and functors, we are not juststudying a class of mathematical structures and the mor-phisms between them; we are studying the relationshipsbetween various classes of mathematical structures. Thisis a fundamental idea, which rst surfaced in algebraictopology. Dicult topological questions can be trans-lated into algebraic questions which are often easier tosolve. Basic constructions, such as the fundamental groupor the fundamental groupoid of a topological space, canbe expressed as functors to the category of groupoids inthis way, and the concept is pervasive in algebra and itsapplications.

2.2.3 Natural transformationsMain article: Natural transformation

Abstracting yet again, some diagrammatic and/or sequen-tial constructions are often naturally related a vaguenotion, at rst sight. This leads to the clarifying conceptof natural transformation, a way to map one functor toanother. Many important constructions in mathematicscan be studied in this context. Naturality is a princi-ple, like general covariance in physics, that cuts deeperthan is initially apparent. An arrow between two functorsis a natural transformation when it is subject to certainnaturality or commutativity conditions.Functors and natural transformations ('naturality') are thekey concepts in category theory.[3]

2.3 Categories, objects, and mor-phisms

Main articles: Category (mathematics) and Morphism

2.3.1 CategoriesA category C consists of the following three mathematicalentities:

A class ob(C), whose elements are called objects; A class hom(C), whose elements are called

morphisms or maps or arrows. Each morphism fhas a source object a and target object b.

6 CHAPTER 2. CATEGORY THEORY

The expression f : a b, would be verbally statedas "f is a morphism from a to b".The expression hom(a, b) alternatively ex-pressed as homC(a, b), mor(a, b), or C(a, b) denotes the hom-class of all morphisms from a to b.

A binary operation , called composition of mor-phisms, such that for any three objects a, b, and c,we have hom(b, c) hom(a, b) hom(a, c). Thecomposition of f : a b and g : b c is written asg f or gf,[4] governed by two axioms:

Associativity: If f : a b, g : b c and h : c d then h (g f) = (h g) f, and

Identity: For every object x, there exists a mor-phism 1x : x x called the identity morphismfor x, such that for every morphism f : a b,we have 1b f = f = f 1a.

From the axioms, it can be provedthat there is exactly one identitymorphism for every object. Someauthors deviate from the denitionjust given by identifying each ob-ject with its identity morphism.

2.3.2 Morphisms

Relations among morphisms (such as fg = h) are of-ten depicted using commutative diagrams, with points(corners) representing objects and arrows representingmorphisms.Morphisms can have any of the following properties. Amorphism f : a b is a:

monomorphism (or monic) if f g1 = f g2 impliesg1 = g2 for all morphisms g1, g2 : x a.

epimorphism (or epic) if g1 f = g2 f implies g1= g2 for all morphisms g1, g2 : b x.

bimorphism if f is both epic and monic. isomorphism if there exists a morphism g : b a

such that f g = 1b and g f = 1a.[5]

endomorphism if a = b. end(a) denotes the class ofendomorphisms of a.

automorphism if f is both an endomorphism and anisomorphism. aut(a) denotes the class of automor-phisms of a.

retraction if a right inverse of f exists, i.e. if thereexists a morphism g : b a with f g = 1b.

section if a left inverse of f exists, i.e. if there existsa morphism g : b a with g f = 1a.

Every retraction is an epimorphism, and every section is amonomorphism. Furthermore, the following three state-ments are equivalent:

f is a monomorphism and a retraction; f is an epimorphism and a section; f is an isomorphism.

2.4 FunctorsMain article: Functor

Functors are structure-preserving maps between cate-gories. They can be thought of as morphisms in the cate-gory of all (small) categories.A (covariant) functor F from a category C to a categoryD, written F : C D, consists of:

for each object x in C, an object F(x) in D; and for each morphism f : x y in C, a morphism F(f)

: F(x) F(y),

such that the following two properties hold:

For every object x in C, F(1x) = 1Fx; For all morphisms f : x y and g : y z, F(g f)

= F(g) F(f).

A contravariant functor F: C D, is like a covariantfunctor, except that it turns morphisms around (re-verses all the arrows). More specically, every mor-phism f : x y in C must be assigned to a morphismF(f) : F(y) F(x) in D. In other words, a contravari-ant functor acts as a covariant functor from the oppositecategory Cop to D.

2.5 Natural transformationsMain article: Natural transformation

A natural transformation is a relation between two func-tors. Functors often describe natural constructions andnatural transformations then describe natural homomor-phisms between two such constructions. Sometimes twoquite dierent constructions yield the same result; thisis expressed by a natural isomorphism between the twofunctors.If F andG are (covariant) functors between the categoriesC and D, then a natural transformation from F to Gassociates to every object X in C a morphism X : F(X)

2.6. OTHER CONCEPTS 7

G(X) in D such that for every morphism f : X Y inC, we have Y F(f) = G(f) X; this means that thefollowing diagram is commutative:

Commutative diagram dening natural transformations

The two functors F and G are called naturally isomorphicif there exists a natural transformation from F to G suchthat X is an isomorphism for every object X in C.

2.6 Other concepts

2.6.1 Universal constructions, limits, andcolimits

Main articles: Universal property and Limit (categorytheory)

Using the language of category theory, many areas ofmathematical study can be categorized. Categories in-clude sets, groups and topologies.Each category is distinguished by properties that all itsobjects have in common, such as the empty set or theproduct of two topologies, yet in the denition of a cat-egory, objects are considered to be atomic, i.e., we donot know whether an object A is a set, a topology, or anyother abstract concept. Hence, the challenge is to denespecial objects without referring to the internal structureof those objects. To dene the empty set without refer-ring to elements, or the product topology without refer-ring to open sets, one can characterize these objects interms of their relations to other objects, as given by themorphisms of the respective categories. Thus, the taskis to nd universal properties that uniquely determine theobjects of interest.Indeed, it turns out that numerous important construc-tions can be described in a purely categorical way. Thecentral concept which is needed for this purpose is calledcategorical limit, and can be dualized to yield the notionof a colimit.

2.6.2 Equivalent categoriesMain articles: Equivalence of categories andIsomorphism of categories

It is a natural question to ask: under which conditions cantwo categories be considered to be essentially the same,in the sense that theorems about one category can readilybe transformed into theorems about the other category?The major tool one employs to describe such a situationis called equivalence of categories, which is given by ap-propriate functors between two categories. Categoricalequivalence has found numerous applications in mathe-matics.

2.6.3 Further concepts and resultsThe denitions of categories and functors provide onlythe very basics of categorical algebra; additional impor-tant topics are listed below. Although there are stronginterrelations between all of these topics, the given ordercan be considered as a guideline for further reading.

The functor category DC has as objects the functorsfrom C to D and as morphisms the natural transfor-mations of such functors. The Yoneda lemma is oneof the most famous basic results of category theory;it describes representable functors in functor cate-gories.

Duality: Every statement, theorem, or denition incategory theory has a dual which is essentially ob-tained by reversing all the arrows. If one statementis true in a category C then its dual will be true in thedual category Cop. This duality, which is transparentat the level of category theory, is often obscured inapplications and can lead to surprising relationships.

Adjoint functors: A functor can be left (or right)adjoint to another functor that maps in the oppo-site direction. Such a pair of adjoint functors typi-cally arises from a construction dened by a univer-sal property; this can be seen as a more abstract andpowerful view on universal properties.

2.6.4 Higher-dimensional categoriesMany of the above concepts, especially equivalence ofcategories, adjoint functor pairs, and functor categories,can be situated into the context of higher-dimensionalcategories. Briey, if we consider a morphism betweentwo objects as a process taking us from one objectto another, then higher-dimensional categories allowus to protably generalize this by considering higher-dimensional processes.For example, a (strict) 2-category is a category togetherwith morphisms between morphisms, i.e., processes

8 CHAPTER 2. CATEGORY THEORY

which allow us to transform one morphism into another.We can then compose these bimorphisms both hor-izontally and vertically, and we require a 2-dimensionalexchange law to hold, relating the two compositionlaws. In this context, the standard example is Cat, the2-category of all (small) categories, and in this example,bimorphisms of morphisms are simply natural transfor-mations of morphisms in the usual sense. Another basicexample is to consider a 2-category with a single object;these are essentially monoidal categories. Bicategoriesare a weaker notion of 2-dimensional categories in whichthe composition of morphisms is not strictly associative,but only associative up to an isomorphism.This process can be extended for all natural numbers n,and these are called n-categories. There is even a notionof -category corresponding to the ordinal number .Higher-dimensional categories are part of the broadermathematical eld of higher-dimensional algebra, a con-cept introduced by Ronald Brown. For a conversationalintroduction to these ideas, see John Baez, 'A Tale of n-categories (1996).

2.7 Historical notesIn 194245, Samuel Eilenberg and Saunders Mac Laneintroduced categories, functors, and natural transfor-mations as part of their work in topology, especiallyalgebraic topology. Their work was an important partof the transition from intuitive and geometric homologyto axiomatic homology theory. Eilenberg and Mac Lanelater wrote that their goal was to understand natural trans-formations; in order to do that, functors had to be dened,which required categories.Stanislaw Ulam, and some writing on his behalf, haveclaimed that related ideas were current in the late 1930sin Poland. Eilenberg was Polish, and studied mathematicsin Poland in the 1930s. Category theory is also, in somesense, a continuation of the work of Emmy Noether (oneof Mac Lanes teachers) in formalizing abstract processes;Noether realized that in order to understand a type ofmathematical structure, one needs to understand the pro-cesses preserving that structure. In order to achieve thisunderstanding, Eilenberg and Mac Lane proposed an ax-iomatic formalization of the relation between structuresand the processes preserving them.The subsequent development of category theory waspowered rst by the computational needs of homologicalalgebra, and later by the axiomatic needs of algebraic ge-ometry, the eld most resistant to being grounded in ei-ther axiomatic set theory or the Russell-Whitehead viewof united foundations. General category theory, an exten-sion of universal algebra having many new features allow-ing for semantic exibility and higher-order logic, camelater; it is now applied throughout mathematics.Certain categories called topoi (singular topos) can even

serve as an alternative to axiomatic set theory as a foun-dation of mathematics. A topos can also be consideredas a specic type of category with two additional toposaxioms. These foundational applications of category the-ory have been worked out in fair detail as a basis for, andjustication of, constructive mathematics. Topos theoryis a form of abstract sheaf theory, with geometric origins,and leads to ideas such as pointless topology.Categorical logic is now a well-dened eld based ontype theory for intuitionistic logics, with applications infunctional programming and domain theory, where acartesian closed category is taken as a non-syntactic de-scription of a lambda calculus. At the very least, categorytheoretic language claries what exactly these related ar-eas have in common (in some abstract sense).Category theory has been applied in other elds aswell. For example, John Baez has shown a link betweenFeynman diagrams in Physics and monoidal categories.[6]Another application of category theory, more speci-cally: topos theory, has been made in mathematical mu-sic theory, see for example the book The Topos of Music,Geometric Logic of Concepts, Theory, and Performanceby Guerino Mazzola.More recent eorts to introduce undergraduates to cat-egories as a foundation for mathematics include thoseof William Lawvere and Rosebrugh (2003) and Law-vere and Stephen Schanuel (1997) and Mirroslav Yotov(2012).

2.8 See also Group theory Domain theory Enriched category theory Glossary of category theory Higher category theory Higher-dimensional algebra Important publications in category theory Outline of category theory Timeline of category theory and related mathemat-

ics

2.9 Notes[1] Awodey 2006

[2] Geroch, Robert (1985). Mathematical physics ([Repr.]ed.). Chicago: University of Chicago Press. p. 7. ISBN0-226-28862-5. Retrieved 20 August 2012. Note thattheorem 3 is actually easier for categories in general than

2.10. REFERENCES 9

it is for the special case of sets. This phenomenon is byno means rare.

[3] Mac Lane 1998, p. 18: As Eilenberg-Mac Lane rst ob-served, 'category' has been dened in order to be able todene 'functor' and 'functor' has been dened in order tobe able to dene 'natural transformation' "

[4] Some authors compose in the opposite order, writing fg orf g for g f. Computer scientists using category theoryvery commonly write f ; g for g f

[5] Note that a morphism that is both epic and monic is notnecessarily an isomorphism! An elementary counterex-ample: in the category consisting of two objects A and B,the identity morphisms, and a single morphism f from Ato B, f is both epic and monic but is not an isomorphism.

[6] Baez, J.C.; Stay, M. (2009). Physics, topology, logic andcomputation: A Rosetta stone (PDF). arXiv:0903.0340.

2.10 References Admek, Ji; Herrlich, Horst; Strecker, George E.

(1990). Abstract and concrete categories. John Wi-ley & Sons. ISBN 0-471-60922-6.

Awodey, Steve (2006). Category Theory. OxfordLogic Guides 49. Oxford University Press. ISBN978-0-19-151382-4.

Barr, Michael; Wells, Charles (2012), Category The-ory for Computing Science, Reprints in Theory andApplications of Categories 22 (3rd ed.).

Barr, Michael; Wells, Charles (2005), Toposes,Triples and Theories, Reprints in Theory and Ap-plications of Categories 12 (revised ed.), MR2178101.

Borceux, Francis (1994). Handbook of categoricalalgebra. Encyclopedia of Mathematics and its Ap-plications 50-52. Cambridge University Press.

Bucur, Ion; Deleanu, Aristide (1968). Introductionto the theory of categories and functors. Wiley.

Freyd, Peter J. (1964). Abelian Categories. NewYork: Harper and Row.

Freyd, Peter J.; Scedrov, Andre (1990). Categories,allegories. North Holland Mathematical Library 39.North Holland. ISBN 978-0-08-088701-2.

Goldblatt, Robert (2006) [1979]. Topoi: The Cat-egorial Analysis of Logic. Studies in logic and thefoundations of mathematics 94 (Reprint, reviseded.). Dover Publications. ISBN 978-0-486-45026-1.

Hatcher, William S. (1982). Ch. 8. The logicalfoundations of mathematics. Foundations & philos-ophy of science & technology (2nd ed.). PergamonPress.

Herrlich, Horst; Strecker, George E. (2007), Cate-gory Theory (3rd ed.), Heldermann Verlag Berlin,ISBN 978-3-88538-001-6.

Kashiwara, Masaki; Schapira, Pierre (2006).Categories and Sheaves. Grundlehren der Mathema-tischen Wissenschaften 332. Springer. ISBN 978-3-540-27949-5.

Lawvere, F. William; Rosebrugh, Robert (2003).Sets for Mathematics. Cambridge University Press.ISBN 978-0-521-01060-3.

Lawvere, F. W.; Schanuel, Stephen Hoel (2009)[1997]. Conceptual Mathematics: A First Introduc-tion to Categories (2nd ed.). Cambridge UniversityPress. ISBN 978-0-521-89485-2.

Leinster, Tom (2004). Higher operads, higher cat-egories. London Math. Society Lecture Note Se-ries 298. Cambridge University Press. ISBN 978-0-521-53215-0.

Leinster, Tom (2014). Basic Category Theory.Cambridge University Press.

Lurie, Jacob (2009). Higher topos theory. Annals ofMathematics Studies 170. Princeton, NJ: PrincetonUniversity Press. arXiv:math.CT/0608040. ISBN978-0-691-14049-0. MR 2522659.

Mac Lane, Saunders (1998). Categories for theWorking Mathematician. Graduate Texts in Math-ematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. MR 1712872.

Mac Lane, Saunders; Birkho, Garrett (1999)[1967]. Algebra (2nd ed.). Chelsea. ISBN 0-8218-1646-2.

Martini, A.; Ehrig, H.; Nunes, D. (1996).Elements of basic category theory. Technical Re-port (Technical University Berlin) 96 (5).

May, Peter (1999). A Concise Course in AlgebraicTopology. University of Chicago Press. ISBN 0-226-51183-9.

Guerino, Mazzola (2002). The Topos ofMusic, Geo-metric Logic of Concepts, Theory, and Performance.Birkhuser. ISBN 3-7643-5731-2.

Pedicchio, Maria Cristina; Tholen, Walter, eds.(2004). Categorical foundations. Special topics inorder, topology, algebra, and sheaf theory. Ency-clopedia of Mathematics and Its Applications 97.Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.

Pierce, Benjamin C. (1991). Basic Category Theoryfor Computer Scientists. MIT Press. ISBN 978-0-262-66071-6.

10 CHAPTER 2. CATEGORY THEORY

Schalk, A.; Simmons, H. (2005). An introductionto Category Theory in four easy movements (PDF).Notes for a course oered as part of the MSc. inMathematical Logic, Manchester University.

Simpson, Carlos. Homotopy theory of higher cate-gories. arXiv:1001.4071., draft of a book.

Taylor, Paul (1999). Practical Foundations of Math-ematics. Cambridge Studies in Advanced Mathe-matics 59. Cambridge University Press. ISBN 978-0-521-63107-5.

Turi, Daniele (19962001). Category Theory Lec-ture Notes (PDF). Retrieved 11 December 2009.Based on Mac Lane 1998.

2.11 Further reading Jean-Pierre Marquis (2008). From a GeometricalPoint of View: A Study of the History and Philosophyof Category Theory. Springer Science & BusinessMedia. ISBN 978-1-4020-9384-5.

2.12 External links Theory and Application of Categories, an electronic

journal of category theory, full text, free, since1995.

nLab, a wiki project on mathematics, physics andphilosophy with emphasis on the n-categorical pointof view.

Andr Joyal, CatLab, a wiki project dedicated to theexposition of categorical mathematics.

Category Theory, a web page of links to lecturenotes and freely available books on category theory.

Hillman, Chris, A Categorical Primer, CiteSeerX:10 .1 .1 .24 .3264, a formal introduction to categorytheory.

Adamek, J.; Herrlich, H.; Stecker, G. Abstract andConcrete Categories-The Joy of Cats (PDF).

Category Theory entry by Jean-Pierre Marquis inthe Stanford Encyclopedia of Philosophy with an ex-tensive bibliography.

List of academic conferences on category theory Baez, John (1996). The Tale of n-categories.

An informal introduction to higher order categories. WildCats is a category theory package for

Mathematica. Manipulation and visualization ofobjects, morphisms, categories, functors, naturaltransformations, universal properties.

The catsterss channel on YouTube, a channel aboutcategory theory.

Category Theory at PlanetMath.org. Video archive of recorded talks relevant to cate-

gories, logic and the foundations of physics.

Interactive Web page which generates examples ofcategorical constructions in the category of nitesets.

Category Theory for the Sciences, an instruction oncategory theory as a tool throughout the sciences.

Chapter 3

Endomorphism

This article is about the mathematical concept. For otheruses, see Endomorphic.

In mathematics, an endomorphism is a morphism (or

mPv

Pu w = Pw Px

v

u

x

Orthogonal projection onto a line m is a linear operator on theplane. This is an example of an endomorphism that is not anautomorphism.

homomorphism) from a mathematical object to itself.For example, an endomorphism of a vector space V isa linear map f: V V, and an endomorphism of a groupG is a group homomorphism f: G G. In general, wecan talk about endomorphisms in any category. In thecategory of sets, endomorphisms are functions from a setS to itself.In any category, the composition of any two endomor-phisms of X is again an endomorphism of X. It followsthat the set of all endomorphisms of X forms a monoid,denoted End(X) (or EndC(X) to emphasize the categoryC).

3.1 AutomorphismsMain article: Automorphism

An invertible endomorphism of X is called anautomorphism. The set of all automorphisms is asubset of End(X) with a group structure, called theautomorphism group of X and denoted Aut(X). In thefollowing diagram, the arrows denote implication:

3.2 Endomorphism ring

Main article: Endomorphism ring

Any two endomorphisms of an abelian group A can beadded together by the rule (f + g)(a) = f(a) + g(a). Un-der this addition, the endomorphisms of an abelian groupform a ring (the endomorphism ring). For example, theset of endomorphisms of Zn is the ring of all n n matri-ces with integer entries. The endomorphisms of a vectorspace or module also form a ring, as do the endomor-phisms of any object in a preadditive category. The en-domorphisms of a nonabelian group generate an algebraicstructure known as a near-ring. Every ring with one isthe endomorphism ring of its regular module, and so is asubring of an endomorphism ring of an abelian group,[1]however there are rings which are not the endomorphismring of any abelian group.

3.3 Operator theory

In any concrete category, especially for vector spaces, en-domorphisms are maps from a set into itself, and may beinterpreted as unary operators on that set, acting on theelements, and allowing to dene the notion of orbits ofelements, etc.Depending on the additional structure dened for the cat-egory at hand (topology, metric, ...), such operators canhave properties like continuity, boundedness, and so on.More details should be found in the article about operatortheory.

11

12 CHAPTER 3. ENDOMORPHISM

3.4 EndofunctionsAn endofunction is a function whose domain is equal toits codomain. A homomorphic endofunction is an endo-morphism.Let S be an arbitrary set. Among endofunctions on S onends permutations of S and constant functions associat-ing to each x S a given c S. Every permutation of Shas the codomain equal to its domain and is bijective andinvertible. A constant function on S, if S has more than 1element, has a codomain that is a proper subset of its do-main, is not bijective (and non invertible). The functionassociating to each natural integer n the oor of n/2 hasits codomain equal to its domain and is not invertible.Finite endofunctions are equivalent to directed pseudo-forests. For sets of size n there are nn endofunctions onthe set.Particular bijective endofunctions are the involutions, i.e.the functions coinciding with their inverses.

3.5 See also Adjoint endomorphism Frobenius endomorphism

3.6 Notes[1] Jacobson (2009), p. 162, Theorem 3.2.

3.7 References Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.),

Dover, ISBN 978-0-486-47189-1

3.8 External links Hazewinkel, Michiel, ed. (2001), Endomorphism,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Endomorphism at PlanetMath.org.

Chapter 4

Epimorphism

This article is about the mathematical function. For thebiological phenomenon, see Epimorphosis.In category theory, an epimorphism (also called an epic

morphism or, colloquially, an epi) is a morphism f : XY that is right-cancellative in the sense that, for all mor-phisms g1, g2 : Y Z,

g1 f = g2 f ) g1 = g2:

Epimorphisms are analogues of surjective functions, butthey are not exactly the same. The dual of an epimor-phism is a monomorphism (i.e. an epimorphism in acategory C is a monomorphism in the dual category Cop).Many authors in abstract algebra and universal algebradene an epimorphism simply as an onto or surjectivehomomorphism. Every epimorphism in this algebraicsense is an epimorphism in the sense of category theory,but the converse is not true in all categories. In this arti-cle, the term epimorphism will be used in the sense ofcategory theory given above. For more on this, see thesection on Terminology below.

4.1 ExamplesEvery morphism in a concrete category whose underlyingfunction is surjective is an epimorphism. In many con-crete categories of interest the converse is also true. Forexample, in the following categories, the epimorphismsare exactly those morphisms which are surjective on theunderlying sets:

Set, sets and functions. To prove that every epimor-phism f: X Y in Set is surjective, we compose itwith both the characteristic function g1: Y {0,1}of the image f(X) and the map g2: Y {0,1} thatis constant 1.

Rel, sets with binary relations and relation preserv-ing functions. Here we can use the same proofas for Set, equipping {0,1} with the full relation{0,1}{0,1}.

Pos, partially ordered sets and monotone functions.If f : (X,) (Y,) is not surjective, pick y0 in Y\ f(X) and let g1 : Y {0,1} be the characteristicfunction of {y | y0 y} and g2 : Y {0,1} thecharacteristic function of {y | y0 < y}. These mapsare monotone if {0,1} is given the standard ordering0 < 1.

Grp, groups and group homomorphisms. The resultthat every epimorphism inGrp is surjective is due toOtto Schreier (he actually proved more, showing thatevery subgroup is an equalizer using the free prod-uct with one amalgamated subgroup); an elementaryproof can be found in (Linderholm 1970).

FinGrp, nite groups and group homomorphisms.Also due to Schreier; the proof given in (Linderholm1970) establishes this case as well.

Ab, abelian groups and group homomorphisms. K-Vect, vector spaces over a eld K and K-linear

transformations. Mod-R, right modules over a ring R and module ho-

momorphisms. This generalizes the two previousexamples; to prove that every epimorphism f: X Y in Mod-R is surjective, we compose it with boththe canonical quotient map g 1: Y Y/f(X) and thezero map g2: Y Y/f(X).

Top, topological spaces and continuous functions.To prove that every epimorphism in Top is surjec-tive, we proceed exactly as in Set, giving {0,1} theindiscrete topology which ensures that all consideredmaps are continuous.

HComp, compact Hausdor spaces and continuousfunctions. If f: X Y is not surjective, let y in Y-fX. Since fX is closed, by Urysohns Lemma thereis a continuous function g1:Y [0,1] such that g1 is0 on fX and 1 on y. We compose f with both g1 andthe zero function g2: Y [0,1].

13

14 CHAPTER 4. EPIMORPHISM

However there are also many concrete categories of in-terest where epimorphisms fail to be surjective. A fewexamples are:

In the category of monoids, Mon, the inclusion mapN Z is a non-surjective epimorphism. To see this,suppose that g1 and g2 are two distinct maps from Zto some monoid M. Then for some n in Z, g1(n) g2(n), so g1(-n) g2(-n). Either n or -n is in N, sothe restrictions of g1 and g2 to N are unequal.

In the category of algebras over commutative ringR,take R[N] R[Z], where R[G] is the group ring ofthe group G and the morphism is induced by the in-clusion N Z as in the previous example. This fol-lows from the observation that 1 generates the alge-bra R[Z] (note that the unit in R[Z] is given by 0 ofZ), and the inverse of the element represented by nin Z is just the element represented by -n. Thus anyhomomorphism from R[Z] is uniquely determinedby its value on the element represented by 1 of Z.

In the category of rings, Ring, the inclusion map Z Q is a non-surjective epimorphism; to see this,note that any ring homomorphism on Q is deter-mined entirely by its action on Z, similar to the pre-vious example. A similar argument shows that thenatural ring homomorphism from any commutativering R to any one of its localizations is an epimor-phism.

In the category of commutative rings, a nitely gen-erated homomorphism of rings f : R S is an epi-morphism if and only if for all prime ideals P ofR, the ideal Q generated by f(P) is either S or isprime, and if Q is not S, the induced map Frac(R/P) Frac(S/Q) is an isomorphism (EGA IV 17.2.6).

In the category of Hausdor spaces, Haus, the epi-morphisms are precisely the continuous functionswith dense images. For example, the inclusion mapQ R, is a non-surjective epimorphism.

The above diers from the case of monomorphismswhere it is more frequently true that monomorphisms areprecisely those whose underlying functions are injective.As to examples of epimorphisms in non-concrete cate-gories:

If a monoid or ring is considered as a category witha single object (composition of morphisms given bymultiplication), then the epimorphisms are preciselythe right-cancellable elements.

If a directed graph is considered as a category (ob-jects are the vertices, morphisms are the paths,composition of morphisms is the concatenation ofpaths), then every morphism is an epimorphism.

4.2 PropertiesEvery isomorphism is an epimorphism; indeed only aright-sided inverse is needed: if there exists a morphismj : Y X such that fj = idY , then f is easily seen to be anepimorphism. A map with such a right-sided inverse iscalled a split epi. In a topos, a map that is both a monicmorphism and an epimorphism is an isomorphism.The composition of two epimorphisms is again an epi-morphism. If the composition fg of two morphisms is anepimorphism, then f must be an epimorphism.As some of the above examples show, the property of be-ing an epimorphism is not determined by the morphismalone, but also by the category of context. If D is asubcategory of C, then every morphism in D which is anepimorphism when considered as a morphism in C is alsoan epimorphism in D; the converse, however, need nothold; the smaller category can (and often will) have moreepimorphisms.As for most concepts in category theory, epimorphismsare preserved under equivalences of categories: given anequivalence F : C D, then a morphism f is an epi-morphism in the category C if and only if F(f) is an epi-morphism in D. A duality between two categories turnsepimorphisms into monomorphisms, and vice versa.The denition of epimorphism may be reformulated tostate that f : X Y is an epimorphism if and only if theinduced maps

Hom(Y; Z) ! Hom(X;Z)g 7! gf

are injective for every choice of Z. This in turn is equiv-alent to the induced natural transformation

Hom(Y;) ! Hom(X;)being a monomorphism in the functor category SetC .Every coequalizer is an epimorphism, a consequence ofthe uniqueness requirement in the denition of coequal-izers. It follows in particular that every cokernel is anepimorphism. The converse, namely that every epimor-phism be a coequalizer, is not true in all categories.In many categories it is possible to write every morphismas the composition of a monomorphism followed by anepimorphism. For instance, given a group homomor-phism f : G H, we can dene the group K = im(f) =f(G) and then write f as the composition of the surjectivehomomorphism G K which is dened like f, followedby the injective homomorphism K H which sends eachelement to itself. Such a factorization of an arbitrary mor-phism into an epimorphism followed by a monomorphismcan be carried out in all abelian categories and also in allthe concrete categories mentioned above in the Examplessection (though not in all concrete categories).

4.5. SEE ALSO 15

4.3 Related conceptsAmong other useful concepts are regular epimorphism,extremal epimorphism, strong epimorphism, and split epi-morphism. A regular epimorphism coequalizes some par-allel pair of morphisms. An extremal epimorphism isan epimorphism that has no monomorphism as a sec-ond factor, unless that monomorphism is an isomorphism.A strong epimorphism satises a certain lifting prop-erty with respect to commutative squares involving amonomorphism. A split epimorphism is a morphismwhich has a right-sided inverse.A morphism that is both a monomorphism and an epi-morphism is called a bimorphism. Every isomorphismis a bimorphism but the converse is not true in general.For example, the map from the half-open interval [0,1) tothe unit circle S1 (thought of as a subspace of the complexplane) which sends x to exp(2ix) (see Eulers formula) iscontinuous and bijective but not a homeomorphism sincethe inverse map is not continuous at 1, so it is an instanceof a bimorphism that is not an isomorphism in the cate-gory Top. Another example is the embedding Q R inthe category Haus; as noted above, it is a bimorphism,but it is not bijective and therefore not an isomorphism.Similarly, in the category of rings, the map Z Q is abimorphism but not an isomorphism.Epimorphisms are used to dene abstract quotient objectsin general categories: two epimorphisms f1 : X Y1and f2 : X Y2 are said to be equivalent if there existsan isomorphism j : Y1 Y2 with j f1 = f2. This isan equivalence relation, and the equivalence classes aredened to be the quotient objects of X.

4.4 TerminologyThe companion terms epimorphism and monomorphismwere rst introduced by Bourbaki. Bourbaki uses epimor-phism as shorthand for a surjective function. Early cate-gory theorists believed that epimorphisms were the cor-rect analogue of surjections in an arbitrary category, sim-ilar to how monomorphisms are very nearly an exact ana-logue of injections. Unfortunately this is incorrect; strongor regular epimorphisms behave much more closely tosurjections than ordinary epimorphisms. Saunders MacLane attempted to create a distinction between epimor-phisms, which were maps in a concrete category whoseunderlying set maps were surjective, and epic morphisms,which are epimorphisms in the modern sense. However,this distinction never caught on.It is a common mistake to believe that epimorphisms areeither identical to surjections or that they are a better con-cept. Unfortunately this is rarely the case; epimorphismscan be very mysterious and have unexpected behavior. Itis very dicult, for example, to classify all the epimor-phisms of rings. In general, epimorphisms are their own

unique concept, related to surjections but fundamentallydierent.

4.5 See also List of category theory topics

4.6 References Admek, Ji, Herrlich, Horst, & Strecker, George

E. (1990). Abstract and Concrete Categories (4.2MBPDF). Originally publ. John Wiley & Sons. ISBN0-471-60922-6. (now free on-line edition)

Bergman, George M. (1998), An Invitation to Gen-eral Algebra and Universal Constructions, HarryHelson Publisher, Berkeley. ISBN 0-9655211-4-1.

Hazewinkel, Michiel, ed. (2001), Epimorphism,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Linderholm, Carl (1970). A Group Epimorphism isSurjective. American Mathematical Monthly 77, pp.176177. Proof summarized by Arturo Magidin in.

Lawvere & Rosebrugh: Sets for Mathematics, Cam-bridge university press, 2003. ISBN 0-521-80444-2.

Chapter 5

Equivalence of categories

In category theory, an abstract branch of mathematics,an equivalence of categories is a relation between twocategories that establishes that these categories are es-sentially the same. There are numerous examples ofcategorical equivalences from many areas of mathemat-ics. Establishing an equivalence involves demonstratingstrong similarities between the mathematical structuresconcerned. In some cases, these structures may appearto be unrelated at a supercial or intuitive level, makingthe notion fairly powerful: it creates the opportunity totranslate theorems between dierent kinds of mathe-matical structures, knowing that the essential meaning ofthose theorems is preserved under the translation.If a category is equivalent to the opposite (or dual) ofanother category then one speaks of a duality of cate-gories, and says that the two categories are dually equiv-alent.An equivalence of categories consists of a functor be-tween the involved categories, which is required to havean inverse functor. However, in contrast to the situa-tion common for isomorphisms in an algebraic setting,the composition of the functor and its inverse is notnecessarily the identity mapping. Instead it is sucientthat each object be naturally isomorphic to its image un-der this composition. Thus one may describe the functorsas being inverse up to isomorphism. There is indeed aconcept of isomorphism of categories where a strict formof inverse functor is required, but this is of much lesspractical use than the equivalence concept.

5.1 DenitionFormally, given two categories C and D, an equivalenceof categories consists of a functor F : C D, a functorG : D C, and two natural isomorphisms : FGIDand : ICGF. Here FG: DD and GF: CC, denotethe respective compositions of F and G, and IC: CCand ID: DD denote the identity functors on C and D,assigning each object and morphism to itself. If F andG are contravariant functors one speaks of a duality ofcategories instead.One often does not specify all the above data. For in-

stance, we say that the categories C and D are equivalent(respectively dually equivalent) if there exists an equiva-lence (respectively duality) between them. Furthermore,we say that F is an equivalence of categories if an in-verse functor G and natural isomorphisms as above exist.Note however that knowledge of F is usually not enoughto reconstruct G and the natural isomorphisms: there maybe many choices (see example below).

5.2 Equivalent characterizationsOne can show that a functor F : C D yields an equiva-lence of categories if and only if it is simultaneously:

full, i.e. for any two objects c1 and c2 of C, the mapHomC(c1,c2) HomD(Fc1,Fc2) induced by F issurjective;

faithful, i.e. for any two objects c1 and c2 of C, themap HomC(c1,c2) HomD(Fc1,Fc2) induced by Fis injective; and

essentially surjective (dense), i.e. each object d in Dis isomorphic to an object of the form Fc, for c in C.

This is a quite useful and commonly applied criterion, be-cause one does not have to explicitly construct the in-verse G and the natural isomorphisms between FG, GFand the identity functors. On the other hand, though theabove properties guarantee the existence of a categori-cal equivalence (given a suciently strong version of theaxiom of choice in the underlying set theory), the miss-ing data is not completely specied, and often there aremany choices. It is a good idea to specify the missing con-structions explicitly whenever possible. Due to this cir-cumstance, a functor with these properties is sometimescalled a weak equivalence of categories (unfortunatelythis conicts with terminology from homotopy theory).There is also a close relation to the concept of adjointfunctors. The following statements are equivalent forfunctors F : C D and G : D C:

There are natural isomorphisms from FG to ID andIC to GF.

16

5.4. PROPERTIES 17

F is a left adjoint of G and both functors are full andfaithful.

G is a right adjoint of F and both functors are fulland faithful.

One may therefore view an adjointness relation betweentwo functors as a very weak form of equivalence. As-suming that the natural transformations for the adjunc-tions are given, all of these formulations allow for an ex-plicit construction of the necessary data, and no choiceprinciples are needed. The key property that one has toprove here is that the counit of an adjunction is an isomor-phism if and only if the right adjoint is a full and faithfulfunctor.

5.3 Examples Consider the category C having a single object c

and a single morphism 1c , and the category Dwith two objects d1 , d2 and four morphisms: twoidentity morphisms 1d1 , 1d2 and two isomorphisms : d1 ! d2 and : d2 ! d1 . The categories Cand D are equivalent; we can (for example) have Fmap c to d1 and G map both objects of D to c andall morphisms to 1c .

By contrast, the category C with a single object anda single morphism is not equivalent to the categoryE with two objects and only two identity morphismsas the two objects therein are not isomorphic.

Consider a category C with one object c , and twomorphisms 1c; f : c ! c . Let 1c be the identitymorphism on c and set f f = 1 . Of course, Cis equivalent to itself, which can be shown by taking1c in place of the required natural isomorphisms be-tween the functor IC and itself. However, it is alsotrue that f yields a natural isomorphism from IC toitself. Hence, given the information that the identityfunctors form an equivalence of categories, in thisexample one still can choose between two naturalisomorphisms for each direction.

The category of sets and partial functions is equiv-alent to but not isomorphic with the category ofpointed sets and point-preserving maps.[1]

Consider the category C of nite-dimensional realvector spaces, and the category D = Mat(R) of allreal matrices (the latter category is explained in thearticle on additive categories). Then C and D areequivalent: The functor G : D ! C which mapsthe object An of D to the vector space Rn and thematrices in D to the corresponding linear maps isfull, faithful and essentially surjective.

One of the central themes of algebraic geometry isthe duality of the category of ane schemes and thecategory of commutative rings. The functor G as-sociates to every commutative ring its spectrum, thescheme dened by the prime ideals of the ring. Itsadjoint F associates to every ane scheme its ringof global sections.

In functional analysis the category of commutativeC*-algebras with identity is contravariantly equiva-lent to the category of compact Hausdor spaces.Under this duality, every compact Hausdor spaceX is associated with the algebra of continuouscomplex-valued functions onX , and every commu-tative C*-algebra is associated with the space of itsmaximal ideals. This is the Gelfand representation.

In lattice theory, there are a number of duali-ties, based on representation theorems that connectcertain classes of lattices to classes of topologicalspaces. Probably the most well-known theoremof this kind is Stones representation theorem forBoolean algebras, which is a special instance withinthe general scheme of Stone duality. Each Booleanalgebra B is mapped to a specic topology on theset of ultralters of B . Conversely, for any topol-ogy the clopen (i.e. closed and open) subsets yielda Boolean algebra. One obtains a duality betweenthe category of Boolean algebras (with their homo-morphisms) and Stone spaces (with continuous map-pings). Another case of Stone duality is Birkhosrepresentation theorem stating a duality between -nite partial orders and nite distributive lattices.

In pointless topology the category of spatial localesis known to be equivalent to the dual of the categoryof sober spaces.

For two rings R and S, R-ModS-Mod is equivalentto (RS)-Mod.

Any category is equivalent to its skeleton.

5.4 PropertiesAs a rule of thumb, an equivalence of categories preservesall categorical concepts and properties. If F : C D isan equivalence, then the following statements are all true:

the object c of C is an initial object (or terminal ob-ject, or zero object), if and only if Fc is an initialobject (or terminal object, or zero object) of D

the morphism in C is a monomorphism (orepimorphism, or isomorphism), if and only if F is amonomorphism (or epimorphism, or isomorphism)in D.

18 CHAPTER 5. EQUIVALENCE OF CATEGORIES

the functor H : I C has limit (or colimit) l if andonly if the functor FH : I D has limit (or colimit)Fl. This can be applied to equalizers, products andcoproducts among others. Applying it to kernels andcokernels, we see that the equivalence F is an exactfunctor.

C is a cartesian closed category (or a topos) if andonly if D is cartesian closed (or a topos).

Dualities turn all concepts around": they turn initial ob-jects into terminal objects, monomorphisms into epimor-phisms, kernels into cokernels, limits into colimits etc.If F : C D is an equivalence of categories, and G1 andG2 are two inverses of F, then G1 and G2 are naturallyisomorphic.If F : C D is an equivalence of categories, and if C is apreadditive category (or additive category, or abelian cat-egory), then D may be turned into a preadditive category(or additive category, or abelian category) in such a waythat F becomes an additive functor. On the other hand,any equivalence between additive categories is necessar-ily additive. (Note that the latter statement is not true forequivalences between preadditive categories.)An auto-equivalence of a category C is an equivalence F: C C. The auto-equivalences of C form a group undercomposition if we consider two auto-equivalences that arenaturally isomorphic to be identical. This group capturesthe essential symmetries of C. (One caveat: if C is not asmall category, then the auto-equivalences of C may forma proper class rather than a set.)

5.5 See also Equivalent denitions of mathematical structures

5.6 References[1] Lutz Schrder (2001). Categories: a free tour. In Jr-

gen Koslowski and Austin Melton. Categorical Perspec-tives. Springer Science & Business Media. p. 10. ISBN978-0-8176-4186-3.

Hazewinkel, Michiel, ed. (2001), Equivalence ofcategories, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Mac Lane, Saunders (1998). Categories for theworking mathematician. New York: Springer. pp.xii+314. ISBN 0-387-98403-8.

Chapter 6

Identity function

Not to be confused with Null function or Empty function.In mathematics, an identity function, also called an

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

x

Graph of the identity function on the real numbers

identity relation or identity map or identity transfor-mation, is a function that always returns the same valuethat was used as its argument. In equations, the functionis given by f(x) = x.

6.1 DenitionFormally, if M is a set, the identity function f on M isdened to be that function with domain and codomain Mwhich satises

f(x) = x for all elements x in M.[1]

In other words, the function assigns to each element x ofM the element x of M.The identity function f on M is often denoted by idM.In set theory, where a function is dened as a particularkind of binary relation, the identity function is given bythe identity relation, or diagonal of M.

6.2 Algebraic propertyIf f : M N is any function, then we have f idM = f= idN f (where "" denotes function composition). Inparticular, idM is the identity element of the monoid ofall functions from M to M.Since the identity element of a monoid is unique, one canalternately dene the identity function on M to be thisidentity element. Such a denition generalizes to the con-cept of an identity morphism in category theory, wherethe endomorphisms of M need not be functions.

6.3 Properties The identity function is a linear operator, when ap-

plied to vector spaces.[2]

The identity function on the positive integers is acompletely multiplicative function (essentially mul-tiplication by 1), considered in number theory.[3]

In an n-dimensional vector space the identity func-tion is represented by the identity matrix In, regard-less of the basis.[4]

In a metric space the identity is trivially an isometry.An object without any symmetry has as symmetrygroup the trivial group only containing this isometry(symmetry type C1).[5]

6.4 See also Inclusion map

6.5 References[1] Knapp, Anthony W. (2006), Basic algebra, Springer,

ISBN 978-0-8176-3248-9

[2] Anton, Howard (2005), Elementary Linear Algebra (Ap-plications Version) (9th ed.), Wiley International

19

20 CHAPTER 6. IDENTITY FUNCTION

[3] D. Marshall, E. Odell, and M. Starbird (2007). Num-ber Theory through Inquiry. Mathematical Association ofAmerica Textbooks. Mathematical Assn of Amer. ISBN978-0883857519.

[4] T. S. Shores (2007). Applied Linear Algebra and MatrixAnalysis. Undergraduate Texts in Mathematics. Springer.ISBN 038-733-195-6.

[5] James W. Anderson, Hyperbolic Geometry, Springer2005, ISBN 1-85233-934-9

Chapter 7

Isomorphism

This article is about mathematics. For other uses, seeIsomorphism (disambiguation).

0

+i

i

1 +1

72

108

The group of fth roots of unity under multiplicationis isomorphic to the group of rotations of the regularpentagon under composition.

In mathematics, an isomorphism (from the AncientGreek: isos equal, and morphe shape)is a homomorphism (or more generally a morphism) thatadmits an inverse.[note 1] Two mathematical objects areisomorphic if an isomorphism exists between them. Anautomorphism is an isomorphism whose source and tar-get coincide. The interest of isomorphisms lies in the factthat two isomorphic objects cannot be distinguished byusing only the properties used to dene morphisms; thusisomorphic objects may be considered the same as longas one considers only these properties and their conse-quences.

For most algebraic structures, including groups and rings,a homomorphism is an isomorphism if and only if it isbijective.In topology, where the morphisms are continuous func-tions, isomorphisms are also called homeomorphisms orbicontinuous functions. In mathematical analysis, wherethe morphisms are dierentiable functions, isomorphismsare also called dieomorphisms.A canonical isomorphism is a canonical map that is anisomorphism. Two objects are said to be canonicallyisomorphic if there is a canonical isomorphism betweenthem. For example, the canonical map from a nite-dimensional vector space V to its second dual space is acanonical isomorphism; on the other hand, V is isomor-phic to its dual space but not canonically in general.Isomorphisms are formalized using category theory. Amorphism f : X Y in a category is an isomorphism if itadmits a two-sided inverse, meaning that there is anothermorphism g : Y X in that category such that gf = 1Xand fg = 1Y , where 1X and 1Y are the identity morphismsof X and Y, respectively.[1]

7.1 Examples

7.1.1 Logarithm and exponentialLet R+ be the multiplicative group of positive real num-bers, and let R be the additive group of real numbers.The logarithm function log : R+ ! R satiseslog(xy) = logx + log y for all x; y 2 R+ , so itis a group homomorphism. The exponential functionexp : R ! R+ satises exp(x + y) = (expx)(exp y)for all x; y 2 R , so it too is a homomorphism.The identities log expx = x and exp log y = y showthat log and exp are inverses of each other. Since log isa homomorphism that has an inverse that is also a homo-morphism, log is an isomorphism of groups.Because log is an isomorphism, it translates multiplica-tion of positive real numbers into addition of real num-bers. This facility makes it possible to multiply real num-bers using a ruler and a table of logarithms, or using a

21

22 CHAPTER 7. ISOMORPHISM

slide rule with a logarithmic scale.

7.1.2 Integers modulo 6Consider the group (Z6;+) , the integers from 0 to 5 withaddition modulo 6. Also consider the group (Z2Z3;+), the ordered pairs where the x coordinates can be 0 or1, and the y coordinates can be 0, 1, or 2, where addi-tion in the x-coordinate is modulo 2 and addition in they-coordinate is modulo 3.These structures are isomorphic under addition, if youidentify them using the following scheme:

(0,0) 0(1,1) 1(0,2) 2(1,0) 3(0,1) 4(1,2) 5

or in general (a,b) (3a + 4b) mod 6.For example note that (1,1) + (1,0) = (0,1), which trans-lates in the other system as 1 + 3 = 4.Even though these two groups look dierent in that thesets contain dierent elements, they are indeed isomor-phic: their structures are exactly the same. More gen-erally, the direct product of two cyclic groups Zm andZn is isomorphic to (Zmn;+) if and only if m and n arecoprime.

7.1.3 Relation-preserving isomorphismIf one object consists of a set X with a binary relationR and the other object consists of a set Y with a binaryrelation S then an isomorphism from X to Y is a bijectivefunction : X Y such that:[2]

S(f(u); f(v)) () R(u; v)S is reexive, irreexive, symmetric, antisymmetric,asymmetric, transitive, total, trichotomous, a partial or-der, total order, strict weak order, total preorder (weak or-der), an equivalence relation, or a relation with any otherspecial properties, if and only if R is.For example, R is an ordering and S an ordering v ,then an isomorphism from X to Y is a bijective function: X Y such that

f(u) v f(v) () u v:Such an isomorphism is called an order isomorphism or(less commonly) an isotone isomorphism.If X = Y, then this is a relation-preserving automorphism.

7.2 Isomorphism vs. bijective mor-phism

In a concrete category (that is, roughly speaking, a cate-gory whose objects are sets and morphisms are mappingsbetween sets), such as the category of topological spacesor categories of algebraic objects like groups, rings, andmodules, an isomorphism must be bijective on the un-derlying sets. In algebraic categories (specically, cate-gories of varieties in the sense of universal algebra), anisomorphism is the same as a homomorphism which isbijective on underlying sets. However, there are concretecategories in which bijective morphisms are not neces-sarily isomorphisms (such as the category of topologicalspaces), and there are categories in which each object ad-mits an underlying set but in which isomorphisms neednot be bijective (such as the homotopy category of CW-complexes).

7.3 ApplicationsIn abstract algebra, two basic isomorphisms are dened:

Group isomorphism, an isomorphism betweengroups

Ring isomorphism, an isomorphism between rings.(Note that isomorphisms between elds are actuallyring isomorphisms)

Just as the automorphisms of an algebraic structure forma group, the isomorphisms between two algebras sharinga common structure form a heap. Letting a particular iso-morphism identify the two structures turns this heap intoa group.In mathematical analysis, the Laplace transform is an iso-morphism mapping hard dierential equations into easieralgebraic equations.In category theory, Iet the category C consist of twoclasses, one of objects and the other of morphisms. Thena general denition of isomorphism that covers the pre-vious and many other cases is: an isomorphism is a mor-phism : a b that has an inverse, i.e. there exists a mor-phism g: b a with g = 1b and g = 1a. For example,a bijective linear map is an isomorphism between vectorspaces, and a bijective continuous function whose inverseis also continuous is an isomorphism between topologicalspaces, called a homeomorphism.In graph theory, an isomorphism between two graphs Gand H is a bijective map f from the vertices of G to thevertices of H that preserves the edge structure in thesense that there is an edge from vertex u to vertex v in Gif and only if there is an edge from (u) to (v) in H. Seegraph isomorphism.

7.4. RELATION WITH EQUALITY 23

In mathematical analysis, an isomorphism between twoHilbert spaces is a bijection preserving addition, scalarmultiplication, and inner product.In early theories of logical atomism, the formal relation-ship between facts and true propositions was theorized byBertrand Russell and Ludwig Wittgenstein to be isomor-phic. An example of this line of thinking can be found inRussells Introduction to Mathematical Philosophy.In cybernetics, the Good Regulator or Conant-Ashby the-orem is stated Every Good Regulator of a system mustbe a model of that system. Whether regulated or self-regulating an isomorphism is required between regulatorpart and the processing part of the system.

7.4 Relation with equalitySee also: Equality (mathematics)

In certain areas of mathematics, notably category theory,it is valuable to distinguish between equality on the onehand and isomorphism on the other.[3] Equality is whentwo objects are exactly the same, and everything thatstrue about one object is true about the other, while anisomorphism implies everything thats true about a des-ignated part of one objects structure is true about theothers. For example, the sets

A = fx 2 Z j x2 < 2g and B = f1; 0; 1g

are equal; they are merely dierent presentationstherst an intensional one (in set builder notation), andthe second extensional (by explicit enumeration)of thesame subset of the integers. By contrast, the sets {A,B,C}and {1,2,3} are not equalthe rst has elements that areletters, while the second has elements that are numbers.These are isomorphic as sets, since nite sets are deter-mined up to isomorphism by their cardinality (number ofelements) and these both have three elements, but thereare many choices of isomorphismone isomorphism is

A 7! 1;B 7! 2;C 7! 3; while another is A 7!3;B 7! 2;C 7! 1;

and no one isomorphism is intrinsically better than anyother.[note 2][note 3] On this view and in this sense, thesetwo sets are not equal because one cannot consider themidentical: one can choose an isomorphism between them,but that is a weaker claim than identityand valid onlyin the context of the chosen isomorphism.Sometimes the isomorphisms can seem obvious and com-pelling, but are still not equalities. As a simple exam-ple, the genealogical relationships among Joe, John, andBobby Kennedy are, in a real sense, the same as thoseamong the American football quarterbacks in the Man-ning family: Archie, Peyton, and Eli. The father-son

pairings and the elder-brother-younger-brother pairingscorrespond perfectly. That similarity between the twofamily structures illustrates the origin of the word iso-morphism (Greek iso-, same, and -morph, form orshape). But because the Kennedys are not the samepeople as the Mannings, the two genealogical structuresare merely isomorphic and not equal.Another example is more formal and more directly illus-trates the motivation for distinguishing equality from iso-morphism: the distinction between a nite-dimensionalvector space V and its dual space V* = { : V K} oflinear maps from V to its eld of scalars K. These spaceshave the same dimension, and thus are isomorphic as ab-stract vector spaces (since algebraically, vector spaces areclassied by dimension, just as sets are classied by car-dinality), but there is no natural choice of isomorphismV!V . If one chooses a basis for V, then this yields an

isomorphism: For all u. v V,

v7! v 2 V that such v(u) = vTu

This corresponds to transforming a column vector (ele-ment of V) to a row vector (element of V*) by transpose,but a dierent choice of basis gives a dierent isomor-phism: the isomorphism depends on the choice of ba-sis. More subtly, there is a map from a vector space V toits double dual V** = { x: V* K} that does not dependon the choice of basis: For all v V and V*,

v7! xv 2 V that such xv() = (v)

This leads to a third notion, that of a natural isomorphism:while V and V** are dierent sets, there is a naturalchoice of isomorphism between them. This intuitive no-tion of an isomorphism that does not depend on an ar-bitrary choice is formalized in the notion of a naturaltransformation; briey, that one may consistently identify,or more generally map from, a vector space to its doubledual, V !V , for any vector space in a consistent way.Formalizing this intuition is a motivation for the develop-ment of category theory.However, there is a case where the distinction betweennatural isomorphism and equality is usually not made.That is for the objects that may be characterized by auniversal property. In fact, there is a unique isomorphism,necessarily natural, between two objects sharing the sameuniversal property. A typical example is the set of realnumbers, which may be dened through innite decimalexpansion, innite binary expansion, Cauchy sequences,Dedekind cuts and many other ways. Formally these con-structions dene dierent objects, which all are solutionsof the same universal property. As these objects haveexactly the same properties, one may forget the methodof construction and considering them as equal. This iswhat everybody does when talking of "the set of the realnumbers. The same occurs with quotient spaces: they

24 CHAPTER 7. ISOMORPHISM

are commonly constructed as sets of equivalence classes.However, talking of set of sets may be counterintuitive,and quotient spaces are commonly considered as a pair ofa set of undetermined objects, often called points, anda surjective map onto this set.If one wishes to draw a distinction between an arbitraryisomorphism (one that depends on a choice) and a naturalisomorphism (one that can be done consistently), one maywrite for an unnatural isomorphism and for a naturalisomorphism, as in V V* and V V**. This conven-tion is not universally followed, and authors who wish todistinguish between unnatural isomorphisms and naturalisomorphisms will generally explicitly state the distinc-tion.Generally, saying that two objects are equal is reservedfor when there is a notion of a larger (ambient) space thatthese objects live in. Most often, one speaks of equalityof two subsets of a given set (as in the integer set exampleabove), but not of two objects abstractly presented. Forexample, the 2-dimensional unit sphere in 3-dimensionalspace

S2 := f(x; y; z) 2 R3 j x2 + y2 + z2 = 1gand the Riemann sphere bC

which can be presented as the one-point compactica-tion of the complex plane C {} or as the complexprojective line (a quotient space)

P1C := (C2 n f(0; 0)g)/(C)

are three dierent descriptions for a mathematical object,all of which are isomorphic, but not equal because theyare not all subsets of a single space: the rst is a subset ofR3, the second is C R2[note 4] plus an additional point,and the third is a subquotient of C2

In the context of category theory, objects are usually atmost isomorphicindeed, a motivation for the develop-ment of category theory was showing that dierent con-structions in homology theory yielded equivalent (iso-morphic) groups. Given maps between two objects Xand Y, however, one asks if they are equal or not (theyare both elements of the set Hom(X, Y), hence equalityis the proper relationship), particularly in commutativediagrams.

7.5 See also Bisimulation Heap (mathematics) Isometry Isomorphism class

Isomorphism theorem Universal property

7.6 Notes[1] For clarity, by inverse is meant inverse homomorphism or

inverse morphism respectively, not inverse function.

[2] The careful reader may note that A, B, C have a conven-tional order, namely alphabetical order, and similarly 1, 2,3 have the order from the integers, and thus one particularisomorphism is natural, namely

A 7! 1;B 7! 2;C 7! 3

More formally, as sets these are isomorphic, but not nat-urally isomorphic (there are multiple choices of isomor-phism), while as ordered sets they are naturally isomorphic(there is a unique isomorphism, given above), since nitetotal orders are uniquely determined up to unique isomor-phism by cardinality. This intuition can be formalized bysaying that any two nite totally ordered sets of the samecardinality have a natural isomorphism, the one that sendsthe least element of the rst to the least element of the sec-ond, the least element of what remains in the rst to theleast element of what remains in the second, and so forth,but in general, pairs of sets of a given nite cardinality arenot naturally isomorphic because there is more than onechoice of mapexcept if the cardinality is 0 or 1, wherethere is a unique choice.

[3] In fact, there are precisely 3! = 6 dierent isomorphismsbetween two sets with three elements. This is equal tothe number of automorphisms of a given three-element set(which in turn is equal to the order of the symmetric groupon three letters), and more generally one has that the set ofisomorphisms between two objects, denoted Iso(A;B);is a torsor for the automorphism group of A, Aut(A) andalso a torsor for the automorphism group of B. In fact, au-tomorphisms of an object are a key reason to be concernedwith the distinction between isomorphism and equality, asdemonstrated in the eect of change of