Category theory abFrom Wikipedia, the free encyclopedia

Contents

1 Automorphism 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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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ii CONTENTS

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 Identity function 165.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Isomorphism 186.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.1.1 Logarithm and exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.1.2 Integers modulo 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.1.3 Relation-preserving isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Isomorphism vs. bijective morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 Relation with equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Mathematical structure 237.1 Example: the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CONTENTS iii

8 Monomorphism 258.1 Relation to invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.5 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 Section (category theory) 279.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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 Definition

The exact definition 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 definition 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 definition 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 group

If the automorphisms of an object X 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 definition.

Inverses: by definition 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 field.

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

1

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2 CHAPTER 1. AUTOMORPHISM

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 finite-dimensional, the automorphism groupof V is the same as the general linear group, GL(V).

A field automorphism is a bijective ring homomor-phism from a field to itself. In the cases of therational numbers (Q) and the real numbers (R)there are no nontrivial field automorphisms. Somesubfields of R have nontrivial field 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 infinitely (uncountably) many wild au-tomorphisms (assuming the axiom of choice).[2][3]Field automorphisms are important to the theory offield extensions, in particular Galois extensions. Inthe case of a Galois extension L/K the subgroup ofall automorphisms of L fixing K pointwise is calledthe Galois group of the extension.

The field 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 differentiable manifoldM is a diffeomorphism from M to itself. Theautomorphism group is sometimes denotedDiff(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 sufficient fora morphism to be bijective to be an isomor-phism.

1.4 History

One 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 fifth root of unity, con-nected with the former fifth 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 definition holds in any unital ring or algebrawhere a is any invertible element. For Lie algebras thedefinition is slightly different.

1.6 See also

Endomorphism ring

Antiautomorphism

Frobenius automorphism

Morphism

Characteristic subgroup

https://en.wikipedia.org/wiki/Kernel_(algebra)https://en.wikipedia.org/wiki/Center_(group_theory)https://en.wikipedia.org/wiki/Trivial_grouphttps://en.wikipedia.org/wiki/Linear_algebrahttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Linear_transformationhttps://en.wikipedia.org/wiki/General_linear_grouphttps://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Ring_homomorphismhttps://en.wikipedia.org/wiki/Ring_homomorphismhttps://en.wikipedia.org/wiki/Field_(mathematics)https://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Complex_conjugatehttps://en.wikipedia.org/wiki/Uncountablehttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Field_extensionhttps://en.wikipedia.org/wiki/Galois_extensionhttps://en.wikipedia.org/wiki/Subgrouphttps://en.wikipedia.org/wiki/Galois_grouphttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Graph_automorphismhttps://en.wikipedia.org/wiki/Isomorphism#A_relation-preserving_isomorphismhttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Order_automorphismhttps://en.wikipedia.org/wiki/Geometryhttps://en.wikipedia.org/wiki/Motion_(geometry)https://en.wikipedia.org/wiki/Metric_geometryhttps://en.wikipedia.org/wiki/Isometryhttps://en.wikipedia.org/wiki/Isometry_grouphttps://en.wikipedia.org/wiki/Riemann_surfacehttps://en.wikipedia.org/wiki/Biholomorphyhttps://en.wikipedia.org/wiki/Conformal_maphttps://en.wikipedia.org/wiki/Riemann_spherehttps://en.wikipedia.org/wiki/M%C3%B6bius_transformationhttps://en.wikipedia.org/wiki/M%C3%B6bius_transformationhttps://en.wikipedia.org/wiki/Manifoldhttps://en.wikipedia.org/wiki/Diffeomorphismhttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Homeomorphism_grouphttps://en.wikipedia.org/wiki/Homeomorphism_grouphttps://en.wikipedia.org/wiki/William_Rowan_Hamiltonhttps://en.wikipedia.org/wiki/William_Rowan_Hamiltonhttps://en.wikipedia.org/wiki/Icosian_calculushttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Inner_automorphismhttps://en.wikipedia.org/wiki/Normal_subgrouphttps://en.wikipedia.org/wiki/Goursat%2527s_lemmahttps://en.wikipedia.org/wiki/Outer_automorphismhttps://en.wikipedia.org/wiki/Outer_automorphismhttps://en.wikipedia.org/wiki/Quotient_grouphttps://en.wikipedia.org/wiki/Unital_algebrahttps://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Algebra_over_a_fieldhttps://en.wikipedia.org/wiki/Unit_(ring_theory)https://en.wikipedia.org/wiki/Lie_algebrahttps://en.wikipedia.org/wiki/Endomorphism_ringhttps://en.wikipedia.org/wiki/Antiautomorphismhttps://en.wikipedia.org/wiki/Frobenius_automorphismhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/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), Clifford 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.

http://books.google.com/?id=kvoaoWOfqd8C&pg=PA376https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-67995-2https://en.wikipedia.org/wiki/Special:BookSources/3-540-67995-2http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/PaulBYale.pdfhttp://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/PaulBYale.pdfhttps://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.2307%252F2689301https://en.wikipedia.org/wiki/JSTORhttps://www.jstor.org/stable/2689301https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-00551-5https://en.wikipedia.org/wiki/William_Rowan_Hamiltonhttp://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdfhttp://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdfhttps://en.wikipedia.org/wiki/Philosophical_Magazinehttp://www.encyclopediaofmath.org/index.php/Automorphismhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/Automorphism.htmlhttps://en.wikipedia.org/wiki/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 differently from their uses in therest of mathematics. In category theory, a morphismobeys a set of conditions specific 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 significant 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 fields 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 first 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.

4

https://en.wikipedia.org/wiki/Mathematical_structurehttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Identity_functionhttps://en.wikipedia.org/wiki/Abstractionshttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Ring_theoryhttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Category_of_setshttps://en.wikipedia.org/wiki/Category_of_setshttps://en.wikipedia.org/wiki/Theoretical_computer_sciencehttps://en.wikipedia.org/wiki/Data_typehttps://en.wikipedia.org/wiki/Mathematical_physicshttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Samuel_Eilenberghttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Algebraic_topologyhttps://en.wikipedia.org/wiki/Algebraic_topologyhttps://en.wikipedia.org/wiki/Abstract_nonsensehttps://en.wikipedia.org/wiki/Homological_algebrahttps://en.wikipedia.org/wiki/Abstract_algebra

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 first 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-cific defining commutativity conditions. Functors can de-

fine (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 first category a morphism in the second.In fact, what we have done is define 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 first surfaced in algebraictopology. Difficult 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 transformations

Main article: Natural transformation

Abstracting yet again, some diagrammatic and/or sequen-tial constructions are often naturally related a vaguenotion, at first 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 Categories

A category C consists of the following three mathematicalentities:

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

A class hom(C), whose elements are calledmorphisms or maps or arrows. Each morphism fhas a source object a and target object b.

https://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Mathematical_structureshttps://en.wikipedia.org/wiki/Class_(set_theory)https://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Mathematical_proofhttps://en.wikipedia.org/wiki/Theoremhttps://en.wikipedia.org/wiki/Identity_elementhttps://en.wikipedia.org/wiki/Identity_elementhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Group_homomorphismhttps://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Smooth_functionhttps://en.wikipedia.org/wiki/Smooth_functionhttps://en.wikipedia.org/wiki/Manifold_theoryhttps://en.wikipedia.org/wiki/Pointed_topological_spacehttps://en.wikipedia.org/wiki/Finitary_relationhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Allegory_(category_theory)https://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Adjoint_functors#Motivationhttps://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Diagram_chasinghttps://en.wikipedia.org/wiki/Algebraic_topologyhttps://en.wikipedia.org/wiki/Algebraic_topologyhttps://en.wikipedia.org/wiki/Fundamental_grouphttps://en.wikipedia.org/wiki/Fundamental_groupoidhttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Groupoidhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/General_covariancehttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Class_(set_theory)https://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Map_(mathematics)

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 definitionjust 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 asuch 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 Functors

Main 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 specifically, 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 transformations

Main 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 different constructions yield the same result; thisis expressed by a natural isomorphism between the twofunctors.If F and G 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)

https://en.wikipedia.org/wiki/Binary_operationhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Identity_morphismhttps://en.wikipedia.org/wiki/Identity_morphismhttps://en.wikipedia.org/wiki/Identity_morphismhttps://en.wikipedia.org/wiki/Commutative_diagramhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Monomorphismhttps://en.wikipedia.org/wiki/Epimorphismhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Endomorphismhttps://en.wikipedia.org/wiki/Automorphismhttps://en.wikipedia.org/wiki/Retract_(category_theory)https://en.wikipedia.org/wiki/Section_(category_theory)https://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Opposite_categoryhttps://en.wikipedia.org/wiki/Opposite_categoryhttps://en.wikipedia.org/wiki/Natural_transformation

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 defining 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 definition 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 definespecial objects without referring to the internal structureof those objects. To define 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 find 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 categories

Main 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 results

The definitions 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 definition 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 defined by a univer-sal property; this can be seen as a more abstract andpowerful view on universal properties.

2.6.4 Higher-dimensional categories

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

https://en.wikipedia.org/wiki/Commutative_diagramhttps://en.wikipedia.org/wiki/Universal_propertyhttps://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Product_topologyhttps://en.wikipedia.org/wiki/Universal_propertyhttps://en.wikipedia.org/wiki/Limit_(category_theory)https://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Isomorphism_of_categorieshttps://en.wikipedia.org/wiki/Equivalence_of_categories#Exampleshttps://en.wikipedia.org/wiki/Functor_categoryhttps://en.wikipedia.org/wiki/Yoneda_lemmahttps://en.wikipedia.org/wiki/Dual_(category_theory)https://en.wikipedia.org/wiki/Adjoint_functorshttps://en.wikipedia.org/wiki/2-category

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 field 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 notes

In 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 defined,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 first by the computational needs of homologicalalgebra, and later by the axiomatic needs of algebraic ge-ometry, the field 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 flexibility 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 specific 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, andjustification 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-defined field 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 clarifies what exactly these related ar-eas have in common (in some abstract sense).Category theory has been applied in other fields aswell. For example, John Baez has shown a link betweenFeynman diagrams in Physics and monoidal categories.[6]Another application of category theory, more specifi-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 efforts 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

https://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Monoidal_categoryhttps://en.wikipedia.org/wiki/Bicategoryhttps://en.wikipedia.org/wiki/Natural_numberhttps://en.wikipedia.org/wiki/N-categoryhttps://en.wikipedia.org/wiki/Quasi-categoryhttps://en.wikipedia.org/wiki/Ordinal_numberhttps://en.wikipedia.org/wiki/%CE%A9_(ordinal_number)https://en.wikipedia.org/wiki/Higher-dimensional_algebrahttps://en.wikipedia.org/wiki/Ronald_Brown_(mathematician)http://math.ucr.edu/home/baez/week73.htmlhttp://math.ucr.edu/home/baez/week73.htmlhttps://en.wikipedia.org/wiki/Samuel_Eilenberghttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Algebraic_topologyhttps://en.wikipedia.org/wiki/Homology_(mathematics)https://en.wikipedia.org/wiki/Axiomhttps://en.wikipedia.org/wiki/Homology_theoryhttps://en.wikipedia.org/wiki/Stanislaw_Ulamhttps://en.wikipedia.org/wiki/Emmy_Noetherhttps://en.wikipedia.org/wiki/Homological_algebrahttps://en.wikipedia.org/wiki/Homological_algebrahttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Russell-Whiteheadhttps://en.wikipedia.org/wiki/Universal_algebrahttps://en.wikipedia.org/wiki/Semantichttps://en.wikipedia.org/wiki/Higher-order_logichttps://en.wikipedia.org/wiki/Toposhttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Constructivism_(mathematics)https://en.wikipedia.org/wiki/Toposhttps://en.wikipedia.org/wiki/Sheaf_(mathematics)https://en.wikipedia.org/wiki/Pointless_topologyhttps://en.wikipedia.org/wiki/Categorical_logichttps://en.wikipedia.org/wiki/Type_theoryhttps://en.wikipedia.org/wiki/Intuitionistic_logichttps://en.wikipedia.org/wiki/Functional_programminghttps://en.wikipedia.org/wiki/Domain_theoryhttps://en.wikipedia.org/wiki/Cartesian_closed_categoryhttps://en.wikipedia.org/wiki/Lambda_calculushttps://en.wiktionary.org/wiki/abstracthttps://en.wikipedia.org/wiki/John_Baezhttps://en.wikipedia.org/wiki/Feynman_diagramshttps://en.wikipedia.org/wiki/Guerino_Mazzolahttps://en.wikipedia.org/wiki/William_Lawverehttps://en.wikipedia.org/wiki/Stephen_Schanuelhttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Domain_theoryhttps://en.wikipedia.org/wiki/Enriched_categoryhttps://en.wikipedia.org/wiki/Glossary_of_category_theoryhttps://en.wikipedia.org/wiki/Higher_category_theoryhttps://en.wikipedia.org/wiki/Higher-dimensional_algebrahttps://en.wikipedia.org/wiki/List_of_publications_in_mathematics#Category_theoryhttps://en.wikipedia.org/wiki/Outline_of_category_theoryhttps://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematicshttps://en.wikipedia.org/wiki/Timeline_of_category_theory_and_related_mathematicshttps://en.wikipedia.org/wiki/Category_theory#CITEREFAwodey2006http://press.uchicago.edu/ucp/books/book/chicago/M/bo4158035.htmlhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-226-28862-5

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 first ob-served, 'category' has been defined in order to be able todefine 'functor' and 'functor' has been defined in order tobe able to define '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; Birkhoff, 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.

https://en.wikipedia.org/wiki/Category_theory#CITEREFMac_Lane1998http://math.ucr.edu/home/baez/rosetta.pdfhttp://math.ucr.edu/home/baez/rosetta.pdfhttps://en.wikipedia.org/wiki/ArXivhttps://arxiv.org/abs/0903.0340http://katmat.math.uni-bremen.de/acc/acc.htmhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-60922-6https://en.wikipedia.org/wiki/Steve_Awodeyhttp://books.google.com/books?id=IK_sIDI2TCwChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-19-151382-4https://en.wikipedia.org/wiki/Michael_Barr_(mathematician)https://en.wikipedia.org/wiki/Charles_Wells_(mathematician)http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.htmlhttp://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.htmlhttps://en.wikipedia.org/wiki/Michael_Barr_(mathematician)https://en.wikipedia.org/wiki/Charles_Wells_(mathematician)http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.htmlhttp://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.htmlhttps://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2178101https://en.wikipedia.org/wiki/Peter_J._Freydhttp://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.htmlhttp://books.google.com/books?id=fCSJRegkKdoChttp://books.google.com/books?id=fCSJRegkKdoChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-08-088701-2http://books.google.com/books?id=AwLc-12-7LMChttp://books.google.com/books?id=AwLc-12-7LMChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-486-45026-1https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-45026-1http://books.google.com/books?id=qNXuAAAAMAAJhttp://books.google.com/books?id=qNXuAAAAMAAJhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-88538-001-6http://books.google.com/books?id=K-SjOw_2gXwChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-27949-5https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-27949-5http://books.google.com/books?id=h3_7aZz9ZMoChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-01060-3http://books.google.com/books?id=h0zOGPlFmcQChttp://books.google.com/books?id=h0zOGPlFmcQChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-89485-2http://www.maths.gla.ac.uk/~tl/book.htmlhttp://www.maths.gla.ac.uk/~tl/book.htmlhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-53215-0https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-53215-0https://en.wikipedia.org/wiki/ArXivhttps://arxiv.org/abs/math.CT/0608040https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-691-14049-0https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2522659https://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematicianhttps://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematicianhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-98403-8https://en.wikipedia.org/wiki/Special:BookSources/0-387-98403-8https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=1712872https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-8218-1646-2https://en.wikipedia.org/wiki/Special:BookSources/0-8218-1646-2http://citeseer.ist.psu.edu/martini96element.htmlhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-226-51183-9https://en.wikipedia.org/wiki/Special:BookSources/0-226-51183-9https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-7643-5731-2https://en.wikipedia.org/wiki/Cambridge_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-83414-7https://en.wikipedia.org/wiki/Special:BookSources/0-521-83414-7https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:1034.18001http://books.google.com/books?id=ezdeaHfpYPwChttp://books.google.com/books?id=ezdeaHfpYPwChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-262-66071-6https://en.wikipedia.org/wiki/Special:BookSources/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 offered 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 Geometrical

Point 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 forMathematica. 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 finitesets.

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

http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdfhttp://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdfhttps://en.wikipedia.org/wiki/Mathematical_Logichttps://en.wikipedia.org/wiki/Manchester_Universityhttps://en.wikipedia.org/wiki/ArXivhttps://arxiv.org/abs/1001.4071http://books.google.com/books?id=iSCqyNgzamcChttp://books.google.com/books?id=iSCqyNgzamcChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-63107-5https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-63107-5http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdfhttp://www.dcs.ed.ac.uk/home/dt/CT/categories.pdfhttps://en.wikipedia.org/wiki/Category_theory#CITEREFMac_Lane1998https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-9384-5http://www.tac.mta.ca/tac/http://ncatlab.org/nlabhttps://en.wikipedia.org/wiki/Andr%C3%A9_Joyalhttp://ncatlab.org/nlabhttp://www.logicmatters.net/categories/https://en.wikipedia.org/wiki/CiteSeer#CiteSeerXhttp://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.3264http://katmat.math.uni-bremen.de/acc/acc.pdfhttp://katmat.math.uni-bremen.de/acc/acc.pdfhttp://plato.stanford.edu/entries/category-theoryhttps://en.wikipedia.org/wiki/Stanford_Encyclopedia_of_Philosophyhttp://www.mta.ca/~cat-dist/http://math.ucr.edu/home/baez/week73.htmlhttp://wildcatsformma.wordpress.com/https://en.wikipedia.org/wiki/Mathematicahttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Functorhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Universal_propertieshttps://www.youtube.com/user/TheCatstershttps://en.wikipedia.org/wiki/YouTubehttp://planetmath.org/?op=getobj&from=objects&id=5622https://en.wikipedia.org/wiki/PlanetMathhttp://categorieslogicphysics.wikidot.com/eventshttp://www.j-paine.org/cgi-bin/webcats/webcats.phphttp://category-theory.mitpress.mit.edu/index.html

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 Automorphisms

Main 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 define the notion of orbits ofelements, etc.Depending on the additional structure defined 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

https://en.wikipedia.org/wiki/Endomorphic_(disambiguation)https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Orthogonal_projectionhttps://en.wikipedia.org/wiki/Linear_operatorhttps://en.wikipedia.org/wiki/Automorphismhttps://en.wikipedia.org/wiki/Homomorphismhttps://en.wikipedia.org/wiki/Mathematical_objecthttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Linear_maphttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Group_homomorphismhttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Category_of_setshttps://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Monoidhttps://en.wikipedia.org/wiki/Automorphismhttps://en.wikipedia.org/wiki/Inverse_elementhttps://en.wikipedia.org/wiki/Automorphismhttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Automorphism_grouphttps://en.wikipedia.org/wiki/Endomorphism_ringhttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Endomorphism_ringhttps://en.wikipedia.org/wiki/Module_(mathematics)https://en.wikipedia.org/wiki/Preadditive_categoryhttps://en.wikipedia.org/wiki/Near-ringhttps://en.wikipedia.org/wiki/Regular_modulehttps://en.wikipedia.org/wiki/Concrete_categoryhttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Unary_operatorhttps://en.wikipedia.org/wiki/Action_(group_theory)https://en.wikipedia.org/wiki/Orbit_(group_theory)https://en.wikipedia.org/wiki/Topologyhttps://en.wikipedia.org/wiki/Metric_(mathematics)https://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Bounded_functionhttps://en.wikipedia.org/wiki/Operator_theoryhttps://en.wikipedia.org/wiki/Operator_theory

12 CHAPTER 3. ENDOMORPHISM

3.4 Endofunctions

An 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 onefinds 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 floor 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.

https://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Homomorphismhttps://en.wikipedia.org/wiki/Permutationhttps://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Directed_pseudoforesthttps://en.wikipedia.org/wiki/Directed_pseudoforesthttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Adjoint_endomorphismhttps://en.wikipedia.org/wiki/Frobenius_endomorphismhttps://en.wikipedia.org/wiki/Nathan_Jacobsonhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-486-47189-1http://www.encyclopediaofmath.org/index.php?title=p/e035600https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4http://planetmath.org/?op=getobj&from=objects&id=7462https://en.wikipedia.org/wiki/PlanetMath

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 : X Y 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 algebradefine 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 Examples

Every 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, finite 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 field K and K-lineartransformations.

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 Hausdorff 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

https://en.wikipedia.org/wiki/Epimorphosishttps://en.wikipedia.org/wiki/Category_theoryhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Cancellation_propertyhttps://en.wikipedia.org/wiki/Surjective_functionhttps://en.wikipedia.org/wiki/Dual_(category_theory)https://en.wikipedia.org/wiki/Monomorphismhttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Dual_(category_theory)https://en.wikipedia.org/wiki/Abstract_algebrahttps://en.wikipedia.org/wiki/Universal_algebrahttps://en.wikipedia.org/wiki/Surjectivehttps://en.wikipedia.org/wiki/Homomorphismhttps://en.wikipedia.org/wiki/Epimorphism#Terminologyhttps://en.wikipedia.org/wiki/Concrete_categoryhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Surjectivehttps://en.wikipedia.org/wiki/Category_of_setshttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Partially_ordered_sethttps://en.wikipedia.org/wiki/Monotone_functionhttps://en.wikipedia.org/wiki/Category_of_groupshttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Group_homomorphismhttps://en.wikipedia.org/wiki/Otto_Schreierhttps://en.wikipedia.org/wiki/Subgrouphttps://en.wikipedia.org/wiki/Equaliser_(mathematics)https://en.wikipedia.org/wiki/Free_producthttps://en.wikipedia.org/wiki/Free_producthttps://en.wikipedia.org/wiki/Finite_groupshttps://en.wikipedia.org/wiki/Category_of_abelian_groupshttps://en.wikipedia.org/wiki/Abelian_grouphttps://en.wikipedia.org/wiki/Category_of_vector_spaceshttps://en.wikipedia.org/wiki/Vector_spacehttps://en.wikipedia.org/wiki/Field_(mathematics)https://en.wikipedia.org/wiki/Linear_transformationhttps://en.wikipedia.org/wiki/Linear_transformationhttps://en.wikipedia.org/wiki/Module_(mathematics)https://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Module_homomorphismhttps://en.wikipedia.org/wiki/Module_homomorphismhttps://en.wikipedia.org/wiki/Quotient_modulehttps://en.wikipedia.org/wiki/Zero_maphttps://en.wikipedia.org/wiki/Category_of_topological_spaceshttps://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Trivial_topologyhttps://en.wikipedia.org/wiki/Compact_spacehttps://en.wikipedia.org/wiki/Hausdorff_spacehttps://en.wikipedia.org/wiki/Urysohn%2527s_Lemma

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 finitely 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 Hausdorff 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 differs 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 Properties

Every 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 definition 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 definition 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 define the group K = im(f) =f(G) and then write f as the composition of the surjectivehomomorphism G K which is defined 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).

https://en.wikipedia.org/wiki/Monoid_(category_theory)https://en.wikipedia.org/wiki/Inclusion_maphttps://en.wikipedia.org/wiki/Group_ringhttps://en.wikipedia.org/wiki/Category_of_ringshttps://en.wikipedia.org/wiki/Ring_homomorphismhttps://en.wikipedia.org/wiki/Commutative_ringhttps://en.wikipedia.org/wiki/Commutative_ringhttps://en.wikipedia.org/wiki/Localization_of_a_ringhttps://en.wikipedia.org/wiki/Category_of_commutative_ringshttps://en.wikipedia.org/wiki/Finitely_generated_objecthttps://en.wikipedia.org/wiki/Finitely_generated_objecthttps://en.wikipedia.org/wiki/Prime_idealhttps://en.wikipedia.org/wiki/Field_of_fractionshttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9briquehttps://en.wikipedia.org/wiki/Hausdorff_spacehttps://en.wikipedia.org/wiki/Dense_sethttps://en.wikipedia.org/wiki/Injectivehttps://en.wikipedia.org/wiki/Monoidhttps://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Directed_graphhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Section_(category_theory)https://en.wikipedia.org/wiki/Toposhttps://en.wikipedia.org/wiki/Monic_morphismhttps://en.wikipedia.org/wiki/Monic_morphismhttps://en.wikipedia.org/wiki/Subcategoryhttps://en.wikipedia.org/wiki/Equivalence_of_categorieshttps://en.wikipedia.org/wiki/Duality_(category_theory)https://en.wikipedia.org/wiki/Injectivehttps://en.wikipedia.org/wiki/Natural_transformationhttps://en.wikipedia.org/wiki/Functor_categoryhttps://en.wikipedia.org/wiki/Coequalizerhttps://en.wikipedia.org/wiki/Cokernel

4.5. SEE ALSO 15

4.3 Related concepts

Among 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 satisfies 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 define 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 aredefined to be the quotient objects of X.

4.4 Terminology

The companion terms epimorphism and monomorphismwere first 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 difficult, for example, to classify all the epimor-phisms of rings. In general, epimorphisms are their own

unique concept, related to surjections but fundamentallydifferent.

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.

https://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Bimorphismhttps://en.wikipedia.org/wiki/Half-open_intervalhttps://en.wikipedia.org/wiki/Unit_circlehttps://en.wikipedia.org/wiki/Topological_subspacehttps://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Euler%2527s_formulahttps://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Ring_(algebra)https://en.wikipedia.org/wiki/Quotient_objecthttps://en.wikipedia.org/wiki/Equivalence_relationhttps://en.wikipedia.org/wiki/Monomorphismhttps://en.wikipedia.org/wiki/Bourbakihttps://en.wikipedia.org/wiki/Surjective_functionhttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/Saunders_Mac_Lanehttps://en.wikipedia.org/wiki/List_of_category_theory_topicshttp://katmat.math.uni-bremen.de/acc/acc.pdfhttps://en.wikipedia.org/wiki/Special:BookSources/0471609226https://en.wikipedia.org/wiki/Special:BookSources/0471609226https://en.wikipedia.org/wiki/Special:BookSources/0965521141http://www.encyclopediaofmath.org/index.php?title=p/e035890https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/American_Mathematical_Monthlyhttps://en.wikipedia.org/wiki/Special:BookSources/0521804442https://en.wikipedia.org/wiki/Special:BookSources/0521804442

Chapter 5

Identity function

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

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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.

5.1 Definition

Formally, if M is a set, the identity function f on M isdefined