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Category theory is a general theory of
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additio ...
s and their relations that was introduced by Samuel Eilenberg and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces,
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
s, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism composition has similar properties as
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
( associativity and existence of identity morphisms). Morphisms are often some sort of
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
, but this is not always the case. For example, a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
may be viewed as a category with a single object, whose morphisms are the elements of the monoid. The second fundamental concept of category is the concept of a functor, which plays the role of a morphism between two categories $C_1$ and $C_2:$ it maps objects of $C_1$ to objects of $C_2$ and morphisms of $C_1$ to morphisms of $C_2$ in such a way that sources are mapped to sources and targets are mapped to targets (or, in the case of a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
, sources are mapped to targets and ''vice-versa''). A third fundamental concept is a
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
that may be viewed as a morphism of functors.

# Categories, objects, and morphisms

## Categories

A ''category'' ''C'' consists of the following three mathematical entities: * A class ob(''C''), whose elements are called ''objects''; * A class hom(''C''), whose elements are called
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s or
maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althou ...
or ''arrows''.
Each morphism ''f'' has a ''source object a'' and ''target object b''.
The expression , would be verbally stated as "''f'' is a morphism from ''a'' to ''b''".
The expression – alternatively expressed as , , or – denotes the ''hom-class'' of all morphisms from ''a'' to ''b''. * A
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary ope ...
∘, called ''composition of morphisms'', such that
for any three objects ''a'', ''b'', and ''c'', we have ::. :The composition of and is written as or ''gf'', governed by two axioms: ::1. Associativity: If , , and then ::: ::2. Identity: For every object ''x'', there exists a morphism called the ''
identity morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
for x'',
such that :::for every morphism , we have :::. :: From the axioms, it can be proved that there is exactly one
identity morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
for every object. ::Some authors deviate from the definition just given, by identifying each object with its identity morphism.

## Morphisms

Relations among morphisms (such as ) are often depicted using
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s, with "points" (corners) representing objects and "arrows" representing morphisms.
Morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s can have any of the following properties. A morphism is a: *
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
(or ''monic'') if implies for all morphisms . * epimorphism (or ''epic'') if implies for all morphisms . * ''bimorphism'' if ''f'' is both epic and monic. * isomorphism if there exists a morphism such that . * endomorphism if . end(''a'') denotes the class of endomorphisms of ''a''. * automorphism if ''f'' is both an endomorphism and an isomorphism. aut(''a'') denotes the class of automorphisms of ''a''. *
retraction Retraction or retract(ed) may refer to: Academia * Retraction in academic publishing, withdrawals of previously published academic journal articles Mathematics * Retraction (category theory) * Retract (group theory) * Retraction (topology) Huma ...
if a right inverse of ''f'' exists, i.e. if there exists a morphism with . * section if a left inverse of ''f'' exists, i.e. if there exists a morphism with . Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent: * ''f'' is a monomorphism and a retraction; * ''f'' is an epimorphism and a section; * ''f'' is an isomorphism.

# Functors

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. A (covariant) functor ''F'' from a category ''C'' to a category ''D'', written , consists of: * for each object ''x'' in ''C'', an object ''F''(''x'') in ''D''; and * for each morphism in ''C'', a morphism in ''D'', such that the following two properties hold: * For every object ''x'' in ''C'', ; * For all morphisms and , . A contravariant functor is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism in ''C'' must be assigned to a morphism in ''D''. In other words, a contravariant functor acts as a covariant functor from the opposite category ''C''op to ''D''.

# Natural transformations

A ''natural transformation'' is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors. If ''F'' and ''G'' are (covariant) functors between the categories ''C'' and ''D'', then a natural transformation η from ''F'' to ''G'' associates to every object ''X'' in ''C'' a morphism in ''D'' such that for every morphism in ''C'', we have ; this means that the following diagram is commutative: The two functors ''F'' and ''G'' are called ''naturally isomorphic'' if there exists a natural transformation from ''F'' to ''G'' such that η''X'' is an isomorphism for every object ''X'' in ''C''.

# Other concepts

## Universal constructions, limits, and colimits

Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies. Each category is distinguished by properties that all its objects have in common, such as the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we ''do not know'' whether an object ''A'' is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find '' universal properties'' that uniquely determine the objects of interest. Numerous important constructions can be described in a purely categorical way if the ''category limit'' can be developed and dualized to yield the notion of a ''colimit''.

## Equivalent categories

It is a natural question to ask: under which conditions can two categories be considered ''essentially the same'', in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called ''equivalence of categories'', which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.

## Further concepts and results

The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading. * The
functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
''D''''C'' has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories. * Duality: Every statement, theorem, or definition in category theory has a ''dual'' which is essentially obtained by "reversing all the arrows". If one statement is true in a category ''C'' then its dual is true in the dual category ''C''op. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships. * Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.

## Higher-dimensional categories

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of ''higher-dimensional categories''. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes". For example, a (strict)
2-category In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
s of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism. This process can be extended for all
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s ''n'', and these are called ''n''-categories. There is even a notion of '' ω-category'' corresponding to the
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
ω. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, se
John Baez, 'A Tale of ''n''-categories' (1996).

# Historical notes

Whilst specific examples of functors and natural transformations had been given by Samuel Eilenberg and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
in a 1942 paper on
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors (who discussed applications of category theory to the field of algebraic topology). Their work was an important part of the transition from intuitive and geometric homology to homological algebra, Eilenberg and Mac Lane later writing that their goal was to understand natural transformations, which first required the definition of functors, then categories.
Stanislaw Ulam Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapo ...
, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in some sense, a continuation of the work of
Emmy Noether Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
(one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure ( homomorphisms). Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes ( functors) that relate topological structures to algebraic structures (
topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
s) that characterize them. Category theory was originally introduced for the need of homological algebra, and widely extended for the need of modern algebraic geometry (
scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different s ...
). Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additio ...
and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to mathematical logic and
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...
(
categorical abstract machine The categorical abstract machine (CAM) is a model of computation for programs''Cousineau G., Curien P.-L., Mauny M.'' The categorical abstract machine. — LNCS, 201, Functional programming languages computer architecture.-- 1985, pp.~50-64. that ...
) came later. Certain categories called topoi (singular ''topos'') can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of,
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...
.
Topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
is a form of abstract
sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
, with geometric origins, and leads to ideas such as pointless topology. Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense). Category theory has been applied in other fields as well. For example, John Baez has shown a link between Feynman diagrams in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
and monoidal categories. Another application of category theory, more specifically: topos theory, has been made in mathematical music theory, see for example the book ''The Topos of Music, Geometric Logic of Concepts, Theory, and Performance'' by Guerino Mazzola. More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Rosebrugh (2003) and Lawvere and
Stephen Schanuel Stephen H. Schanuel (14 July 1933 – 21 July 2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory. Life While he was a graduate student at University of Chicago, ...
(1997) and Mirroslav Yotov (2012).

* Domain theory * Enriched category theory *
Glossary of category theory This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.) *Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, th ...
*
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
* Higher category theory * Higher-dimensional algebra * Important publications in category theory * Lambda calculus * Outline of category theory *
Timeline of category theory and related mathematics This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: * Categories of abstract algebraic structures including representation theory and universal algebra; * Homological algebra; * Homotopical ...

# References

## Sources

* * . * . * * * * * . * * * * * * * * * * * * * * Notes for a course offered as part of the MSc. in Mathematical Logic,
Manchester University The University of Manchester is a public university, public research university in Manchester, England. The main campus is south of Manchester city centre, Manchester City Centre on Wilmslow Road, Oxford Road. The university owns and operates majo ...
. * , draft of a book. * * Based on .

*

Theory and Application of Categories
an electronic journal of category theory, full text, free, since 1995.
nLab
a wiki project on mathematics, physics and philosophy with emphasis on the ''n''-categorical point of view.
The n-Category Café
essentially a colloquium on topics in category theory.
Category Theory
a web page of links to lecture notes and freely available books on category theory. * , a formal introduction to category theory. * * , with an extensive bibliography.
List of academic conferences on category theory
* — An informal introduction to higher order categories.
WildCats
is a category theory package for Mathematica. Manipulation and visualization of objects,
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s, categories, functors,
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
s, universal properties. * , a channel about category theory. * .
Video archive
of recorded talks relevant to categories, logic and the foundations of physics.
Interactive Web page
which generates examples of categorical constructions in the category of finite sets.

an instruction on category theory as a tool throughout the sciences.
Category Theory for Programmers
A book in blog form explaining category theory for computer programmers.
Introduction to category theory.
{{DEFAULTSORT:Category Theory Higher category theory Foundations of mathematics