Category theory is a general theory of

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

for any three objects ''a'', ''b'', and ''c'', we have ::. :The composition of and is written as or ''gf'', governed by two axioms: ::1.

such that :::for every morphism , we have :::. :: From the axioms, it can be proved that there is exactly one

^{op} to ''D''.

_{''X''} is an isomorphism for every object ''X'' in ''C''.

^{''C''} has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The ^{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.

John Baez, 'A Tale of ''n''-categories' (1996).

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

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

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 additi ...

s and their relations that was introduced by Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to a ...

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. In particular, many constructions of new mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...

s 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 t ...

s, completion, and duality.
A category is formed by two sorts of objects: the object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...

s 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, morphisms ...

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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

and existence of identity morphisms). Morphisms are often some sort of function, 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
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 ...

, 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 ...

, sources are mapped to targets and ''vice-versa''). A third fundamental concept is a natural transformation 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 calledmorphism
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 ...

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 op ...

∘, 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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

: 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 usingcommutative 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, 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, morphisms ...

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 monomorphism ...

(or ''monic'') if implies for all morphisms .
* epimorphism
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 objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f \ ...

(or ''epic'') if implies for all morphisms .
* ''bimorphism'' if ''f'' is both epic and monic.
* isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

if there exists a morphism such that .
* endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

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 if a right inverse of ''f'' exists, i.e. if there exists a morphism with .
* section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...

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

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 ...

s 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
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...

''C''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 ηOther 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 theempty 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. * Thefunctor 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 ...

''D''Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...

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''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 transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...

. 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
In mathematics, especially ( higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Higher-dimensional categories
A f ...

, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, seJohn Baez, 'A Tale of ''n''-categories' (1996).

Historical notes

Whilst specific examples of functors and natural transformations had been given bySamuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to a ...

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
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...

, 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 weapon ...

, 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 (homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

s). Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes (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 ...

s) 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
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...

, and widely extended for the need of modern algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...

(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
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...

, as the latter studies algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

s, 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 additi ...

and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sy ...

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) came later.
Certain categories called topoi
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 ...

(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. 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 In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...

.
Categorical logic is now a well-defined field based on type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

for intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...

s, with applications in functional programming
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...

and domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in compute ...

, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...

. 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
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, applic ...

has shown a link between Feynman diagrams
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...

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 relat ...

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
Guerino Bruno Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as a writer.
Education and career
Mazzola obtained his PhD in mathematics at University of Zürich in 1971 under the supervision of Herbert Groß a ...

.
More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere
Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Biography
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesd ...

and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).
See also

*Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in compute ...

* Enriched category theory In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the ho ...

* 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, t ...

* 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 In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher cate ...

* Higher-dimensional algebra
In mathematics, especially ( higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Higher-dimensional categories
A f ...

* Important publications in category theory
* Lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...

* Outline of category theory
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of ''objec ...

* Timeline of category theory and related mathematics
Notes

References

Citations

Sources

* * . * . * * * * * . * * * * * * * * * * * * * * Notes for a course offered as part of the MSc. inMathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sy ...

, Manchester University
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Unive ...

.
* , draft of a book.
*
* Based on .
Further reading

*External links

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
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...

. 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, morphisms ...

s, categories, 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 ...

s, natural transformations, 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