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Category theory is a general theory of
mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...
s and their relations. It 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 ...
and
Saunders Mac Lane Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near w ...
in the middle of the 20th century in their foundational work on
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. Category theory is used in most areas of mathematics. In particular, many constructions of new
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
also rely on category theory, such as
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarat ...
and
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
. A
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * 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 a ...
s of the category, and the
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s, which relate two objects called the ''source'' and the ''target'' of the morphism. Metaphorically, 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. Morphism composition has similar properties as
function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
(
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
and existence of an
identity morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...
for each object). Morphisms are often some sort of functions, but this is not always the case. For example, a
monoid In abstract algebra, 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 . Monoids are semigroups with identity ...
may be viewed as a category with a single object, whose morphisms are the elements of the monoid. The second fundamental concept of category theory is the concept of a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
, which plays the role of a morphism between two categories \mathcal_1 and \mathcal_2: it maps objects of \mathcal_1 to objects of \mathcal_2 and morphisms of \mathcal_1 to morphisms of \mathcal_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 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 natur ...
that may be viewed as a morphism of functors.


Categories, objects, and morphisms


Categories

A ''category'' \mathcal consists of the following three mathematical entities: * A
class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...
\text(\mathcal), whose elements are called ''objects''; * A class \text(\mathcal), whose elements are called
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s or
maps A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
or ''arrows''.

Each morphism f has a ''source object '' a and ''target object'' b.

The expression f:a \rightarrow b would be verbally stated as "f is a morphism from to ".

The expression \text(a, b) – alternatively expressed as \text_\mathcal(a, b), \text(a, b), or \mathcal(a, b) – 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, a binary operation ...
\circ, called ''composition of morphisms'', such that for any three objects ', ', and ', we have\circ : \text(b, c) \times \text(a, b) \mapsto \text(a, c)The composition of f : a \rightarrow b and g: b \rightarrow c is written as g \circ f or gf, governed by two axioms: *#
Associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
: If f: a \rightarrow b, g: b \rightarrow c, and h: c \rightarrow d then h \circ (g \circ f) = (h \circ g) \circ f *# Identity: For every object , there exists a morphism 1_x : x \rightarrow x (also denoted as \text_x) called the ''
identity morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...
for '', such that for every morphism f: a \rightarrow b, we have1_b \circ f = f = f \circ 1_a

From the axioms, it can be proved that there is exactly one

identity morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...
for every object.


Examples

* The category
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
** As the class of objects \text (\text), we choose the class of all sets. ** As the class of morphisms \text (\text), we choose the class of all functions. Therefore, for two objects and , i.e. sets, we have \text (A,B) to be the class of all functions such that . ** The composition of morphisms is simply the usual
function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
, i.e. for two morphisms and , we have , (g \circ f)(x) = g(f(x)), which is obviously associative. Furthermore, for every object we have the identity morphism \text_A to be the identity map \text_A : A \rightarrow A, \text_A (x) = x on


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, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 monomorphis ...
(or ''monic'') if implies for all morphisms . * an
epimorphism In category theory, an epimorphism 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 \implies g_1 = g_2. Epimorphisms are categorical analo ...
(or ''epic'') if implies for all morphisms . * a ''bimorphism'' if ''f'' is both epic and monic. * an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
if there exists a morphism such that . * an
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''. * an
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
if ''f'' is both an endomorphism and an isomorphism. aut(''a'') denotes the class of automorphisms of ''a''. * a retraction if a right inverse of ''f'' exists, i.e. if there exists a morphism with . * a 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

Functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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^ of a given Category (mathematics), category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal ...
''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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
: 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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
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 i ...
''D''''C'' has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The
Yoneda lemma In mathematics, the Yoneda lemma is a fundamental 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 (viewing a group as a ...
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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
: 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 in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat of all (small) categories, where a 2-morphism is a natural transformation between functors. ...
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 natur ...
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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 leas ...
ω. Higher-dimensional categories are part of the broader mathematical field of
higher-dimensional algebra In mathematics, especially (Higher category theory, higher) category theory, higher-dimensional algebra is the study of Categorification, categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebr ...
, 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 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 ...
and
Saunders Mac Lane Saunders Mac Lane (August 4, 1909 – April 14, 2005), born Leslie Saunders MacLane, was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near w ...
in a 1942 paper on
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, 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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
). 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 (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, 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, 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 (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
(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 morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s). Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes (
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 space ...
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 (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, and widely extended for the need of modern
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
(
scheme theory In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and 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 in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of stud ...
, as the latter studies
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
s, and the former applies to any kind of
mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...
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 Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
and
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
( 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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
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 th ...
.
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 (: sheaves) 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 d ...
, with geometric origins, and leads to ideas such as
pointless topology In mathematics, pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology that avoids mentioning point (mathematics), points, and in which the Lattice (order ...
.
Categorical logic __NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, cate ...
is now a well-defined field based on
type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...
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 ...
s, with applications in
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarat ...
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 computer ...
, where a
cartesian closed category In category theory, a Category (mathematics), category is Cartesian closed if, roughly speaking, any morphism defined on a product (category theory), product of two Object (category theory), objects can be naturally identified with a morphism defin ...
is taken as a non-syntactic description of a
lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
. 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, see applied category theory. 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, ap ...
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 introduced ...
in
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
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, Musicology, musicologist, jazz pianist, and 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 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, he ...
(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 computer ...
*
Categorical theory In mathematical logic, a theory is categorical if it has exactly one model ( up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a f ...
(logic) * 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 group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
*
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 morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
*
Higher-dimensional algebra In mathematics, especially (Higher category theory, higher) category theory, higher-dimensional algebra is the study of Categorification, categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebr ...
* Important publications in category theory *
Lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
* Outline of category theory * Timeline of category theory and related mathematics * Applied category theory


Notes


References


Citations


Sources

* * * . * . * * * * Joseph Goguen., "A Categorical Manifesto", ''Mathematical Structures in Computer Science'' 1 (1): 49–67 (1991). * * . * * * * * * * * * * * * * Notes for a course offered as part of the MSc. in
Mathematical Logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
,
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 of Manchester is c ...
. * . * , draft of a book. * * Based on .


Further reading

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External links


Theory and Application of Categories
an electronic journal of category theory, full text, free, since 1995.
Cahiers de Topologie et Géométrie Différentielle Catégoriques
an electronic journal of category theory, full text, free, funded in 1957.
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 (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
. Manipulation and visualization of objects,
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s, categories,
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s,
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 natur ...
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