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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology. Samuel Eilenberg ( September 30, 1913 — January 30, 1998) was a Polish and American Mathematician of Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

Category theory has several faces known not just to specialists, but to other mathematicians. "General abstract nonsense" refers, perhaps not entirely affectionately, to its high level of abstraction, compared to more classical branches of mathematics. Abstract nonsense, or general abstract nonsense, alternatively general nonsense, is a popular term used by Mathematicians to describe certain kinds of arguments Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Diagram chasing is a visual method of arguing with abstract 'arrows'. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points

Contents

Background

The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

Consider the following example. The class Grp of groups consists of all objects having a "group structure". In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject One can proceed to prove theorems about groups by making logical deductions from the set of axioms. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements For example, it is immediately proved from the axioms that the identity element of a group is unique. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that

Instead of focusing merely on the individual objects (e. g. groups) possessing a given structure, category theory emphasizes the morphisms — the structure-preserving mappings — between these objects. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and It turns out that by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function A group homomorphism between two groups "preserves the group structure" in a precise sense — it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps between topological spaces in topology and the study of diffeomorphisms in manifold theory. Continuity may refer to In mathematics: Continuous probability distribution or random variable in probability and statistics For Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable The notion of a category is an axiomatic formulation of this idea of relating mathematical structures to the structure-preserving functions between them. A systematic study of categories then allows us to prove general results from the axioms of a category.

A category is itself a type of mathematical structure, so we can look for 'processes' which preserve this structure in some sense. Such a process is called a functor. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories It associates to every object of one category an object of another category; and to every morphism in the first category a morphism in the second. By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them, we are studying the relationships between various classes of mathematical structures. This is a fundamental idea, which first surfaced in algebraic topology. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Difficult topological questions can be translated into algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group of a topological space, can be expressed as functors in this way. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

Constructions are often "naturally related", a vague notion at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal Many important constructions in mathematics can be studied in this context. 'Naturality' is a principle, like general covariance in physics, that cuts deeper than is initially apparent. In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the

Historical notes

In 1942-45, Samuel Eilenberg and Saunders Mac Lane were the first to introduce categories, functors, and natural transformations, as part of their work in topology, especially algebraic topology. Samuel Eilenberg ( September 30, 1913 — January 30, 1998) was a Polish and American Mathematician of Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Their work was an important part of the transition from intuitive and geometric homology to axiomatic homology theory. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations; in order to do that, functors had to be defined, which required categories.

Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland. Stanisław Marcin Ulam ( April 13, 1909 &ndash May 13, 1984) was a Polish Mathematician who participated in the Manhattan Eilenberg was Polish and studied mathematics there in the 1930s. Category theory is also, in some sense, a continuation of Emmy Noether's (one of Mac Lane's teachers) work in formalizing abstract processes. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving this structure. In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic formalization of the relation between structures and the processes preserving them. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject

The subsequent development of category theory was powered first by the computational needs of homological algebra; and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from In Mathematics, higher-order logic is distinguished from First-order logic in a number of ways

Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets 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 More recent efforts to introduce undergraduates to categories as a foundation for mathematics include Lawvere and Rosebrugh (2003) and Lawvere and Schanuel (1997). Francis William Lawvere (b February 9 1937 in Muncie Indiana is a Mathematician known for his work in Category theory, topos theory and the Philosophy

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. Categorical logic is a branch of Category theory within Mathematics, adjacent to Mathematical logic but in fact more notable for its connections to In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. In Category theory, a category is cartesian closed if roughly speaking any Morphism defined on a product of two objects can be naturally identified with a morphism In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense).

Categories, objects and morphisms

Main articles: category (mathematics) and morphism

A category C consists of the following three mathematical entities:

  • Associativity: If f : a → b, g : b → c and h : c → d then h\circ(g\circ f)=(h\circ g)\circ f, and
  • Identity: For every object x, there exists a morphism 1x : x → x called the identity morphism for x, such that for every morphism f : a → b, we have {\rm 1}_b\circ f=f=f\circ{\rm 1}_a. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

From these axioms, it can be proved that there is exactly one identity morphism for every object. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Some authors deviate from the definition just given by identifying each object with its identity morphism.

Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also The influence of commutative diagrams has been such that "arrow" and morphism are now synonymous. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and This article deals with the general meaning of the term "synonym"

Properties of morphisms

Some morphisms have important properties. A morphism f : a → b is:

Functors

Main article: functor

Functors are structure-preserving maps between categories. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Category theory, a branch of Mathematics, a functor is a special type of mapping 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 F:C → D, consists of:

such that the following two properties hold:

A contravariant functor F: C → D, is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism f : x → y in C must be assigned to a morphism F(f) : F(y) → F(x) in D. In other words, a contravariant functor is a covariant functor from the opposite category Cop to D. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

Natural transformations and isomorphisms

Main article: natural transformation

A natural transformation is a relation between two functors. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal 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 from F to G associates to every object x in C a morphism ηx : F(x) → G(x) in D such that for every morphism f : x → y in C, we have ηy o F(f) = G(f) o ηx; this means that the following diagram is commutative:

Commutative diagram defining natural transformations

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηx is an isomorphism for every object x in C. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also

Universal constructions, limits, and colimits

Main articles: universal property and limit (category theory)

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural Yet, in the definition of a category, objects are considered to be 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 these objects. But how can we define the empty set without referring to elements, or the product topology without referring to open sets?

The solution is to 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. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts

Equivalent categories

Main articles: equivalence of categories and isomorphism of categories

It is a natural question to ask, under which conditions two categories can be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G The major tool one employs to describe such a situation is called equivalence of categories. It 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.

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 is a category together with "morphisms between morphisms", i. In Category theory, a 2-category is a category with "morphisms between morphisms" 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. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal 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 C equipped with a Bifunctor &otimes: C Bicategories are a weaker notion of 2-dimensional categories where the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism. In Mathematics, a bicategory is a concept in Category theory used to extend the notion of category to handle the cases where the composition of morphisms

This process can be extended for all natural numbers n, and these are called n-categories. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, n-categories are a high-order generalization of the notion of category. There is even a notion of ω-category corresponding to the ordinal number ω. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. For a conversational introduction to these ideas, see Baez (1996).

See also

Notes

  1. ^ Note that a morphism that is both epic and monic is not necessarily an isomorphism! For example, in the category consisting of two objects A and B, the identity morphisms, and a single morphism f from A to B, f is both epic and monic but is not an isomorphism. This is a list of Category theory topics, by Wikipedia page Specific categories Category of sets Concrete category Algebra Theory of equations Hisab This is a glossary of properties and concepts in Category theory in Mathematics. Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. In Category theory and its applications to Mathematics, an enriched category is a category whose Hom-sets are replaced by objects from some other 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

References

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External links

Dictionary

category theory

-noun

  1. (mathematics) A branch of mathematics which deals with spaces and maps between them in abstraction, taking similar theorems from various disparate more concrete branches of mathematics and unifying them.
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