FACTOID # 12: It's not the government they hate: Washington DC has the highest number of hate crimes per capita in the US.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. 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. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942-1945, in connection with algebraic topology. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...


Category theory has several faces known, not just to specialists, but to other mathematicians. "Generalized abstract nonsense" refers, not entirely affectionately, to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting calculations in abstract algebra. Diagram chasing is a visual method of arguing with abstract 'arrows', and has appeared in a Hollywood film, as Jill Clayburgh proved the snake lemma (at the start of It's My Turn, 1980). Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology. Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... Jill Clayburgh (born April 30, 1944) is an American actress of stage, motion pictures, and television. ... In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ... For discussion of topoi in literary theory, see literary topos. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Pointless 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, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...


Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms that the identity element of a group is unique. In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... This article does not cite its references or sources. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... 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 set. ...


Instead of focusing merely on the individual objects (groups) possessing a given structure, as mathematical theories have traditionally done, category theory emphasizes the morphisms — the structure-preserving processes — between these objects. It turns out that by studying these morphisms, we are able to learn more about the structure of the 'objects' (groups). Here the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a very precise way — 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. In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...


A similar type of investigation occurs in many mathematical theories. 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. 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 structure. This is a fundamental idea, which first surfaced in algebraic topology. Difficult topological questions can be translated into algebraic questions which are much easier to solve. Basic constructions, such as the fundamental group of a topological space, can be expressed as functors in this way. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... Topological spaces are structures that allow one to formalize 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. 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 category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... This article or section is in need of attention from an expert on the subject. ...


Historical notes

Categories, functors and natural transformations were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 (1945) in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the late 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy Noether in formalizing abstract processes in the first half of the 20th-century. Noether realized that in order to understand a type of mathematical structure, one really needs to understand the processes preserving this structure. Eilenberg and Mac Lane gave an axiomatic formalization of this relation between structures and the processes preserving them. Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... This article does not cite its references or sources. ... Stanisław Marcin Ulam (April 13, 1909–May 13, 1984) was a Polish-American mathematician who helped develop the key theory behind the hydrogen bomb. ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a talented German-born mathematician, often said to be the best female mathematician who has ever lived. ... This article does not cite its references or sources. ...


Eilenberg and Mac Lane have said that their goal was to understand natural transformations; in order to do that, functors had to be defined; and to define functors one needed categories.


The subsequent development of the theory was powered first by the computational needs of homological algebra; and then by the axiomatic needs of algebraic geometry, the field most resistant to the Russell-Whitehead view of united foundations. General category theory, an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... The title page of the shortened version of the work, Principia Mathematica to *56. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... In mathematics, higher-order logic is distinguished from first-order logic in a number of ways. ...


Special categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as the foundation of mathematics. These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for constructive mathematics. One can say, in particular, that axiomatic set theory still hasn't been replaced by the category-theoretic commentary on it, in the everyday usage of mathematicians. The idea of bringing category theory into earlier, undergraduate teaching (signified by the difference between the Birkhoff-Mac Lane and later Mac Lane-Birkhoff abstract algebra texts) has hit noticeable opposition. In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ... This article or section is in need of attention from an expert on the subject. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...


Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus. At the very least, the use of category theory language allows one to clarify what exactly these related areas have in common (in an abstract sense). Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Functional programming is a programming paradigm that conceives computation as the evaluation of mathematical functions and avoids state and mutable data. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets 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 defined on one of the factors. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...


Categories, objects, and morphisms

A category C consists of the following three mathematical entities: In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

  • A class ob(C) of objects;
  • A class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: ab, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b) or C(a, b).)
  • A binary operation ○, called composition of morphisms, such that for any three objects a, b, and c, we have hom(a, b) × hom(b, c) → hom(a, c). The composition of f: ab and g: bc is written as gf or gf. (Some authors write fg.),

governed by two axioms: In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

  • Associativity: If f : ab, g : bc and h : cd then h ○ (gf) = (hg) ○ f, and
  • Identity: For every object x, there exists a morphism 1x : xx called the identity morphism for x, such that for every morphism f : ab, we have 1bf = f = f ○ 1a.

From these axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism. In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...


Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms. The influence of commutative diagrams has been such that "arrow" and morphism are now synonymous. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Synonyms (in ancient Greek syn συν = plus and onoma όνομα = name) are different words with similar or identical meanings. ...


Some properties of morphisms

A morphism f : ab is called

  • a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : xa.
  • an epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : bx.
  • an isomorphism if there exists a morphism g : ba with fg = 1b and gf = 1a.[1]
  • an endomorphism if a = b. The class of endomorphisms of a is denoted end(a).
  • an automorphism if f is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a).

In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In category theory an epimorphism (also called an epic morphism or an epi) is a morphism f : X → Y which is right-cancellable in the following sense: g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : Y → Z. Epimorphisms are analogues of surjective functions, but... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

Functors

Main article: functor

Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...


A (covariant) functor F from the category C to the category D

  • associates to each object x in C an object F(x) in D;
  • associates to each morphism f : xy a morphism F(f) : F(x) → F(y)

such that the following two properties hold:

  • F(1x) = 1F(x) for every object x in C.
  • F(gf) = F(g) ○ F(f) for all morphisms f : xy and g : yz.

A contravariant functor F from C to D is a functor that "turns morphisms around" ("reverses all the arrows"). Specifically, F is contravariant if whenever f : xy is a morphism in C, then F(f) : F(y) → F(x). The quickest way to define a contravariant functor is as a covariant functor from the opposite category Cop to D. In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...


Natural transformations and isomorphisms

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. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...


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 : xy in C, we have ηyF(f) = G(f) ○ ηx; this means that the following diagram is commutative: In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

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. Wikipedia does not have an article with this exact name. ...


Universal constructions, limits, and colimits

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. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. 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? In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... 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 of mathematics, like products and inverse limits. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...


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. 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 various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...


Equivalent 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. 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. In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ...


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 DC has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories.
  • Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows". If one statement is true in a category C then its dual will be true in the dual category Cop. 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; it can be seen as a more abstract and powerful view on universal properties.

In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

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.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. 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 category theory, a 2-category is a category with morphisms between morphisms. It can be formally defined as a category enriched over Cat (the category of catetgories and functors, with the monoidal structure induced by the composition). ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). ... In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i. ...


This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω. For a conversational introduction to these ideas, see Baez (1996). In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In mathematics, n-categories are a high-order generalization of the notion of category. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...


See also

Wikibooks
Wikibooks Abstract algebra has more about this subject:
Category theory

Image File history File links Wikibooks-logo-en. ... This is a list of category theory topics, by Wikipedia page. ... This is a list of important publications in mathematics, organized by field. ... This is a glossary of properties and concepts in category theory in mathematics. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ...

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.

References

Available online for free:

  • Awodey, Steve (2005) Category Theory.
  • Adámek, Jiří, Herrlich, Horst, & Strecker, George E. (1990) Abstract and concrete categories. John Wiley & Sons. ISBN 0-471-60922-6.
  • Barr, Michael, & Wells, Charles (2002) Toposes, triples and theories. Revised and corrected translation of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, 1983).
  • Leinster, Tom (2004) Higher operads, higher categories (London Math. Society Lecture Note Series 298). Cambridge Univ. Press.

Other:

  • Borceux, Francis (1994) Handbook of categorical algebra (Encyclopedia of Mathematics and its Applications 50-52). Cambridge Univ. Press.
  • Freyd, Peter J. & Scedrov, Andre, (1990) Categories, allegories (North Holland Mathematical Library 39). North Holland.
  • Hatcher, William S. (1982) The Logical Foundations of Mathematics, 2nd ed. Pergamon. Chpt. 8 is an idiosyncratic introduction to category theory, presented as a natural outgrowth of abstract algebra.
  • Lawvere, William, & Schanuel, Steve (1997) Conceptual mathematics: a first introduction to categories. Cambridge University Press.
  • Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.
  • Pedicchio, Maria Cristina & Tholen, Walter (2004) Categorical foundations (Encyclopedia of Mathematics and its Applications 97). Cambridge Univ. Press.
  • Taylor, Paul, 1999. Practical Foundations of Mathematics. Cambridge University Press.

Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Francis William Lawvere is a mathematician who is known for his work in category theory and the philosophy of mathematics. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... Categories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. ...

External links


  Results from FactBites:
 
Category Theory (Stanford Encyclopedia of Philosophy) (11810 words)
Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.
Category theory is, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures.
From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics.
PlanetMath: category theory (1640 words)
Category theory gives us tools for analyzing such functors: we can talk about natural transformations of functors, and in fact we can use these to assemble the category of functors from one category to another into a category, provided certain set-theoretic constraints are met (universes are a tool used to address these set-theoretic difficulties).
The fundamental theorem of Galois theory is that the functor from a subgroup of the Galois group of a field to its fixed field is an equivalence of categories.
Since categories are able to deal with isomorphisms and functors are compatible with the category structure, functors are generally a good way of attacking the problem “are these two objects isomorphic?”;.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m