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1. Category Theory - Wikipedia, The Free Encyclopedia
Category theory is an area of study in mathematics that deals in an abstract way with mathematical structures and relationships between them it abstracts
http://en.wikipedia.org/wiki/Category_theory

2. Category Theory : Good Math, Bad Math
Remember way back when I started writing about category theory? I said that the reason for doing that was because it s such a useful tool for talking about.
http://scienceblogs.com/goodmath/goodmath/category_theory/

3. Category Theory - Encyclopedia Article - Citizendium
Feb 8, 2010 Category theory is the mathematical field that studies categories, which are a certain kind of mathematical structure.
http://en.citizendium.org/wiki/Category_theory

Extractions: This is a draft article , under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to edit intro Category theory is the mathematical field that studies categories, which are a certain kind of mathematical structure. Categories are found throughout mathematics, and category theory thus has many mathematical applications. It is a basis for intuitionistic type theory, and as such has applications in computer science as a basis for functional programming semantics. To constitute a category, some things (called the morphisms of the category) must have three main features: i) Each morphism should have associated with them two objects, called the source and the target (or sometimes the domain and codomain) of the morphism. We write

4. Category Theory - Definition
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
http://www.wordiq.com/definition/Category_theory

Extractions: Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense ". The use of this phrase does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that a small minority consider it too abstract to be useful or interesting. See list of category theory topics for a breakdown of relevant articles. Contents showTocToggle("show","hide") 1 Background 12 External links A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups . Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms i.e. the structure-preserving maps between these objectsare emphasized. In the example of groups, these are the group homomorphisms . Then it becomes possible to relate different categories by functors , generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the

5. Topic:Category Theory - Wikiversity
Feb 22, 2010 Category theory is a relatively new birth that arose from the study of cohomology in topology and quickly broke free of its shackles to that
http://en.wikiversity.org/wiki/Topic:Category_theory

Extractions: From Wikiversity Jump to: navigation search This department is a part of the Subdivision of Higher Algebra Diagrams are used to simplify long arguments in category theory. This diagram was used to prove the Snake Lemma Category theory is a relatively new birth that arose from the study of cohomology in topology and quickly broke free of its shackles to that area and became a powerful tool that currently challenges set theory as a foundation of mathematics, although category theory requires more mathematical experience to appreciate and cannot in its current state be reasonably used to introduce mathematics. It has also found many applications in the physical sciences. The goal of this department is to familiarize the student with the theorems and goals of modern category theory. Prerequisites to full appreciation of this area of study includes knowledge of topology up to basic homology theory, and some basic idea of cohomology. Monday, August 28, 2006

6. Pullback (category Theory): Facts, Discussion Forum, And Encyclopedia Article
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them it abstracts from sets and functions respectively to
http://www.absoluteastronomy.com/topics/Pullback_(category_theory)

Extractions: Home Discussion Topics Dictionary ... Login Pullback (category theory) Overview In category theory Category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows.... of a diagram Diagram (category theory) In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a...

Nov 18, 2009 Category theory can be helpful in understanding Haskell s type system. There exists a Haskell category , of which the objects are Haskell

Extractions: Relational algebra Category theory can be helpful in understanding Haskell's type system. There exists a "Haskell category", of which the objects are Haskell types, and the morphisms from types a to b are Haskell functions of type a b . Various other Haskell structures can be used to make it a Cartesian closed category.

8. Category Theory - Wikibooks, Collection Of Open-content Textbooks
Introduction. This Wikibook is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group
http://en.wikibooks.org/wiki/Category_theory

Extractions: From Wikibooks, the open-content textbooks collection (Redirected from Category theory Jump to: navigation search Wikipedia has related information at Category Theory Wikiversity has learning materials about Introduction to Category Theory Contents Introduction Categories Functors Natural transformations Universal constructions ... References This Wikibook is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. The book contains many examples drawn from various branches of math. If you are not familiar with some of the kinds of math mentioned, don’t worry. If practically all the examples are unfamiliar, this book may be too advanced for you. A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics . (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that

9. Category Theory - Wikibooks, Collection Of Open-content Textbooks
Jun 25, 2010 This Wikibook is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract
http://en.wikibooks.org/wiki/Category_Theory

Extractions: From Wikibooks, the open-content textbooks collection Jump to: navigation search Wikipedia has related information at Category Theory Wikiversity has learning materials about Introduction to Category Theory Contents Introduction Categories Functors Natural transformations Universal constructions ... References This Wikibook is an introduction to category theory. It is written for those who have some understanding of one or more branches of abstract mathematics, such as group theory, analysis or topology. The book contains many examples drawn from various branches of math. If you are not familiar with some of the kinds of math mentioned, don’t worry. If practically all the examples are unfamiliar, this book may be too advanced for you. A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics . (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that

Category theory can be helpful in understanding Haskell's type system. There exists a Haskell category , of which the objects are Haskell types, and the morphisms from types a

Extractions: Relational algebra Category theory can be helpful in understanding Haskell's type system. There exists a "Haskell category", of which the objects are Haskell types, and the morphisms from types a to b are Haskell functions of type a b . Various other Haskell structures can be used to make it a Cartesian closed category.

11. Haskell/Category Theory - Wikibooks, Collection Of Open-content Textbooks
This article attempts to give an overview of category theory, in so far as it applies to Haskell. To this end, Haskell code will be given alongside the mathematical definitions.

Extractions: fix and recursion edit this chapter This article attempts to give an overview of category theory, in so far as it applies to Haskell. To this end, Haskell code will be given alongside the mathematical definitions. Absolute rigour is not followed; in its place, we seek to give the reader an intuitive feel for what the concepts of category theory are and how they relate to Haskell. A simple category, with three objects A B and C , three identity morphisms i d A i d B and i d C , and two other morphisms and . The third element (the specification of how to compose the morphisms) is not shown. A category is, in essence, a simple collection. It has three components: A collection of objects A collection of morphisms , each of which ties two objects (a source object and a target object ) together. (These are sometimes called

12. Category Theory - Wikidoc
You don't need to be EditorIn-Chief to add or edit content to WikiDoc. You can begin to add to or edit text on this WikiDoc page by clicking on the edit button at the top of this
http://www.wikidoc.org/index.php/Category_theory

Extractions: Jump to: navigation search 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 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. 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

13. Category_theory [the Libarynth]
Jun 8, 2007 Category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among other things,
http://libarynth.org/category_theory

Extractions: the libarynth Trace: http://plato.stanford.edu/entries/category-theory/ Category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among other things, how structures of different kinds are related to one another as well as the universal components of a family of structures of a given kind. The theory is philosophically relevant in more than one way. For one thing, it is considered by many as being an alternative to set theory as a foundation for mathematics. Furthermore, it can be thought of as constituting a theory of concepts. Finally, it sheds a new light on many traditional philosophical questions, for instance on the nature of reference and truth. http://math.ucr.edu/home/baez/categories.html “Basic Category Theory for Computer Scientists”, Benjamin C. Pierce “A Categorical Manifesto” by Goguen http://citeseer.nj.nec.com/goguen91categorical.html http://www.let.uu.nl/esslli/Courses/barr-wells.html http://scienceblogs.com/goodmath/goodmath/category_theory/

14. Kids.Net.Au - Encyclopedia > Category Theory
Kids.Net.Au is a search engine / portal for kids, children, parents, and teachers. The site offers a directory of child / kids safe websites, encyclopedia, dictionary, thesaurus
http://encyclopedia.kids.net.au/page/ca/Category_theory

Extractions: Search the Internet with Kids.Net.Au Article Content Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. Although originally developed in the context of algebraic geometry algebraic topology and universal algebra , it is now also used in various other branches of mathematics. Special categories called topoi can even serve as an alternative to set theory as the foundation of mathematics. Category theory is half-jokingly known as "abstract nonsense". Category theory is also used in a foundational way in functional programming , for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories. A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups . Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms , i.e. the structure preserving maps between these objects, are emphasized. In the example of groups, these are the group homomorphisms . Then it becomes possible to relate different categories by functors , generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the

15. Category Theory - Wiktionary
Jun 16, 2010 category theory. Definition from Wiktionary, the free dictionary Retrieved from http//en.wiktionary.org/wiki/category_theory
http://en.wiktionary.org/wiki/category_theory

16. Category Theory - The Art And Popular Culture Encyclopedia
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them it abstracts from sets and functions respectively to objects linked
http://www.artandpopularculture.com/Category_theory

Extractions: Portrait of Giacomo Casanova made (about In mathematics category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows One of the simplest examples of a category (which is a very important concept in topology ) is that of groupoid , defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types , and mathematical physics where they can be used to describe vector spaces . Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with algebraic topology Category theory has several faces known not just to specialists, but to other mathematicians. A term dating from the 1940s, " general abstract nonsense ", refers to its high level of abstraction, compared to more classical branches of mathematics.

17. File:Pullback (category Theory)1.png - Wikimedia Commons
May 17, 2010 Pullback_(category_theory)1.png (518 � 417 pixels, file size 14 KB, MIME type image/png) Pullback diagram (from Category Theory)
http://commons.wikimedia.org/wiki/File:Pullback_(category_theory)1.png

18. Category Theory
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
http://www.fact-index.com/c/ca/category_theory.html

Extractions: Main Page See live article Alphabetical index Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "abstract nonsense". See list of category theory topics for a breakdown of the relevant Wikipedia pages. Table of contents 1 Background 11 Literature A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups . Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms, i.e. the structure preserving maps between these objects, are emphasized. In the example of groups, these are the group homomorphisms . Then it becomes possible to relate different categories by functors , generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the fundamental group of a topological space , can be expressed as functors. Furthermore, different such constructions are often "naturally related" which leads to the concept of

19. The Math Forum - Math Library - Cat. Theory/Homolgcl Alg.
The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites
http://mathforum.org/library/topics/category_theory/

Extractions: A short article designed to provide an introduction to category theory, a comparatively new field of mathematics that provides a universal framework for discussing fields of algebra and geometry. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus. more>>

20. Category_theory: Книги по ТК
Translate this page 7 2009 5695397, Pierce B.C. Basic category theory for computer scientists.djvu. 4046669 , Saunders Mac Lane. Categories for Working Mathematician.
http://category_theory.livejournal.com/2190.html

Extractions: ./Abstract Algebra: Artin M. Algebra.djvu Dummit D.S., Foote R.M. Abstract algebra.djvu Картан, Эйленберг. Гомологическая Алгебра.djvu ./Abstract Algebra/Groups: Baker A.J. Finite Groups and their Representations.pdf Bechtell. The Theory of Groups.djvu Milne J.S. Group Theory.pdf Polites. An Introduction to the Group Theory.djvu Богопольский. Введение в теорию групп.djvu Дужин, Чеботаревский. От орнаментов до дифференциальных уравнений.djvu Холл. Теория групп.djvu ./Abstract Algebra/Groups/Representations: Barcelo H., Ram A. Combinatorial Representation Theory.pdf Finite Groups.pdf Fulton W., Harris J. Representation theory. A first course .djvu Fulton. Young tableau, representation theory and geometry.djvu

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