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

Category theory From Wikipedia, the free encyclopedia Jump to: navigation search A category with objects X, Y, Z and morphisms f g This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations Please improve this article by introducing more precise citations where appropriate (November 2009) Category theory is an area of study in mathematics that 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

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

Category theory From Citizendium, the Citizens' Compendium Jump to: navigation search addthis_pub = 'citizendium'; addthis_logo = ''; addthis_logo_color = ''; addthis_logo_background = ''; addthis_brand = 'Citizendium'; addthis_options = ''; addthis_offset_top = ''; addthis_offset_left = ''; Main Article Talk Related Articles Bibliography External Links 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. Contents Definition and examples Evolution Role in Contemporary Mathematics Notes Definition and examples 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

Category theory - Definition 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 2 Historical notes 3 Categories 3.1 Definition ... 12 External links Background 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

Topic:Category theory 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 Contents Department description Department news Learning materials and learning projects Offsite Learning Materials ... edit Department description 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. edit Department news 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)

Home Discussion Topics Dictionary ... Login Pullback (category theory) Pullback (category theory) Topic Home Discussion 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.... , a branch of mathematics Mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.... , a pullback (also called a fibered product or Cartesian square ) is the limit Limit (category theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits.... 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... consisting of two morphism Morphism In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...

7. Category Theory - HaskellWiki 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 http://www.haskell.org/haskellwiki/Category_theory

Haskell Recent changes Edit this page Page history ... Printable version Not logged in Log in Help Category theory From HaskellWiki Categories Theoretical foundations Mathematics Haskell theoretical foundations General Mathematics Category theory Research Curry/Howard/Lambek Lambda calculus Alpha conversion ... Lambda abstraction Other Recursion Combinatory logic Chaitin's construction Turing machine ... 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. Contents Definition of a category Axioms Examples of categories Further definitions ... See also The Haskell wikibooks has an introduction to Category theory , written specifically with Haskell programmers in mind. edit 1 Definition of a category A category consists of two collections: Ob , the objects of Ar , the arrows of (which are not the same as Arrows defined in GHC Each arrow f in Ar has a domain, dom

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

Category Theory 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 Contents Introduction What is a category? History Aspects of category theory ... edit 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 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. edit What is a category? 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

Category Theory 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 Contents Introduction What is a category? History Aspects of category theory ... edit 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 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. edit What is a category? 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

10. Category Theory - HaskellWiki 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 http://haskell.org/haskellwiki/Category_theory

Haskell Recent changes Edit this page Page history ... Printable version Not logged in Log in Help Category theory From HaskellWiki Categories Theoretical foundations Mathematics Haskell theoretical foundations General Mathematics Category theory Research Curry/Howard/Lambek Lambda calculus Alpha conversion ... Lambda abstraction Other Recursion Combinatory logic Chaitin's construction Turing machine ... 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. Contents Definition of a category Axioms Examples of categories Further definitions ... See also The Haskell wikibooks has an introduction to Category theory , written specifically with Haskell programmers in mind. edit 1 Definition of a category A category consists of two collections: Ob , the objects of Ar , the arrows of (which are not the same as Arrows defined in GHC Each arrow f in Ar has a domain, dom

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. http://en.wikibooks.org/wiki/Haskell/Category_theory

Haskell/Category theory From Wikibooks, the open-content textbooks collection Haskell This page may need to be reviewed for quality. Jump to: navigation search Category theory Solutions Contents Introduction to categories Category laws Hask, the Haskell category Functors ... Notes Wider Theory Denotational semantics Equational reasoning Program derivation Category theory The Curry-Howard isomorphism 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. edit Introduction to categories 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

Category theory You don't need to be Editor-In-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 page. Next enter or edit the information that you would like to appear here. Once you are done editing, scroll down and click the Save page button at the bottom of the page. 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

the libarynth Trace: Category Theory 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 reading “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

Search the Internet with Kids.Net.Au Encyclopedia > Category theory Article Content Category theory 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

category theory Definition from Wiktionary, the free dictionary Jump to: navigation search edit English Wikipedia has an article on: Category theory Wikipedia edit Noun category theory uncountable 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. edit Translations branch of mathematics Czech: teorie kategorií cs (cs) f Finnish: kategoriateoria Swedish: kategoriteori sv (sv) c Retrieved from " http://en.wiktionary.org/wiki/category_theory Categories English nouns Mathematics ... Category theory Personal tools New features Log in / create account Namespaces Entry Discussion Variants Views Read Edit History Actions Search Navigation Main Page Community portal Preferences Requested entries ... Contact us Toolbox What links here Related changes Upload file Special pages ... Permanent link In other languages Svenska This page was last modified on 16 June 2010, at 17:07. Text is available under the Creative Commons Attribution/Share-Alike License ; additional terms may apply. See for details.

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

var skin = 'monobook';var stylepath = '/skins'; @import "/skins/monobook/artandpop_plus.css"; Category theory From The Art and Popular Culture Encyclopedia Jump to: navigation search Related e Wikipedia Wiktionary Tumblr ... Shop Featured visual 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

File:Pullback (category theory)1.png From Wikimedia Commons, the free media repository Jump to: navigation search File File history ... File usage on other wikis No higher resolution available. Pullback_(category_theory)1.png ‎ (518 × 417 pixels, file size: 14 KB, MIME type: image/png) Public domain Public domain false false This Math image should be recreated using vector graphics as an SVG file . This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is already available, please upload it. After uploading an SVG, replace this template with vector version available new image name Català Česky Dansk Deutsch ... Tiếng Việt Pullback diagram (from Category Theory) Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License , Version 1.2 or any later version published by the Free Software Foundation ; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License www.gnu.org/copyleft/fdl.html

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

Main Page See live article Alphabetical index Category theory 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 2 Historical notes 3 Categories 3.1 Definition ... 11 Literature Background 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/

Browse and Search the Library Home Math Topics Algebra Modern Algebra : Cat. Theory/Homolgcl Alg. Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Category Theory, Homological Algebra - Dave Rusin; The Mathematical Atlas 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>> All Sites - 21 items found, showing 1 to 21 Applied and Computational Category Theory - RISC-Linz, Austria A brief history and description of category theory, and some related links. From the Research�Institute�for�Symbolic�Computation. ...more>> Categories, Quantization, and Much More - John Baez

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

maxim ) wrote in nivanych Эта пиратская подборка, которая нарушает все мыслимые и немыслимые авторские права, будет доступна на openbittorrent.com: Download TORRENT File ./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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