1. Category Theory - Wikipedia, The Free Encyclopedia In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches http://en.wikipedia.org/wiki/Category_theory

Category theory From Wikipedia, the free encyclopedia Jump to: navigation search Category theory Portal 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 Jill Clayburgh proved the snake lemma (at the start of It's My Turn Topos theory is a form of abstract sheaf theory , with geometric origins, and leads to ideas such as pointless topology Contents Background Historical notes Categories, objects and morphisms

2. Category Theory (Stanford Encyclopedia Of Philosophy) category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical http://plato.stanford.edu/entries/category-theory/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Category Theory First published Fri Dec 6, 1996; substantive revision Thu Aug 30, 2007 1. General Definitions, Examples and Applications 2. Brief Historical Sketch 3. Philosophical Significance Bibliography ... Related Entries 1. General Definitions, Examples and Applications 1.1 Definitions et al . 2000, 2001, 2002). The very definition of a category is not without philosophical importance, since one of the objections to category theory as a foundational framework is the claim that since categories are defined Definition : A mapping e will be called an identity if and only if the existence of any product e e implies that e e Definition C is an aggregate Ob of abstract elements, called the objects of C , and abstract elements Map , called mappings of the category. The mappings are subject to the following five axioms: is defined. When either is defined, the associative law are defined. e e is defined, and at least one identity e such that e (C4) The mapping e X corresponding to each object X is an identity.

3. Categories Home Page Web page for the category theory mailing list. http://www.mta.ca/~cat-dist/

Categories List How to use the list Archives Moderator Conferences of interest ... Addresses - electronic and postal General, Seminar-related, and Local Sites Theory and Applications of Categories - refereed electronic journal. TeX Macros for diagrams Using the list: Articles for posting should be sent to categories@mta.ca Administrative items (subscriptions, address changes etc.) should be sent to categories-request@mta.ca Usually, items of this sort sent to `categories@mta.ca' will not be posted. Policy: The moderator will not modify articles except for minor typographical and formatting changes, therefore no offensive or defamatory material should be sent (it will be returned and not posted), and inflammatory posts are discouraged. Nevertheless, wide latitude for vigorous debate is allowed. Please do not send attachments, html or excessively long (greater than about 10K) postings. Postings with attachments will be discarded, any with html may be. Return to top. Archives M. Alsani has created a selected list of CT email, also sorted by thread, from August 1999 to February 2002 at http://north.ecc.edu/alsani/cat-dist2html/index.html

4. Categories In fact, MacLane said I did not invent category theory to talk about functors. I invented it to talk about natural transformations. Huh? Wait and see. http://math.ucr.edu/home/baez/categories.html

Categories, Quantization, and Much More John Baez April 12, 2006 Quantum theory can be thought of as the generalization of classical mechanics you get by dropping the assumption that observable quantities like position and momentum commute. In quantum theory one thus learns to like noncommutative, but still associative, algebras. It is interesting however to note why associativity without commutativity is studied so much more than commutativity without associativity. Basically, because most of our examples of binary operations can be interpreted as composition of functions. For example, if write simply x for the operation of adding x to a real number (where x is a real number), then x + y is just x composed with y. Composition is always associative so the + operation is associative! If we try to generalize the heck out of the concept of a group, keeping associativity as a sacred property, we get the notion of a category. Categories are some of the most basic structures in mathematics. They were created by Samuel Eilenberg and Saunders MacLane. In fact, MacLane said: "I did not invent category theory to talk about functors. I invented it to talk about natural transformations." Huh? Wait and see.

5. Introduction To Category Theory Lecture notes by Maarten M. Fokkinga introducing some important notions from category theory, in particular adjunctions. Proofs are given in a calculational http://wwwhome.cs.utwente.nl/~fokkinga/mmf92b.html

A Gentle Introduction to Category Theory - the calculational approach Maarten M Fokkinga In these notes we present the important notions from category theory. The intention is to provide a fairly good skill in manipulating with those concepts formally. What you probably will not acquire from these notes is the ability to recognise the concepts in your daily work when that differs from algorithmics, since we give only a few examples and those are taken from algorithmics. For such an ability you need to work through many, very many examples, in diverse fields of applications. Full paper reachable via: here (80 pages). Bibtex data

6. Category Theory Authors/titles Recent Submissions category theory. Authors and titles for recent submissions Subjects Algebraic Topology (math.AT); category theory (math.CT) http://arxiv.org/list/math.CT/recent

arXiv.org math math.CT Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Category Theory Authors and titles for recent submissions Wed, 12 Mar 2008 Tue, 11 Mar 2008 Mon, 10 Mar 2008 Fri, 7 Mar 2008 ... Mon, 3 Mar 2008 [ total of 7 entries: [ showing up to 25 entries per page: fewer more Wed, 12 Mar 2008 arXiv:0803.1642 (cross-list from math.GT) [ pdf other Title: Categorifying Coloring Numbers Authors: John Armstrong Comments: To appear in "Interactions between Representation Theory, Quantum Field Theory, Category Theory, and Quantum Information Theory" conference proceedings Subjects: Geometric Topology (math.GT) ; Category Theory (math.CT) Tue, 11 Mar 2008 arXiv:0803.1408 ps pdf other Title: Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras Authors: Thomas M. Fiore Po Hu Igor Kriz Comments: 21 pages. This paper will appear in the Advances in Mathematics Subjects: Category Theory (math.CT) ; High Energy Physics - Theory (hep-th); Algebraic Topology (math.AT) Mon, 10 Mar 2008 arXiv:0803.1133

7. CTO : Category Theory 101 Note this paper tells you Why category theory Matters, but it will not be of much help in actually learning category theory perhaps will help you to http://tunes.org/wiki/category_20theory_20101.html

CTO CLiki for the TUNES project Home Recent Changes About CLiki Text Formatting ... Create New Page Category Theory 101 A Learning Lounge course about Category Theory The basics: A category is a thing with objects and arrows (called morphism s) that lead between the objects. The arrows have heads and tails. They are abstract in the sense that they can represent anything with complex structure or even no structure at all. Many categories are different, and there are types of categories. All categories follow some basic rules. The differences otherwise can be enormous, though: For every object there is an identity arrow over that object that just leads from that object to that object. There may be other identities over that object, but one is distinctly the identity If one arrow leads to an object from which another arrow leads, then those arrows can compose. All such arrows compose, but what you can say about the resulting arrow differs from category to category. Some arrows are the reverse or inverse of others. There are some basic examples of categories: one is Set, whose objects are sets and whose arrows are (total) functions between sets. The category Set's natural composition is therefore function composition.

8. Front: Math.CT Category Theory Section of the eprint arXiv dealing with category theory, including such topics as enriched categories, topoi, abelian categories, monoidal categories, http://front.math.ucdavis.edu/math.CT

Front for the arXiv Fri, 14 Mar 2008 Front math CT search register submit journals ... iFAQ math.CT Category Theory Calendar Search Atom feed Authors: AB CDE FGH IJK ... U-Z Search Author Title/ID Abstract+ Category articles per page Show Search help Recent New articles (last 12) 11 Mar arXiv:0803.1408 Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras. Thomas M. Fiore , Po Hu , Igor Kriz math.CT math.AT physics.hep-th 10 Mar arXiv:0803.1133 Opposite relation on dual polar spaces and half-spin Grassmann spaces. Mariusz Kwiatkowski , Mark Pankov math.CT math.GR 7 Mar arXiv:0803.0853 Girard couples of quantales. J. M. Egger , David Kruml math.CT math.LO math.QA 3 Mar arXiv:0802.4414 Semigroup cohomology as a derived functor. A. A. Kostin , B. V. Novikov Filomat, math.CT math.GR Cross-listings 12 Mar arXiv:0803.1642 Categorifying Coloring Numbers. John Armstrong math.GT math.CT 7 Mar arXiv:0803.0812 Archimedean Type Conditions in Categories. Elemer E. Rosinger math.GM math.CT Revisions 4 Mar arXiv:0708.1741 The inner automorphism 3-group of a strict 2-group. David Roberts , Urs Schreiber math.CT

9. Category Theory category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic http://www.andrew.cmu.edu/course/80-413-713/

Category Theory Spring 2006 Course Information Place: BH 225B Time: TR 3 - 4:20 Instructor: Steve Awodey Office: Baker 152 (mail: Baker 135) Office Hour: Monday 3 - 4, or by appointment Phone: x8947 Email: awodey@andrew Secretary: Baker 135 TA: Michael Warren Office: Baker 135 Office Hour: Fridays 10:30 11:20 in BH 150 Webpage: www.andrew.cmu.edu/course/80-413-713 Overview Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations. This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines. To be followed by a Fall course on categorical logic. Prerequisites Some familiarity with abstract algebra or logic.

10. Centre De Recherche En Théorie Des Catégories -- Montréal category theory Research Centre. Announcements of weekly seminars, conferences, and other research activities in category theory. http://www.math.mcgill.ca/triples/

Category Theory Research Center Translation? List of current seminars Some upcoming meetings What is Category Theory? Category Theory at McGill A personal account by Marta Bunge. Category Theory - an expository article from the Stanford Encyclopedia of Philosophy. Theory page. Categorical Logic , an entry by J. Lambek from the Encyclopaedia of Mathematics (Springer). Category , another article from the Encyclopaedia of Mathematics (Springer). Categories, Quantization, and Much More - an introduction by John Baez. (You might also explore his series "This Week's Finds in Mathematical Physics" which often has references to logic , category theory, and particularly n -category theory and more n -category theory Toposes, Triples and Theories A classic text by M Barr and C Wells, now freely available on-line (TACReprints). This is also available at the authors' own website Category Theory for Computing Science (Ordering information for another classic text by M Barr and C Wells, the current version published by the CRM.) Acyclic Models (Ordering information for a recent book by M Barr.)

11. Alsani's Descent & Category Theory WebPage! Selected category theory Email - from the category theory mailing list; category theory Archives - from the Front for the Mathematics ArXiv site http://north.ecc.edu/alsani/descent.html

D ESCENT A ND C ATEGORY T HEORY C ONNECTIONS! M. Alsani; alsani@ecc.edu This page is merely a launching pad to sites of interest in Descent or Category Theory. Selected Category Theory E-mail from the Category Theory mailing list Category Theory Archives from the Front for the Mathematics ArXiv site Category Theory Stanford Encyclopedia of Philosophy - Archives CATEGORIES HOME PAGE Bob Rosebrugh Ccard 2.0 - or : How to make fun out of something highly abstract. Category Theory for Beginners Steve Easterbrook TEORIA E APPLICAZIONI DELLE CATEGORIE University of Genoa, Italy Kategorielle Methoden in Algebra und Topologie Texte d'Alexandre Grothendieck (in progress) Categorical Geometry Zhaohua Luo page F. William Lawvere page John Duskin page Toposes, Triples and Theories - A classic text by M Barr and C Wells Descent Theory and its Higher Dimensional Analogues Descent theory and Amitsur cohomology of adjoint functors Slides by Dragos Stefan Geometric and Logical Aspects of Descent Theory Oberwolfach 1995. Descent Theory of Coalgebras and Hopf Algebras DESCENT OF COHERENT SHEAVES AND COMPLEXES TO GEOMETRIC INVARIANT THEORY Etale descent for two-primary algebraic K-theory of totally imaginary number fields , by J. Rognes and C. Weibel A DESCENT THEOREM IN TOPOLOGICAL K-THEORY , by Max Karoubi Category Theory at McGill Marta Bunge page W. Tholen Page

12. 18: Category Theory, Homological Algebra category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. http://www.math.niu.edu/~rusin/known-math/index/18-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 18: Category theory, homological algebra Introduction Category theory, a comparatively new field of mathematics, 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 A survey article which discusses the roles of categories and topoi in twentieth-century mathematics. Applications and related fields The word "category" is used to mean something completely different in general topology Subfields General theory of categories and functors Special categories Categories and algebraic theories Categories with structure Abelian categories Categories and geometry Homological algebra, see also 13DXX, 16EXX, 55UXX This is among the smaller areas in the Math Reviews database. Browse all (old) classifications for this area at the AMS.

13. CT2000 CT2000. International category theory Conference. Villa Olmo, Como, July 1622, 2000 International category theory Conference http://www.disi.unige.it/conferences/ct2000/

International Category Theory Conference Villa Olmo, Como, July 16-22, 2000 If you see this, it probably means that your browser does not handle frames. Follow the link to Detailed Information or that to the map of the site . Clicking buttons from there on may open pages in new windows, else remember to use your Back button to return to that page.

14. Computational Category Category Theory Project Computational category theory Project group. People, activities, software. http://www.mcs.le.ac.uk/~ah83/compcat/

Contents Goals and Method Members of CompCat Developments and Information Links to CompCat Member Sites Universita dell' Insubria, Como, Italy Mt. Allison University, Sackville, New Brunswick, Canada School of Mathematics, University of Wales, Bangor, Wales Computing Department, Macquarie University, Sydney, Australia ... MCS, University of Leicester, England Up: Anne's Home page Computational Category Theory Project Goals and Method The aim of this project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures. Although writing on different platforms each group will undertake to make available programs for translating their input and output files to the formats of the other groups. Members R. F. C. Walters Universita dell' Insubria, Como, Italy Bob Rosebrugh Mt. Allison University, Sackville, New Brunswick, Canada ... MCS, University of Leicester, England Developments and Information Here is a link to the list of software and structure definitions To join the mailing list contact You might also like to visit: Author: Anne Heyworth Last updated: 2nd February 2001 Any opinions expressed on this page are those of the author.

15. Category Theory For Computing Science category theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing http://www.case.edu/artsci/math/wells/pub/ctcs.html

Category Theory for Computing Science by Michael Barr and Charles Wells Category Theory for Computing Science is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. You may read the excerpts from the Preface to find out more about it. The third edition is now available from Centre de recherches mathématiques , or by email to crmbooks@crm.umontreal.ca . This edition contains all the material dropped from the second edition (with corrections) and the answers to all the exercises. About earlier editions Some of the chapters in the first edition were dropped from the second edition in order to make room for new material. Revised and corrected versions of the omitted chapters may now be found in an electronic supplement to the text. We also provide corrections and additions to the first edition and corrections to the second edition Preface This book is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. We expound the constructions we feel are basic to category theory in the context of examples and applications to computing science.

16. Centre Of Australian Category Theory, Macquarie University Centre of Australian category theory Macquarie University. http://www.ics.mq.edu.au/CoACT/

CoACT members projects awards ... Staff Centre of Australian Category Theory Welcome to the Centre of Australian Category Theory The Centre of Australian Category Theory (CoACT) develops an algebra of widespread applicability for the synthesis and analysis of systems and processes in fields as diverse as physics and computer science, and also mathematics itself. Although having operated as a coherent group since the founding of the on-going Australian Category Seminar in 1971, CoACT was formally established in 1999. Vision To provide the environment for the top international centre for Higher-dimensional Category Theory and a major international centre for general Category Theory. Mission To pursue vigorously research into those parts of mathematics and computer science which find natural expression and advancement in terms of Category Theory and to train thehighest-quality research students. More information about CoACT can be found in the History Objectives Performance Indicators and Benchmarks and Constitution Further Information We welcome your enquiries about our work - please contact the Director Professor Ross Street Your Privacy Comments?

17. Home Page It will begin with a reception at 6pm on Sunday July 18, 2004, and will end at 1pm on Saturday July 24, 2004. All those interested in category theory and http://www.pims.math.ca/science/2004/CT04/

The Pacific Institute for the Mathematical Sciences launched a new web site on March 31, 2005. If there is any discrepancy between the information on this page and the new site, the information on the new site should be used. Question, Comments? Contact johnm@math.ubc.ca INTERNATIONAL CATEGORY THEORY CONFERENCE (CT04) July 18-24, 2004 University of British Columbia Vancouver, Canada This conference will be held on the University of British Columbia campus. It will begin with a reception at 6pm on Sunday July 18, 2004, and will end at 1pm on Saturday July 24, 2004. All those interested in category theory and its applications are welcome. Sculpture by John Robinson http://www.JohnRobinson.com Pacific Institute for the Mathematical Sciences Last Modified: Sunday, 18-Apr-2004 22:18:18 PDT

18. Basic Category Theory For Computer Scientists - The MIT Press Basic category theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7986

19. An ABC Of Category Theory Here is the page for another category theory course that I gave, more detailed than this one. This page was last modified on 13 December 2004. http://www.maths.gla.ac.uk/~tl/ct/

An ABC of Category Theory Autumn 2004 This course is aimed at potential users of categorical ideas rather than aspiring category theorists. I will skip details wherever I can. There will not be many useful theorems in the course. Rather, the point is to teach you how to think categorically. To this end, I will set a couple of exercises each week, and I strongly suggest that you do them: otherwise, the point is likely to be lost. (I know authors always say "the exercises are an essential part of the text", but I really think it's true here.) Informal introduction (handout, not a live performance: pdf ps ; solutions to exercises: pdf ps Categories and functors (notes: pdf ps ; solutions to exercises: pdf ps Natural transformations and equivalence (notes: pdf ps ; solutions to exercises: pdf ps Adjoints (notes: pdf ps ; solutions to exercises: pdf ps Representability (notes: pdf ps ; solutions to exercises: pdf ps Limits (notes: pdf ps ; solutions to exercises: pdf ps Adjoints, representables and limits (notes: pdf ps ; solutions to exercises: pdf ps Monads (notes: pdf ps ; solutions to exercises: pdf ps Monoidal categories (notes: pdf ps ; solutions to exercises: pdf ps Here is the page for another category theory course that I gave, more detailed than this one.

20. University Of Chicago Category Theory Seminar There will occasionally be category theory talks in the Topology Proseminar Eckhart 203 T Th 130300. Check the calendar to see if there is a topology http://www.math.uchicago.edu/~fiore/1/categoryseminar.html

University of Chicago Category Theory Seminar There will occasionally be category theory talks in the Topology Proseminar Eckhart 203 T Th 1:30-3:00. Check the calendar to see if there is a topology talk or category talk, or both. To request a future speaker or for questions, please contact Tom Fiore (fiore AT math.uchicago.edu). To sign up for electronic announcements of talks, please visit http://zaphod.uchicago.edu:8080/mailman/listinfo/category_theory_seminar Spring 2007 Talks Fall 2006 Talks Fall 2006 Plan This quarter we will have a series of talks on topos theory, model categories, quasicategories, and higher topos theory. These current topics lie at the intersection of algebraic geometry, algebraic topology, logic, and higher category theory, and should be of interest to a wider audience. References Topos Theory Goldblatt, Robert I. Topoi. Studies in Logic and the Foundations of Mathematics, 98. North-Holland Publishing Co., Amsterdam-New York, 1979. Johnstone, Peter T. Sketches of an elephant: a topos theory compendium. Vol. 1. Oxford Logic Guides, 43. The Clarendon Press, Oxford University Press, New York, 2002. xxii+468+71 pp. Johnstone, Peter T. Sketches of an elephant: a topos theory compendium. Vol. 2. Oxford Logic Guides, 44. The Clarendon Press, Oxford University Press, Oxford, 2002. pp. ixxii, 4691089 and I1I71.

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