1. Category Theory JeanPierre Marquis of the University of Montreal introduces the general mathematicaltheory of structures and systems of structures. http://plato.stanford.edu/entries/category-theory/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A ... Z This document uses XHTML/Unicode to format the display. If you think special symbols are not displaying correctly, see our guide Displaying Special Characters last substantive content change FEB The Encyclopedia Now Needs Your Support Please Read How You Can Help Keep the Encyclopedia Free Category Theory 1. General Definitions, Examples and Applications 2. Brief Historical Sketch 3. Philosophical Significance Bibliography ... Related Entries 1. General Definitions, Examples and Applications defined as sets, category theory cannot provide a philosophically enlightening foundation for mathematics. In terms of collections, a category C can be described as a collection Ob , the objects of C , which satisfy the following conditions: For every pair a b of objects, there is a collection Mor a b ), namely, the morphisms from a to b in C (when f is a morphism from a to b , we write f a b For every triple a b and c of objects, there is a partial operation from pairs of morphisms in Mor a b ) X Mor b c ) to morphisms in Mor a c ), called the composition of morphisms in

2. Introduction To Category Theory A Gentle Introduction to category theory the calculational approach by MaartenM. Fokkinga in PostScript. 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 (postscript version): here (80 pages). Bibtex data

3. Comcat The aim of the project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures. http://www.unico.it/~walters/comcat/comcatproj.html

The Computational Category Theory Project 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. (There is a related Categorical Computation Project concerned with a categorical analysis of computers, computation and programming.) The groups currently connected with this project are: Contact R.F.C. Walters, walters@fis.unico.it Mt. Allison University, Sackville, New Brunswick, Canada Contact Bob Rosebrugh, rrosebrugh@mta.ca School of Mathematics, University of Wales, Bangor, Wales Contact Ronnie Brown Computing Department, Macquarie University, Sydney, Australia Contact Mike Johnson MCS, University of Leicester, England Contact Anne Heyworth The organization of the project is as follows: Each group in the project will maintain a home page on the web with details of its own work and with links to the other groups. 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. New versions will be announced on the Categories Mailing List.

4. 18: Category Theory, Homological Algebra category theory, a comparatively new field of mathematics, provides a universalframework A full, wideranging text on category theory is by Borceux, 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.

5. 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. 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

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

Fri 16 Sep 2005 Search Submit Retrieve Subscribe ... iFAQ CT Category Theory Calendar Search Authors: AB CDE FGH IJK ... U-Z New articles (last 12) 15 Sep math.CT/0509318 Lambda-presentable morphisms, injectivity and (weak) factorization systems. Michel Hebert . 16 pages. CT 14 Sep math.CT/0509266 2-C*-categories with non-simple units. Pasquale A. Zito . 47 pages. CT OA 6 Sep math.CT/0509102 Notes on enriched categories with colimits of some class (completed version). G. M. Kelly , V. Schmitt CT 6 Sep math.CT/0509056 On lifting diagrams up to homotopy in Frobenius categories. Matthias Kuenzer CT 30 Aug math.CT/0508565 A definition of triangulated categories following Heller. Matthias Kuenzer CT KT 25 Aug math.CT/0508479 Quotients of Tannakian Categories. J. S. Milne CT AG 23 Aug math.CT/0508379 Support varieties - an ideal approach. Aslak Bakke Buan , Henning Krause Solberg . 25 pages. CT AG RT Cross-listings 31 Aug math.GR/0508617 A simple description of Thompson's group F. Marcelo Fiore , Tom Leinster . 14 pages. GR CT 29 Aug math.LO/0508531 Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem. Benno van den Berg , Federico De Marchi LO CT 23 Aug math.AT/0508416

7. Centre De Recherche En Théorie Des Catégories -- Montréal category theory at McGill A personal account by Marta Bunge. category theory an expository article from the Stanford Encyclopedia of Philosophy. http://www.math.mcgill.ca/triples/

Category Theory Research Center Translation? List of current seminars Some upcoming meetings Octoberfest 2005 (Ottawa) 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. A Brief Description of Category Theory from the RISC-Linz Category Theory page. 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. This text has also appeared in the TACReprints series, also as a free download. 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.) Texte d'Alexandre Grothendieck (in progress) (This is not an introductory text, but rather is a "historic" document from a legendary category theorist.)

8. 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.

9. Computational Category Theory An implementation of concepts and constructions from category theory in thefunctional programming language Standard ML. Documentation and code. http://www.cs.man.ac.uk/~david/categories/

Welcome to the Computational Category Theory Project Computational Category Theory is an implementation of concepts and constructions from category theory in the functional programming language Standard ML. The Manual For full details of the project, there is a copy of the manual (in PDF or Postscript) available here The Programs To access the Computational Category Theory programs go to this directory . There is a Readme file giving instructions on compiling and running the programs. Any comments? Please email me at david @ cs.man.ac.uk This page is maintained by myself, email address above. Last updated 21-05-03

10. Category Theory category theory. category theory now occupies a central position not only in contemporary mathematics, but also in theoretical computer science and http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Archival Information For "Category Theory" More results from plato.stanford.edu category theory Wikipedia, the free encyclopediacategory theory is a mathematical theory that deals in an abstract way with See list of category theory topics for a breakdown of relevant articles. http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=category-theor

12. Theory And Applications Of Categories An electronic journal of category theory. Full text, free. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Structure (category Theory) - Wikipedia, The Free Encyclopedia More results from en.wikipedia.org Theory and Applications of CategoriesAn electronic journal of category theory. Full text, free. http://en.wikipedia.org/wiki/Structure_(category_theory)

Structure (category theory) From Wikipedia, the free encyclopedia. In mathematics , progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. In fact this is a normal way of proceeding; in the absence of recognisable structure (which might though be hidden) problems tend to fall into the combinatorics classification of matters requiring special arguments. In category theory structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structures . Starting with a given class of algebraic structure, such as groups , one can build the category in which the objects are groups and the morphisms are group homomorphisms : that is, of structures on one type, and mappings respecting that structure. Starting with a category C given abstractly, the challenge is to infer what structure it is on the objects that the morphisms 'preserve'. The term structure was much used in connection with the Bourbaki group's approach. There is even a definition. Structure must definitely include topological space as well as the standard abstract algebra notions. Structure in this sense is probably commensurate with the idea of

14. Structuralism, Category Theory And Philosophy Of Mathematics By Richard Stefanik (Washington MSG Press,1994). http://www.mmsysgrp.com/strctcat.htm

Structuralism, Category Theory and Philosophy of Mathematics by Richard Stefanik (Washington: MSG Press,1994) Bibliography Bell,J.L."Category Theory and the Foundations of Mathematics", British Journal of Philosophy of Science , vol.32, 1981. Bell, J.L. Toposes and Local Set Theory , Clarendon Press, Oxford, 1988. Benaceraf, Paul."What Numbers Could Not Be", Philosophical review ,vol.74, 1965 Chihara, Charles. Constructibility and Mathematical Existence ,Clarendon Press, Oxford, 1990. Corry, Leo."Nicholas Bourbaki and the Concept of Mathematical Structure", Synthese ,vol.92,1992 Goldblatt, Robert. Topoi, A Categorial Analysis of Logic , North Holland, New York, 1984 Harman, Gilbert."Identifying Numbers", Analysis Jubien, Michael."Ontology and Mathematical Truth", Nous , vol.11, 1977 Katzner, Donald. Analysis Without Measurement , Cambridge University Press, Cambridge,1974 MacLane, Saunders. Mathematics: Form and Function , Springer-Verlag, new York, 1986 Resnik, Michael."Mathematics as a Science of Patterns: Ontology and Reference", Nous , vol.15, 1981

15. Categories John Baez and James Dolan, Categorification, in Higher category theory, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Operads In Higher-dimensional Category Theory Theory and Applications of Categories, Vol. 12, 2004, No. 3, pp 73194.http//www.tac.mta.ca/tac/volumes/12/3/12-03.dvi http://www.tac.mta.ca/tac/volumes/12/3/12-03abs.html

Operads in higher-dimensional category theory Tom Leinster The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n is less than or equal to 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures. Keywords: n-category, operad, higher-dimensional category 2000 MSC: 18D05, 18D50, 18F99, 18A99 Theory and Applications of Categories, Vol. 12, 2004, No. 3, pp 73-194. http://www.tac.mta.ca/tac/volumes/12/3/12-03.dvi http://www.tac.mta.ca/tac/volumes/12/3/12-03.ps http://www.tac.mta.ca/tac/volumes/12/3/12-03.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/12/3/12-03.dvi ... TAC Home

17. 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. Categories Home Page The Applied and Computational category theory Group page at The followinghave information on regular seminars in category theory. http://www.mta.ca/~cat-dist/categories.html

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. 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 A subject-sorted list of postings June 1994-December 1999 is at www.mta.ca/~cat-dist/catlist/

19. Alex Simpson: Home Page (Old) University of Edinburgh category theory, domain theory, logic, type theory. http://www.dcs.ed.ac.uk/~als/

Alex Simpson: Home Page My home page has moved. You will shortly be redirected to the new location http://homepages.inf.ed.ac.uk/als/ Please update any links.

20. Categories Home Page Descent and category theory Connections maintained by M. Alsani at http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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