1. Category Theory This expository article is an entry in the Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/category-theory/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses XHTML-1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. last substantive content change FEB 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 C (when f a b and g b c g o f a c is their composition);

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

Fri 4 Jun 2004 Search Submit Retrieve Subscribe ... iFAQ CT Category Theory Calendar Search Authors: AB CDE FGH IJK ... U-Z New articles (last 12) 4 Jun math.CT/0406061 Tours de torseurs, geometrie differentielle des suites de fibres principaux, et theorie des cordes. Aristide Tsemo CT 13 May math.CT/0405226 Deformation theory of abelian categories. Wenty T. Lowen , Michel Van den Bergh . 44 pages. CT KT 23 Apr math.CT/0404405 Deligne Localized Functors. Maurizio Cailotto . 23 pages. CT AG 23 Apr math.CT/0404399 Isomorphisms in pro-categories. J. Dydak , F. R. Ruiz del Portal CT AT Cross-listings 3 Jun math.GR/0406044 On the Zappa-Szep Product. Matthew G. Brin . 29 pages. GR CT 28 May math.QA/0405517 Fiber Functors on Temperley-Lieb Categories. Yamagami Shigeru . 19 pages. QA CT 25 May math.AG/0405453 Michel Hickel . 45 pages. AG CT 13 May math.KT/0405227 Hochschild cohomology of abelian categories and ringed spaces. W. T. Lowen , M. Van den Bergh . 38 pages. KT CT 30 Apr math.RA/0404522 Algebra generators and information theory. T. Kopf , R. Otahalova . 5 pages. RA CT 29 Apr math.QA/0404504 An analogue of Radford's S^4 formula for finite tensor categories. Pavel Etingof , Dmitri Nikshych , Viktor Ostrik . 14 pages. QA CT 28 Apr math.AT/0404470

3. Introduction To Category Theory A Gentle Introduction to category theory the calculational approach. Maarten M Fokkinga In these notes we present the important notions from category theory. 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

4. 18: Category Theory, Homological Algebra category theory, a comparatively new field of mathematics, provides a universal framework for discussing full, wideranging text on category theory is by Borceux, Francis "Handbook 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! This page is merely a launching pad to sites of interest in Descent or category theory. Picture of some people interested in category theory 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. 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

7. MATHS: Category Theory category theory. Motivation. category theory is a way for talking about the relationships between the classes computer science has found that category theory allowed them to express a http://www.csci.csusb.edu/dick/maths/math_25_Categories.html

CSUSB CNS Comp Sci Dept R J Botting ... Comment ] Fri Feb 20 12:05:34 PST 2004 Contents Category Theory : Motivation : Trivial Example : Trivial Exercise ... Index of defined terms etc. Category Theory Motivation Category Theory is a way for talking about the relationships between the classes of objects modeled by mathematics and logic. It is a model of a collection of things with some structural similarity. It is a comparatively recent abstraction from the various abstract algebras developed in the early part of the 20th century. The best source for detailed information is still Madc Lane's classic graduate text The original use of the term category was in the idea of a 'categorical' axiom system - an axiom system which defined its objects so exactly that all objects that satisfied the axioms were isomorphic - they mapped into each other, one-to-one, preserving all the axioms and structure. This is important, because if a logic is categorical and there exists a simple (or cheaply implemented) example then that model can become the standard and all others are handled in terms of this standard. For example binary numbers have all the properties that one can expect of objects that satisfy the rules that describe a "natural number" and are cheap to emulate using electronics. The name for such ideal systems has been changed several times in this century - categorical, free, universal, initial,... Many times category theorists have discovered that some results that they have uncovered have been discovered within a totally different area - say the theory of languages and automata. Equally often an enterprising researcher in mathematics of computer science has found that category theory allowed them to express a specific property they had observed in more general terms. The more general veiw then leads to shorter and simpler proofs of more results. This in turn often illuminates other problems.

8. Applications Of Category Theory To Computer Science Workshop Applications of category theory to Computer Science. June 812, 1998 Mount Allison University, Sackville, NB, Canada. In http://www.mta.ca/~cat-dist/ctss98/

Workshop: Applications of Category Theory to Computer Science June 8-12, 1998 Mount Allison University, Sackville, NB, Canada In conjunction with the Category Theory Session at the Canadian Mathematical Society's Summer 1998 Meeting, see camel.math.ca/CMS/Events/summer98/ , there will be a workshop on the Applications of Category Theory to Computer Science, directed towards graduate students and young researchers. The arrival day Sunday, June 7, 1998 - residence accommodation will be available from June 6. The invited instructors are M. Barr (McGill) and R.F.C. Walters (Sydney). Residence accommodation will be available at Mount Allison University at a cost of $27.60/person/night for a single room $24.30/person/night for a shared double room $23.00/person/night for either of the above for students upon presentation of a student card. (All prices are in Canadian dollars and include taxes.) Bookings can be made at http://www.mta.ca/conference/overnigh.htm There will be a registration fee of $50 for the workshop. To preregister send e-mail to ct95@mscs.dal.ca

9. Categories Home Page The Centre of Australian category theory (CoACT) based at Macquarie University has a page at http//www a postal mailing list (not updated) for category theory. At York University 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. 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/

10. Comcat The aim of the project is the development of software on a wide variety of platforms for computing 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.

11. Categories In fact, MacLane said "I did not invent category theory to talk about functors go to objects and morphisms go to morphisms. category theory is popular among algebraic topologists http://math.ucr.edu/home/baez/categories.html

Categories, Quantization, and Much More John Baez August 7, 1992 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.) What is a category? Well, a category consists of a set of

12. Computational Category Category Theory Project Computational category theory Project. Goals and Method. The aim of this project is the development of software on a wide variety 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.

13. Centre De Recherche En Théorie Des Catégories -- Montréal Centre de Recherche en Théorie des Catégories Montréal category theory Research Center. Translation? Some upcoming meetings. What is 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. 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. 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.) (A translation into English, in progress, of Grothendieck's "Reflections".)

14. School On Category Theory And Applications SCHOOL ON category theory AND APPLICATIONS. This school is being organized by the category theory Group of the University of Coimbra. http://www.mat.uc.pt/~scta/

15. Categories Home Page The following sites include links to general information on category theory, including much not found on this page Descent and category theory Connections 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/

16. Applied And Computational Category Theory A brief description of category theory, and some useful links. http://www.risc.uni-linz.ac.at/research/category/

17. Category Theory - Wikipedia, The Free Encyclopedia category theory is a mathematical theory that deals in an abstract way with mathematical broadlybased foundational applications of category theory are contentious; but they have http://www.wikipedia.org/wiki/Category_theory

Category theory From Wikipedia, the free encyclopedia. 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 ... edit 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

18. Category Theory - Wikipedia, The Free Encyclopedia category theory. category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. http://en.wikipedia.org/wiki/Category_theory

Category theory From Wikipedia, the free encyclopedia. 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 ... edit 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. Alex Simpson: Home Page University of Edinburgh category theory, domain theory, logic, type theory. http://www.dcs.ed.ac.uk/~als/

Alex Simpson: Home Page Dr. Alex Simpson Lecturer and EPSRC Advanced Research Fellow Laboratory for Foundations of Computer Science (LFCS) School of Informatics, University of Edinburgh JCMB, King's Buildings Edinburgh EH9 3JZ, UK. Email: Alex.Simpson@ed.ac.uk Phone: Fax: Research papers Talks

20. Theory And Applications Of Categories Theory and Applications of Categories. ISSN 1201 561X. 3. Operads in higher-dimensional category theory Tom Leinster, 73-194 abstract dvi ps pdf. http://www.tac.mta.ca/tac/

Editors Policy Subscriptions Authors - ... - Canada Theory and Applications of Categories ISSN 1201 - 561X Volume 12 - 2004 Table of contents also available in .dvi or .ps or .pdf format. Baer invariants in semi-abelian categories I: General theory T. Everaert and T. Van der Linden, 1-33 abstract dvi ps pdf ... Simplicial approximation J.F. Jardine, 34-72 abstract dvi ps pdf ... Operads in higher-dimensional category theory Tom Leinster, 73-194 abstract dvi ps pdf ... Baer invariants in semi-abelian categories II: Homology T. Everaert and T. Van der Linden, 195-224 abstract dvi ps pdf ... Notions of flatness relative to a Grothendieck topology Panagis Karazeris, 225-236 abstract dvi ps pdf ... Moore categories Diana Rodelo, 237-247 abstract dvi ps pdf ... Change of base for relational variable sets Susan Niefield, 248-261 abstract dvi ps pdf ... On subgroups of the Lambek pregroup Michael Barr, 262-269 abstract dvi ps pdf ... Algebraically closed and existentially closed substructures in categorical context Michel Hebert, 269-298 abstract dvi ps pdf ... Vertically iterated classical enrichment Stefan Forcey, 299-325 abstract dvi ps pdf ... Return to top.

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