41. MATHS: Category Theory 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 http://www.csci.csusb.edu/dick/maths/math_25_Categories.html

Skip Navigation CSUSB CNS Comp Sci Dept ... Contact ] [Search Tue Sep 18 15:18:21 PDT 2007 Contents Category Theory : Motivation : Trivial Example : Trivial Exercise ... Glossary 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.

42. Category Theory Authors/titles Recent Submissions Subjects Representation Theory (math.RT); category theory (math.CT) AT); High Energy Physics Theory (hep-th); category theory (math. http://aps.arxiv.org/list/math.CT/recent

aps.arXiv.org math math.CT Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text 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 License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/ 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 License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Subjects: Category Theory (math.CT)

43. Category Theory -- From Wolfram MathWorld The objects studied in category theory are called categories. SEE ALSO Category. Pages Linking Here. CITE THIS AS. Weisstein, Eric W. category theory. http://mathworld.wolfram.com/CategoryTheory.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Category Theory Category Theory The branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, sets, topological spaces, etc.) of the same type, subject to the constraint that the collections contain the identity mapping and are closed with respect to compositions of mappings. The objects studied in category theory are called categories SEE ALSO: Category CITE THIS AS: Weisstein, Eric W. "Category Theory." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/CategoryTheory.html Contact the MathWorld Team Wolfram Research, Inc. Other Wolfram Sites: Wolfram Research Demonstrations Site Integrator Tones Functions Site Wolfram Science Mathematica Home Page Show off your math savvy with a MathWorld T-shirt

44. Category Theory And Biology | The N-Category Café The forefather of biological category theory is Robert Rosen. I haven’t had a chance to look at his work yet, but for an easy (for Café regulars) http://golem.ph.utexas.edu/category/2007/11/category_theory_and_biology.html

@import url("/category/styles-site.css"); A group blog on math, physics and philosophy Skip to the Main Content Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main November 15, 2007 Category Theory and Biology Posted by David Corfield Memory Evolutive Systems and Gerhard Mack (somewhere near Urs in Hamburg) on Universal Dynamics, a Unified Theory of Complex Systems: Emergence, Life and Death . Climbing the n -category ladder, Nils Baas who has ideas on abstract matter higher-order cellular automata The forefather of biological category theory is Robert Rosen Organizational invariance and metabolic closure . In particular, it discusses the representation of enzyme metabolism by arrows in categories, taking into account the thesis: Organisms are closed to efficient causes.

45. Downloadable Publications Andrea Asperti and Giuseppe Longo. Categories, Types and Structures. category theory for the working computer scientist. M.I.T. Press, 1991 (pp. 1 300) http://www.di.ens.fr/users/longo/download.html

Selected publications by Giuseppe Longo available as .dvi, .ps (some gzipped) or .pdf files by ftp 1 - Mathematical Logic and Computer Science; 2 - Cognition and Foundations of Mathematical Knowledge; 3 - Morphological Information and Complexity; 4 - Minima philosophica. 1 - Mathematical Logic and Computer Science (for about 70 more papers see publicationsNov06.pdf Title Download in dvi gz or pdf format Download in ps or ps gz format Book: Andrea Asperti and Giuseppe Longo. Categories, Types and Structures . Category Theory for the working computer scientist. M.I.T. Press, 1991 (pp. 1 - 300) (currently out of print and downloadable upon kind permission of the M.I.T. Press The entire book (.pdf) The entire book (.ps-tar) Chapter-by-chapter: Ch.1 Ch.2 Ch.3 Ch.4 ... Ch.12 Giuseppe Longo and Eugenio Moggi A Category-theoretic characterization of functional completeness. . Theoretical Computer Science , 70 (2), 1990, pp.193-211. funct-compl.ps.gz Luca Cardelli and Giuseppe Longo A semantic basis for Quest.

46. Categories In Context: Historical, Foundational, And Philosophical -- Landry And The recognition that category theory was more than ‘a handy language’ came with According to Mac Lane, category theory became an independent field of http://philmat.oxfordjournals.org/cgi/content/full/13/1/1

@import "/resource/css/hw.css"; @import "/resource/css/philmat.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Volume 13, Number 1 Pp. 1-43 Next Article Philosophia Mathematica 2005 13(1):1-43; doi:10.1093/philmat/nki005 This Article Abstract FREE Full Text (PDF) Alert me when this article is cited ... Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Articles by Landry, E. Articles by Marquis, J.-P. Search for Related Content Philosophia Mathematica Categories in Context: Historical, Foundational, and Philosophical Elaine Landry and Jean-Pierre Marquis Department of Philosophy, University of Calgary Calgary, Alta. T2N 1N4 Canada Département de philosophie, Université de Montréal Montréal (Québec) H3C 3J7 Canada Abstract Top Abstract 1. History 2. Categorical Foundations

47. Category Theory And Homotopy Theory School of Informatics, category theory and Homotopy Theory. http://www.informatics.bangor.ac.uk/public/mathematics/research/cathom/cathom1.h

University of Wales, Bangor - Department of Mathematics Mathematics Research Groups Category Theory and Homotopy Theory Personnel: Prof Tim Porter Prof Ronnie Brown Dr Magnus Forrester-Barker Collaborators: Prof Heiner Kamps Prof George Janelidze (Georgia) Dr Manuela Sobral (Coimbra) Dr Manuel Bullejos Prof Tony Bak (Bielefeld) Dr Gabriel Minian (Max Planck, Bonn) Introduction: Category theory was introduced in 1947 to give a richer language than that of set theory, which would be better able to express the structures of homotopy and homology theory then being revealed in the work of Cartan, Eilenberg, Mac Lane, Whitehead and others. In addition to the objects in a category (corresponding to the elements in a set), one also has arrows or "morphisms" between them. Thus for instance the collection of all sets and functions between them forms a category, the category of sets. This language and theory was soon found to have great usefulness in other branches of pure mathematics such as algebra, algebraic geometry, logic and more recently in computer science. The basic areas of research in category theory at Bangor are directed towards achieving a greater understanding of the categorical structure and interrelationships between the various objects studied by algebraic topology and homological algebra. Recent work in these areas has resulted in a large group of fascinating new structures. These have not yet revealed all their categorical structure nor have all the potential applications of these objects been fully investigated.

48. Intermediate Depth Representations Key Words knowledge representation, deep knowledge, category theory, basic level, qualitative reasoning, qualitative disease history, http://www.coiera.com/papers/aimj2/aimj.doc.html

Intermediate Depth Representations Enrico Coiera Hewlett-Packard LaboratoriesFilton Rd., Stoke GiffordBristol, BS12 6QZ United Kingdom This paper appeared in Artificial Intelligence in Medicine Back to List of Publications Abstract Key Words : knowledge representation, deep knowledge, category theory, basic level, qualitative reasoning, qualitative disease history, qualitative superposition, multilevel model. Table of Contents 1 Introduction 2 The Shallow/Deep Paradigm 2.1 Limitations of Model-Based Systems 2.2 Multi-Level Representations ... 6 Conclusion 1 Introduction The notion that medical knowledge can be represented at varying levels of depth is now generally accepted. It was in part the limitations of shallow representations like production systems that drove AI researchers to focus on deeper representations of knowledge. While these deep representations do seem to solve some of the problems associated with shallower ones, they come at a computational cost. This paper focuses on the computational and representational advantages that may exist in using representations whose depth is intermediate, trading off the advantages and disadvantages of shallow and deep representations. In particular, for medical diagnostic systems that reason about time varying aspects of disease, it is proposed that qualitative disease histories are a good candidate for an intermediate representation, lying between shallow disease patterns and deeper qualitative models. It is recognised however, that no single representation will provide complete coverage of a problem domain. For many applications, a multilevel representation of knowledge will be necessary and the current interest in hybrid systems reflects this perception. This paper considers how one could construct, in a principled way, a reasoning system that uses multiple representations. It explores the trade-offs that occur when selecting a particular representational level, and explores the conditions under which a reasoner can decide to switch representations.

49. Octoberfest'06: Local Information category theory OCTOBERFEST 06 University of Ottawa, October 2122, 2006. Local Information. Mathematics Department, R. Blute 613-562-5800 ext. http://aix1.uottawa.ca/~scpsg/Octoberfest06/Octoberfest06.prelim2.html

CATEGORY THEORY OCTOBERFEST '06 University of Ottawa, October 21-22, 2006 Local Information Mathematics Department, R. Blute 613-562-5800 ext. 3535 (rblute@mathstat.uottawa) University of Ottawa, P. Scott 613-562-5800 ext. 3502 (phil@site.uottawa.ca) 585 King Edward Ave (#10 on the maps). (Old) Area Map (click to enlarge) Conference Saturday + Sunday, Oct. 21-22 Octoberfest > Final Revised Schedule and Abstracts NOTE: See the Map on the right. Phil Scott's Ottawa Restaurant Guide will be available during registration. Campus Map (click to enlarge) Here is a campus map The Quality Hotel Downtown is on the Corner of King Edward and Rideau (several blocks north of Laurier): 290 Rideau Street, Ottawa, ON K1N 5Y3 (Phone) 613-789-7511). We reserved a block of rooms, for Friday and Saturday nights. The group number is 105626 and the group name is "Ottawa U-Math Dept". The rate is the University of Ottawa Friends and Family rate of $99.00 plus tax. Guests may phone the hotel directly to reserve and may quote either the group name or number to get the preferred rate. We recommend to reserve ASAP. Parking : $9.50/night hotel rate (or see bottom of this page). Short walk (12 minutes) to math dept.

50. Constructive Category Theory - Huet, Ibi (ResearchIndex) This paper give the full transcript of the Coq axiomatisation. In this note we develop one possible axiomatisation of the notion of category by modeling http://citeseer.ist.psu.edu/255049.html

51. Category Theory (Stanford Encyclopedia Of Philosophy/Summer 2004 Edition) category theory now occupies a central position not only in contemporary mathematics, but also in theoretical computer science and even in mathematical http://www.science.uva.nl/~seop/archives/sum2004/entries/category-theory/

This is a file in the archives of the Stanford Encyclopedia of Philosophy version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... 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 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

52. Como Category Theory News Part of the perception that category theory is foundations (in the pejorative sense of being remote from applications and development) is due to a http://categorytheorynews.blogspot.com/

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?targetBlogID=23821966"); Como Category Theory News Tuesday, July 18, 2006 WORKSHOP in CATEGORICAL ALGEBRA, Milano INTERNAL ACTIONS and INTERNAL STRUCTURES WORKSHOP in CATEGORICAL ALGEBRA 26th-27th-28th october 2006, Milano, Italy Webpage posted by Robert Walters at 6:34 AM 0 comments Friday, July 14, 2006 Visitors In December 2006 we will have a visit from Matias Menni. posted by Robert Walters at 10:09 PM 0 comments Friday, March 31, 2006 F.W. Lawvere: WHY ARE WE CONCERNED? III Categories Mailing List Fri, 31 Mar 2006 09:15:28 -0500 (EST) WHY ARE WE CONCERNED? III The second main misconception about category theory Part of the perception that category theory is "foundations" (in the pejorative sense of being remote from applications and development) is due to a preoccupation with huge size. Since such perceptions hold back the learning of category theory, and hence facilitate its misuse as a mystifying shield, they are among our concerns. We need to deal with the size preoccupation head on. Experience has shown that we cannot build up or construct mathematical concepts from nothing. On the contrary, centuries of experience become concentrated in concepts such as "there must be a group of all rotations" and we then place ourselves conceptually within that creation; we state succinctly the properties which that creation as a structure seems to have, and then develop rigorously the consequences of those properties taken as axioms. The notion of category arose in that way, and in turn serves as a powerful instrument for guiding further such developments. Placing ourselves conceptually within the metacategory of categories, we routinely make use of the leap which idealizes the category of all finite sets as an object. The question is, what more? Of course we make use of the experience of those who have labored to justify mathematics, and it is fortunate that ultimately our results are compatible with theirs. (Mac Lane's use of the term metacategory is not

53. Category Theory - Wiktionary edit Noun. category theory (uncountable). Wikipedia has an article on. category theory Wikipedia. (mathematics) A branch of mathematics which deals http://en.wiktionary.org/wiki/category_theory

category theory From Wiktionary Jump to: navigation search edit English Wikipedia has an article on: Category theory Wikipedia edit Noun Singular category theory Plural uncountable category theory uncountable mathematics A branch of mathematics which deals with spaces and maps between them in extreme abstraction , taking similar theorems from various disparate more concrete branches of mathematics and unifying them. edit Translations Czech: teorie kategori­ f Finnish: kategoriateoria Swedish: kategoriteori c Retrieved from " http://en.wiktionary.org/wiki/category_theory Categories English nouns Mathematics ... Category theory Views Entry Discussion Edit History Personal tools Log in / create account Navigation Main Page Community portal Requested entries Recent changes ... Contact us Search Toolbox What links here Related changes Upload file Special pages ... Permanent link In other languages Svenska This page was last modified 10:59, 21 February 2008. Content is available under GNU Free Documentation License About Wiktionary

54. CoACT.html Those seminars, with their stream of foreign visitors and frequent collaborative missions abroad, are in fact the centre of world category theory and should http://www.maths.mq.edu.au/~street/CoACT.html

Use these links for further information CoACT logo Personnel and visitors with titles and abstracts of talks in the Australian Category Seminar; CoACT Managerial Statements (Vision, Mission, Justification, Objectives, Performance Indicators, Benchmarks, and Constitution); Scott Russell Johnson Memorial Scholarships Macquarie University Calendar entry. A Progress Report CoACT Annual Report for 2001 CoACT Annual Report for 2002 ... CoACT Annual Report for 2003 (pdf) CENTRE OF AUSTRALIAN CATEGORY THEORY (CoACT) A Macquarie University Research Centre PROFILE ADVISORY BOARD Director Professor Ross Street PhD, FAustMS, FAA Professor of Mathematics, Mathematics Department , Macquarie University Associate Director Professor Michael Johnson PhD Director of the Macquarie ICT Innovations Centre, Division of ICS , Macquarie University External Member Professor G. Max Kelly PhD Cambridge , FAA Professorial Fellow and Emeritus Professor, School of Mathematics and Statistics , University of Sydney External Member Dr Wesley Phoa BSc ANU

55. Categories For Software Engineering This book applies littleknown yet quite powerful formal tools from category theory to software structures designs, architectures, patterns, and styles. http://www.fiadeiro.org/jose/CATBook/

"This book applies little-known yet quite powerful formal tools from category theory to software structures: designs, architectures, patterns, and styles. Rather than focus on issues at the level of computational models and semantics, it instead applies these tools to some of the problems facing the sophisticated software architect. The terminology and mind set (Part 1 and 2), while different from many common approaches, can provide startlingly concise expression of key properties of software systems (Part 3), and give rigorous meaning to entire families of box-and-line architecture drawings. It is applicable to the formal specification, decompostion, and composition of service-oriented architectures", Desdmond D'Souza, Kinetium "For computer scientists, this unique book presents category theory in a manner tailored to their interests and with examples to which they can relate", Ira Forman, IBM Ordering The author in academia Bergen CMU Copenhagen Lisbon, ... Nottingham Research Category Theory Institutions Algebraic Specification Formal Methods ... Software Architecture "The meaning of things lies not in the things themselves, but in our attitude towards them."

56. Introduction To Category Theory - Wikiversity Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. This term is believed to http://en.wikiversity.org/wiki/Introduction_to_Category_Theory

Introduction to Category Theory From Wikiversity Jump to: navigation search Welcome to the learning project Introduction to Category Theory. Contents Learning Project Summary What is Category Theory? Scope of the course Contents ... edit Learning Project Summary Project code: Suggested Prerequisites: None Time investment: Assessment suggestions: Portal Mathematics School Mathematics Department: Stream Level:Undergraduate edit What is Category Theory? Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. This term is believed to have been coined by the mathematician Norman Steenrod, himself one of the developers of the categorical point of view. This term is used by practitioners as an indication of mathematical sophistication or coolness rather than as a derogatory designation. Certain ideas and constructions in mathematics display a uniformity throughout many domains. The unifying theme is category theory. Rather than enter an elaborate discussion on particulars of arguments, mathematicians will use the expression such and such is true by abstract nonsense. Typical instances are arguments involving diagram chasing, application of the definition of universal property, definition of natural transformations between functors, use of the Yoneda lemma and so on. This course is an introduction to abstract nonsense.

57. Philosophy Of Real Mathematics: Category Theory And Ontology I ve long harboured the hope that the rise of category theory will push philosophy in interesting directions, and this in two ways (1) Providing http://www.dcorfield.pwp.blueyonder.co.uk/2006/02/category-theory-and-ontology.h

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?blogID=19102847"); Send As SMS BlogThis! Philosophy of Real Mathematics To see what I mean by Philosophy of Real Mathematics look here Friday, February 03, 2006 Category theory and ontology I've long harboured the hope that the rise of category theory will push philosophy in interesting directions, and this in two ways: (1) Providing metaphysics with new tools; (2) Making us think harder about the nature of enquiry if a 4000 year old discipline continues to seek to improve its basic language. Concerning (1), it wouldn't surprise me if much of the category theoretic 'metaphysics' gets done by computer science people and physicists. After all, database theorists have to deal with quesions of type and identity all the time. During the IMA n-categories workshop Michael Johnson gave John Baez and me a fascinating lesson in all this along the banks of the Mississippi. Today the ArXiv has on offer Towards a Definition of an Algorithm by Noson S. Yanofsky. Defining the concept of an algorithm is surprisingly hard. It's easy enough to give examples of programs you'd want to say carried out the same algorithm, but how would you make the equivalence relations of sameness explicit in the following schema?

58. The Math Forum - Math Library - Cat. Theory/Homolgcl Alg. A short article designed to provide an introduction to category theory, a comparatively new field of mathematics that provides a universal framework for 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 - 19 items found, showing 1 to 19 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

59. Category Theory 2008 The International category theory Conference 2008 will take place at the Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville at the Université http://saxo.univ-littoral.fr/CT08/

60. Category Theory Does category theory link different mathematical structures? Can you give a superficial example of how this has been done (eg. http://www.physicsforums.com/showthread.php?p=1575152

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