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1. 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

Extractions: Tue Sep 18 15:18:21 PDT 2007 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.

2. 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

Extractions: [ showing up to 25 entries per page: fewer more 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)

3. 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

Extractions: 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

4. 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

Extractions: @import url("/category/styles-site.css"); A group blog on math, physics and philosophy 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 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.

Andrea Asperti and Giuseppe Longo. Categories, Types and Structures. category theory for the working computer scientist. M.I.T. Press, 1991 (pp. 1 300)

6. 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

7. 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

Extractions: 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.

8. 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

Extractions: 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. 1 Introduction 2.2 Multi-Level Representations ... 6 Conclusion 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.

9. 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

Extractions: 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.

10. 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

11. 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/

Extractions: FEB 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

12. 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/

Extractions: 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

13. 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

Extractions: 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. Retrieved from " http://en.wiktionary.org/wiki/category_theory Categories English nouns Mathematics ... Category theory Views Personal tools Navigation Search Toolbox In other languages Svenska This page was last modified 10:59, 21 February 2008. Content is available under GNU Free Documentation License About Wiktionary

14. 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

Extractions: Use these links for further information 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

15. Categories For Software Engineering
This book applies littleknown yet quite powerful formal tools from category theory to software structures designs, architectures, patterns, and styles.

Extractions: "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

16. 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

Extractions: Jump to: navigation search Welcome to the learning project Introduction to Category Theory. Time investment: Assessment suggestions: Portal Mathematics School Mathematics Department: Stream Level:Undergraduate 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.

17. 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

Extractions: @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! To see what I mean by Philosophy of Real Mathematics look here 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?

18. 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/

Extractions: 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>>

19. 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/

20. Category Theory
Does category theory link different mathematical structures? Can you give a superficial example of how this has been done (eg.