21. Dual (category Theory) - Wikipedia, The Free Encyclopedia Dual (category theory). In category theory, an abstract branch of mathematics, the dual of a category is the category formed by reversing all the morphisms of . http://en.wikipedia.org/wiki/Dual_(category_theory)

Dual (category theory) From Wikipedia, the free encyclopedia. In category theory , an abstract branch of mathematics , the dual of a category is the category formed by reversing all the morphisms of . That is, we take to be the category with objects that are those of , but with the morphisms from X to Y in being the morphisms from Y to X in . Hence, the dual of a dual of a category is itself. It is also often called the opposite category . Examples come from reversing the direction of inequalities in a partial order . So if X new by the definition x new y if and only if y x For example, there are opposite pairs child/parent, or descendant/ancestor. This is a special case, since partial orders correspond to a certain kind of category in which Mor( A B ) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice , we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws Generalising that observation

22. Steve Awodey Carnegie Mellon University category theory, logic, history and philosophy of mathematics and logic. http://www.andrew.cmu.edu/user/awodey/

Steve Awodey Associate Professor Department of Philosophy Carnegie Mellon University Research Areas: Category Theory Logic Philosophy of Mathematics History of Logic and Analytic Philosophy Connections: Logic of Types and Computation Selected Publications Modal operators and the formal dual of Birkhoff's completeness theorem. S. Awodey and J. Hughes, Mathematical Structures in Computer Science , vol. 13 (2003), pp. 233-258. Categoricity and completeness: 19th century axiomatics to 21st century semantics. S. Awodey and E. Reck, History and Philosophy of Logic , 23 (2002), pp. 1-30, 77-94. Elementary axioms for local maps of toposes. S. Awodey and L. Birkedal, Journal of Pure and Applied Algebra , 177 (2003), pp. 215-230. Local realizability toposes and a modal logic for computability. S. Awodey, L. Birkedal, D.S. Scott, Mathematical Structures in Computer Science , vol. 12 (2002), pp. 319-334. Topological completeness for higher-order logic. S. Awodey and C. Butz, Journal of Symbolic Logic 65(3), (2000) pp. 116882. Topological representation of the lambda-calculus. Mathematical Structures in Computer Science (2000), vol. 10, pp. 8196.

23. Category Theory From MathWorld category theory from MathWorld The branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/CategoryTheory.htm

24. Categories: Category Theory For Telecommunication Engineers categories category theory for telecommunication engineers. It allowed me to learn many interesting things and I fell in love with category theory. http://north.ecc.edu/alsani/ct02(1-2)/msg00058.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Category theory for telecommunication engineers To categories@mta.ca Subject : categories: Category theory for telecommunication engineers From cfavergeon@free.fr Date : Tue, 5 Feb 2002 18:34:05 +0100 Sender cat-dist@mta.ca Prev by Date: categories: Announcement: TAC Reprints Next by Date: categories: Pursuing Stacks Previous by thread: categories: Announcement: TAC Reprints Next by thread: categories: Pursuing Stacks Index(es): Date Thread

25. CTCS97 List of Participants. CTCS 97 is the 7th conference on category theory and Computer Science. The purpose of the conference series http://www.disi.unige.it/conferences/ctcs97/

CTCS'97, 4-6 September 1997, S. Margherita Ligure, Italy URL "http://www.disi.unige.it/conferences/ctcs97/" CTCS'99 (Edinburgh) Conference Programme and Abstracts of accepted papers Travel Information List of Participants CTCS'97 is the 7th conference on Category Theory and Computer Science . The purpose of the conference series is the advancement of the foundations of computing using the tools of category theory, algebra, geometry and logic. While the emphasis is upon applications of category theory, it is recognized that the area is highly interdisciplinary. The proceedings will be published by Springer as volume 1290 in the LNCS series. The conference will take place at Hotel Regina Elena , a 4 stars hotel with private beach, located in S. Margherita Ligure . This is a beautiful sea resort in Liguria very close to Portofino promontory and about 30 km east of Genova Programme committee S. Abramsky Edinburgh (UK) P.-L. Curien LIENS (France) P. Dybjer Chalmers (Sweden) P. Johnstone Cambridge (UK) G. Longo

26. Ccard V2.0 - A Category Theory Card Game The official site for this abstract mathematical card game. You can download the deck as a gzipped postscript. http://www.verify-it.de/sub/ccard/index.html

This page is part of the Mozilla Open Directory project Ccard 2.0 or: How to make fun out of something highly abstract. Ccard is a card game. You can download the cards as gzipped postscript here (to be printed 2-sided) or here (single sided) with extra backside It was born in an area of distress in May 1999, kicked of by the Summer School in Semantics (at BRICS, Aarhus University, Denmark) and in particular the course about category theory there. How to play? There are some simple "rules" I made up for two or more players (but you are of course free to change them). The seven suits are organized by a increasing number of "circles" which are meant to reflect the "difficulty" of the facts within. The number of circles/triangles of the suite symbol determines the rank of this suite. Every suite has nine cards. The highest card of one suit is the "aleph"_lambda (resembles a shaky N), followed by "omega", "infinity", then 11, 7, 5, 3, 2 (I like to stick with prime numbers) and finally the empty set (or "naught"). Each of 2 (or possibly more) players gets six cards, the rest is left as a pile on the table.

27. Untitled Document Pace University. Problem decomposition and theory reformulation, integrated cognitive architectures for autonomous robots, distributed constraint satisfaction problems, semigroup theory and dynamical systems, category theory in software design. http://csis.pace.edu/~benjamin/

28. Interactions Between Representation Theories, Knot Theory, Topology, Quantum Fie SUNY Potsdam, NY, USA; 26 June 2003. http://www2.potsdam.edu/mahdavk/Conf.htm

29. CATEGORY THEORY AT MCGILL BACK. category theory at McGill. Silvia Bunge 78. category theory Category theorists are conceptual mathematicians of a special kind. What http://www.math.mcgill.ca/bunge/ctatmcgill.html

Category Theory at McGill Silvia Bunge and Carl Christian Mikkelsen, Montreal '78 Category Theory Category theorists are conceptual mathematicians of a special kind. What binds them together is that they approach mathematical problems with a point of view that is radically different from that on which traditional mathematics is based, and which emphasizes interactions between mathematical objects over their individual constituents. Their results are often surprising, provide new insights, and are obtained by the invention of sophisticated notions, theories, and techniques. Category Theory is only little more than 50 years old (dating it back to the work of S. Eilenberg and S. MacLane in 1945) yet, its impact on several branches of mathematics has been considerable, in spite of the reluctance to recognize it as a revolutionary independent field dealing with foundational questions, very different from Set Theory. Category Theory at McGill The Montreal Categories Center Current Research Areas in Category Theory at McGill Three areas deserve attention because of the novelties they bring and because they are part of a truly international joint effort. Let us refer to them as "Grothendieck's Program", "Lawvere's Program", and "Computational Category Theory". Although not pairwise disjoint, their objectives are different and can be briefly described as follows.

30. Theory And Semantics Group Centred around mathematical models of a variety of languages and logics, using techniques such as structural operational semantics, linear logic, domain theory and category theory. Strong links with Logic and Set Theory in the Pure Mathematics Department. http://www.cl.cam.ac.uk/Research/TSG/

Theory and Semantics Group University of Cambridge Computer Laboratory The work of the Theory and Semantics Group is centred around mathematical models of a variety of languages and logics. These models are intended to be used as a basis for specification and verification, and as a tool for clarifying programming concepts. We use techniques such as structural operational semantics, linear logic, domain theory and category theory. Work is in progress on the underlying mathematical structures of these, and on their application to the study of higher order typed programming languages such as Standard ML, to object-based languages, to foundational languages for concurrent, distributed and mobile computation, to hardware description languages, and to security problems. Work is also being undertaken on the analysis of programming languages in the setting of abstract interpretation and on practical optimising compilation for imperative and functional languages. Related research is undertaken within the Automated Reasoning Group . We also have links with the Logic Seminar at DPMMS (Dept of Pure Mathematics and Mathematical Statistics).

31. CTCS'02 category theory and Computer Science (CTCS 02) August 15th17th, 2002 University of Ottawa. and Graduate Student Preconference August 12-14, 2002. http://www.mathstat.uottawa.ca/lfc/ctcs2002/

Category Theory and Computer Science (CTCS'02) August 15th-17th, 2002 University of Ottawa and Graduate Student Preconference August 12-14, 2002 Pictures from the conference. Thanks to all participants for a successful conference! Several people have taken pictures at the conference, and I will link to them from this page. Pictures by Masahito Hasegawa Pictures by Anjayan Puvananathan The purpose of this conference series is the advancement of the foundations of computing, using the tools of category theory. While the emphasis is on applications of category theory, it is recognized that the area is highly interdisciplinary. Category theory, after having played a major role in the development of mathematics, e.g. in algebraic geometry, has been widely applied by logicians to obtain concise interpretations of many logical concepts. On the other hand, links between logic and computer science have been developped now for over twenty years, notably via the Curry-Howard isomorphism, which identifies programs with proofs. Together, the triangle category theory-logic-computation presents a rich world of interconnections. It is the primary purpose of the CTCS conference series to explore these interconnections. In addition to the usual three day conference, there will be a three day "preconference", which is designed to prepare students, both graduate and undergraduate, to participate in the conference. The preconference will take place from August 12-14.

32. Mathematical Structures In Computer Science Focuses on the application of areas such as logic, algebra, geometry and category theory to theoretical computer science. http://uk.cambridge.org/journals/msc/

Home Journals Features Related Journals Journals By Title By Subject Highlights New Journals 2004 ... Advanced Search Cambridge Alerts Free journal TOC alerts New title information alerts Mathematical Structures in Computer Science Edited by G. Longo CNRS and Ecole Normale Supérieure, Paris, France Editorial Board Instructions for Contributors Advertising Rates Links Aims and Scope Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. Print ISSN: 0960-1295 Online ISSN: 1469-8072 Full pricing details Current volume: 14:1 - 14:6, 2004 All issues View a free sample of this journal Cambridge University Press 2004.

33. Category Theory category theory. category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among http://setis.library.usyd.edu.au/stanford/archives/fall1997/entries/category-the

This is a file in the archives of the Stanford Encyclopedia of Philosophy Stanford Encyclopedia of Philosophy A B C D ... Z Category Theory Category theory is a general mathematical theory of structures and sytems of structures. It allows us to see, among other things, how structures of different kinds are related to one another as well as the universal components of a family of structures of a given kind. The theory is philosophically relevant in more than one way. For one thing, it is considered by many as being an alternative to set theory as a foundation for mathematics. Furthermore, it can be thought of as constituting a theory of concepts. Finally, it sheds a new light on many traditional philosophical questions, for instance on the nature of reference and truth. General Definitions Brief Historical Sketch Philosophical Significance Bibliography ... Related Entries General Definitions Category theory is a generalized mathematical theory of structures. One of its goals is to reveal the universal properties of structures of a given kind via their relationships with one another. Formally, 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

34. Robert Rosebrugh - Home Page Mount Allison University Higher dimensional category theory, computational category theory and theory of database systems. http://www.mta.ca/~rrosebru/index.html

Robert Rosebrugh Professor of Mathematics and Computer Science , at Mount Allison University in Sackville, NB, Canada. Email: rrosebrugh@mta.ca . Here are some photos. Current Atlantic Time Research Interests: higher dimensional category theory, computational category theory, theory of database systems. Publications: Recent Abstracts. Direct to ftp archive Electronic Publishing: Managing Editor of the electronic journal on category theory: Theory and Applications of Categories Moderator of categories -the Internet mailing list on category theory. Software: Graphical Database for Category Theory (GDCT) Java application - Version 2.0 (July 2002) mathcs.mta.ca/research/rosebrugh/gdct/ Category Theory Database Tools Java applet - limited, 1998 version of GDCT at www.mta.ca/~rrosebru/mathcs/javasource/index.htm A Database of Categories - a menu-based C program (1995). Member of the Computational Category Theory Project For local information on the project see www.mta.ca/~rrosebru/compcat/compcat.html

35. Part III Category Theory Part III category theory. This is the main page for the Part III category theory course given in Cambridge in the academic year 20002001. http://www.dpmms.cam.ac.uk/~leinster/categories/

Part III Category Theory Michaelmas/Autumn/Fall 2000, 24 lectures This is the main page for the Part III Category Theory course given in Cambridge in the academic year 2000-2001. All the documents produced for the course were placed here, and all of them apart from the synopsis are in the PostScript (.ps) format. Here is the lecture synopsis and recommended reading list. Here is an informal introduction to Category Theory (10 pages). The first two-thirds of this is a set of notes from the very first lecture. What's the Yoneda Lemma all about? (9 pages) A few more applications of the General Adjoint Functor Theorem , to add to the one given in lectures (one page). Section E2: The Special Adjoint Functor Theorem (7 pages, examinable!). Section F4: The Monadicity Theorem (10 pages, also examinable). Problem sheets: Sheet 1 Sheet 2 Sheet 3 (limits) Sheet 4 Here is a sheet telling you roughly what background mathematical knowledge I'm going to assume you have when I'm setting the exam. Here is a list of known mistakes in the lecture notes and problem sheets, last updated 9 December 2000. Please

36. Pitts, Andrew University of Cambridge Applications of mathematical logic and category theory to computer science, semantics of programming languages and type theories, formal logics for reasoning about program properties. http://www.cl.cam.ac.uk/users/amp12/

Andrew Pitts Picture Professor of Theoretical Computer Science Fellow of Darwin College Research My research is in applications of mathematical logic and category theory to computer science. I am especially interested in the semantics of programming languages and type theories, formal logics for reasoning about program properties, and metaprogramming languages and the foundations of machine-assisted reasoning systems. On-line publications: listing BibTeX database Recent talks The FreshML research project. I participate in the Cambridge Theory and Semantics Group Journals I am associated with: Applied Categorical Structures Chicago Journal of Theoretical Computer Science Higher-Order and Symbolic Computation Mathematical Structures in Computer Science Teaching Lecture notes for 2002/2003 courses: Regular Languages and Finite Automata (CST Part IA ) Computation Theory (CST Part IB/II(G)/Diploma) Types (CST Part II) Lecture notes for old courses: Semantics of Programming Languages (2001/02 CST Part IB ) Denotational Semantics (1998/99 CST Part II) Professor Andrew M Pitts University of Cambridge Computer Laboratory William Gates Building JJ Thomson Avenue Cambridge CB3 0FD, UK

37. Category Theory Seminar Archive category theory Seminar Archive. http://www.dpmms.cam.ac.uk/Seminars/Category/archive.html

Department of Pure Mathematics and Mathematical Statistics DPMMS Research Archive Category Theory Category Theory Seminar Archive 2002 Michaelmas Term 2003 Lent Term 2003 Michaelmas Term Last modified: 00:06 Sat Jan 31 2004 Information provided by webmaster@dpmms.cam.ac.uk

38. 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 School of Informatics Research Groups Personnel: Prof Tim Porter Prof Ronnie Brown Mr Alinor Abdul Kadir Mr 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.

39. Category Theory For Computing Science category theory for Computing Science. These are basic constructions in category theory that allow the formation of equationally defined subtypes and quotients. http://www.cwru.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 sales@crm.umontreal.ca . This edition contains all the material dropped from the second edition (with corrections) and the answers to all the exercises. It is cheaper, too; it costs only about US$ , postpaid (surface mail) anywhere in the world. 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 beread 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.

40. Martin Hofmann's Home Page University of Edinburgh Type theory, principles of programming languages, semantics, category theory, mathematical logic, formal methods. http://www.dcs.ed.ac.uk/~mxh/

Martin Hofmann I have moved to Munich. Please visit my new homepage

Page 2     21-40 of 123    Back | 1  | 2  | 3  | 4  | 5  | 6  | 7  | Next 20