February 2007 - Categories - categories.org.au

Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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terminology: dagger and involution
by Marco Grandis 03 Mar '07

03 Mar '07

Dear Peter, what you are calling a "dagger-category", i.e. a category equipped with a contravariant involutive endofunctor, which is the identity on objects, has been called "a category with involution", at least from Burgin 1969 to Lambek 2001. "Involutive category" has also been used, if less. (The main object of these papers, or most of them, is: categories of relations.) I think it would be better to come back to the old term, which is meaningful, translatable, and old. With best regards Marco

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Yuri Matiyasevich - 60! Call for papers.
by Maxim Vsemirnov 01 Mar '07

01 Mar '07

Sorry for possible cross-posting Dear Academic Community, In March 2007 Yuri Matiyasevich will be 60. In addition to his outstanding scientific achievements, Yuri Matiyasevich was the editor of several issues of the journal "Zapiski Nauchnykh Seminarov POMI" ("Notes of Mathematical Seminars of St.Petersburg Department of V.A.Steklov Institute of Mathematics"). He remains an active author of our journal until now. We decided to devote the next issue of the journal to Yuri Matiyasevich. Please find the call for papers below. We will be grateful if you distribute this call for papers among your colleagues. The deadline is set to May 10, 2007. ****** If you are unable to meet the deadline ****** but strongly wish to contribute, please communicate a more appropriate date and an abstract to any of the editors at the addresses listed below. We have a tight schedule for the publication process (see below). Faithfully yours, Anatol Slissenko and Maxim Vsemirnov %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CALL FOR PAPERS Notes of Mathematical Seminars of St.Petersburg Department of V.A.Steklov Institute of Mathematics http://www.pdmi.ras.ru/znsl/index.html Subseries: Studies in Constructive Mathematics and Mathematical Logic. Issue XI TOPICS. Papers (either describing an original research or surveying a topic) are solicited in all areas related to the interests of Yuri Matiyasevich. They include (but not limited to) all aspects of the mathematical logic, Diophantine equations and discrete mathematics. LANGUAGES. Submissions can be made either in English or in Russian. TRANSLATION AND PUBLICATION. The issue containing the original papers will appear in the printed form as well as on the Web. English translations (or original papers, for papers submitted in English) will later appear in Journal of Mathematical Sciences published by Springer. SUBMISSION and other contacts. Please send your Postscript or PDF file to either of the editors: Anatol Slissenko slissenko(a)univ-paris12.fr Maxim Vsemirnov vsemir(a)pdmi.ras.ru IMPORTANT DATES. Submission: by May 10, 2007. First referee reports: July-August 2007. Revised (final) version if needed: August-September 2007. Final decision: October 2007. Publication: November 2007.

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SECA4
by Jerome Scherer 28 Feb '07

28 Feb '07

Second announcement SECA4, "SEminar on Category theory and Applications" Barcelona (CRM): June 6 - 9, 2007. This is the fourth of a series of meetings devoted to category theory and applications. It is intended as meeting point for specialists in category theory and related areas as homological algebra, representation theory, or homotopy theory. Plenary speakers: Denis-Charles Cisinski, Université Paris 13 José L. García Hernández, Universidad de Murcia Ken Goodearl, University of California at Santa Barbara Joachim Kock, Universitat Autònoma de Barcelona Henning Krause, Universität Paderborn Tom Leinster, University of Glasgow Javier Majadas, Universidade de Santiago de Compostela Ieke Moerdijk, Universiteit Utrecht Antonio Viruel, Universidad de Málaga At the request of the organizing committee, the following have accepted to give survey talks, that are specially addressed to the young participants: Ronnie Brown, University of Wales Carles Casacuberta, Universitat de Barcelona Marco Grandis, Università di Genova You will find more information on the following webpage http://mat.uab.es/~seca4/ Some financial support is available for young participants. The deadline to apply for it is March 27. The registration fee is 125 Euros, but there is a reduced registration fee of 100 Euros for those registrating before April 27. If you have specific questions, please send an email to: seca4(a)mat.uab.es Organizing committee: Pere Ara, Universitat Autonoma de Barcelona Carles Broto, Universitat Autonoma de Barcelona (coordinator) Jose Manuel Casas, Universidad de Vigo Luis Javier Hernandez, Universidad de La Rioja Manuel Ladra, Universidad de Santiago de Compostela Albert Ruiz, Universitat Autonoma de Barcelona Jerome Scherer, Universitat Autonoma de Barcelona Scientific committee: Ronnie Brown, University of Wales Carles Casacuberta, Universitat de Barcelona Jose Luis Gomez Pardo, Universidad de Santiago de Compostela Marco Grandis, Universita di Genova Lionel Schwartz, Universite Paris 13

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preprints available
by Anders Kock 27 Feb '07

27 Feb '07

Dear all, This is to announce the availability of two preprints. 1) "Group valued differential forms revisited" We study the relationship between combinatorial group valued differential forms and classical differential forms with values in the corresponding Lie algebra. In particular, we compare simplicial coboundary and exterior derivative. The results represent strengthening of results I obtained in 1982. This preprint can be downloaded from http://www.imf.au.dk/publs?id=636 or from my home page http://home.imf.au.dk/kock/ 2) "Some matrices with nilpotent entries, and their determinants" We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 (neighbour in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n! This preprint can be downloaded from http://arxiv.org/abs/math.RA/0612435 or from my home page (address as above). Yours Anders

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Thanks from Imogen
by Imogen Kelly 26 Feb '07

26 Feb '07

My family and I sincerely thank all the members of the Category Theory community who, when Max died, sent messages of condolence to us through Ross. It has been a great comfort to us that so many mathematicians have commented on the contribution Max made to your area of mathematics. Many of you, too, made mention of the help and friendship he freely offered. Warm regards to you all, Imogen

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CT2007: Important dates
by Category Theory 2007 23 Feb '07

23 Feb '07

==================== CT2007 ==================== International Category Theory Conference Hotel Tivoli Almansor Carvoeiro, Portugal June 17-23, 2007 Deadlines: Accommodation: March 1, 2007 Registration: March 1, 2007 Abstract submission: May 1, 2007 For more information, please consult the web site http://www.mat.uc.pt/~categ/ct2007 The Organizing Committee.

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A dagger compact closed category with biproducts, but without dagger biproducts (was: categories: Re: Equalisers and coequalisers in categories with a \dag-involution)
by Robin Houston 22 Feb '07

22 Feb '07

On Fri, Feb 16, 2007 at 05:08:06PM -0400, Peter Selinger wrote: > I do not know whether Robin Houston's construction, when applied to a > dagger compact closed category, yields dagger-biproducts. The answer is no (unless I have overlooked something below, which is all too easy to do when cooking up pathological counterexamples). We're looking for a category that: - has a dagger compact-closed structure, - has biproducts - does *not* have dagger-biproducts. I've tried to keep the description quite concrete, perhaps too much so for the taste of some readers of this list. 1. Let A be the category with: - the set of objects is the set of integers; - an arrow is a rational number: A(n,n) = Q, and A(m,n) = {0}, for m != n; - composition is multiplication; - the following compact closed structure: [ The symbol @ denotes tensor below] m @ n := m + n for objects m, n q @ r := q x r for arrows q, r, The tensor unit is the object 0, n* := -n; - the obvious preadditive structure: addition within a hom set is just addition of rational numbers; - the trivial dagger: q^\dagger := q for each arrow q. Note that A is a (preadditive) dagger compact closed category. 2. Let B be the category of matrices over A, i.e. the result of freely adding finite biproducts to A. Concretely, an object of B is a finite tuple of integers (n_i | i<k) for some natural number k. An arrow f: (m_i | i<k) --> (n_j | j<k') is a k x k' matrix of rational numbers, with the property that the (i, j)th entry is 0 unless m_i = n_j. Composition is matrix multiplication. The biproduct of two objects is their concatenation as tuples. The dagger compact closed structure distributes over the formal biproducts, in the standard way. 3. Let C be the full subcategory of B determined by those objects (n_i | i<k) with the property that: for all i<k there exists j<k s.t. n_j = -n_i. Note that C is closed under tensor, * and biproduct. Of course there is a 'standard dagger' (transposition), and a standard compact closed structure that's compatible with it. I'll define a new dagger on C, and exhibit a compact closed structure that's compatible with this new dagger. The purpose of the restriction in (3) is to make it possible -- or at any rate easier -- to define this compact closed structure. Finally I'll show that C does not have dagger-biproducts, with respect to this new dagger. Given a matrix f: (m_i | i<k) --> (n_j | j<k') let f^\dagger be the transpose of f, multiplied by the scalar 2 ^ (max {n_j | j<k'} - max {m_i | i<k}) If either k or k' is zero, this expression is ill-defined, but the matrix is also empty so that doesn't matter. Below I shall write max(n_i) as an abbreviation for max {n_i | j<k}; the intended k will be clear from the context. This dagger is clearly involutive. Also max( (m_i | i<k) @ (n_j | j<k') ) = max {m_i + n_j | i<k, j<k'} = max(m_i) + max(n_j), from which it follows that this dagger is compatible with tensor. It's thus easy to check that C is dagger symmetric monoidal w.r.t this dagger. For the compact closed structure, consider an object A and denote the standard unit eta_A: (0) --> A* @ A. Define eta'_A to be the product of eta_A with the scalar 2 ^ -max(A). By the restriction of (3), we know that max (A* @ A) = 2 max(A), hence (eta'_A)^\dagger = (2 ^ max(A)) x epsilon_A where epsilon_A is the standard counit. Thus the additional scalar factor in (eta'_A)^\dagger is the reciprocal of that in eta'_A, so they cancel out as required. Finally, to see that C does not have dagger biproducts, consider the objects (0) and (-1,1). The injection i: (0) --> (0,-1,1) is a matrix [ q ] for some rational q. Then i^\dagger = [ 2q 0 0 ], [ 0 ] [ 0 ] so i^\dagger . i : (0) -> (0) is the 1x1 matrix [ 2q^2 ]. For this to be the identity, we would need to have 2q^2 = 1, but that equation famously has no solution in the rationals! Can anyone see any errors in the above, or think of a simpler example? Robin

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New Journal: Algebra and Number Theory
by Tom Leinster 20 Feb '07

20 Feb '07

Algebraically-inclined readers may be interested in the following new journal, which embodies all the virtues discussed in the recent thread on publishing as well as boasting an impeccably eminent editorial board. -------- Forwarded Message -------- From: David Eisenbud <de(a)msri.org> Subject: [Eager-gen] New Journal: Algebra and Number Theory Date: Sat, 17 Feb 2007 17:54:02 -0800 Dear number theorists, algebraists, and algebraic geometers, We are thrilled to announce the launch of a new journal: Algebra & Number Theory http://jant.org The purpose of the journal is to provide an alternative to the current range of commercial specialty journals in these fields - an alternative of higher quality and much lower cost. The policies of Algebra & Number Theory are set by the editorial board, a group of working mathematicians, rather than by a profit-oriented company, so they will remain friendly to mathematicians' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. The journal is published by Mathematical Sciences Publishers, using the efficient model that has proved successful for its other journals, such as Geometry & Topology. Please encourage your library to subscribe! And submit your high-quality original articles to us! For more information, see http://jant.org Best, -- David Eisenbud, chair of the editorial board Bjorn Poonen, managing editor on behalf of the editorial board: Georgia Benkart, Dave Benson, Richard E. Borcherds, John H. Coates, Jean-Louis Colliot-Th\'el\`ene, Brian D. Conrad, H\'el\`ene Esnault, Hubert Flenner, Andrew Granville, Joseph Gubeladze, Ehud Hrushovski, Craig Huneke, Mikhail Kapranov, Yujiro Kawamata, J\'anos Koll\'ar, Hendrik W. Lenstra, Yuri Manin, Barry Mazur, Susan Montgomery, Shigefumi Mori, Andrei Okounkov, Raman Parimala, Victor Reiner, Karl Rubin, Peter Sarnak, Michael Singer, Ronald Solomon, Vasudevan Srinivas, J. Tobias Stafford, Richard Stanley, Bernd Sturmfels, Richard Taylor, Ravi Vakil, Michel van den Bergh, Marie-France Vign \'eras, Kei-Ichi Watanabe, Andrei Zelevinsky, and Efim Zelmanov --- David Eisenbud Director, MSRI www.msri.org tel: 510-642-0143 fax: 510-642-8609 _______________________________________________ Eager-gen mailing list Eager-gen(a)euclid.mathematik.uni-kl.de http://www-euclid.mathematik.uni-kl.de/mailman/listinfo/eager-gen -- Tom Leinster <tl(a)maths.gla.ac.uk>

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job opening
by I. Moerdijk 19 Feb '07

19 Feb '07

Dear colleagues, I'd like to draw your attention to an opening for a tenure (track) position at Utrecht, at the level of "universitair docent" (comparable to Lecturer in the UK, Assistant Professor in the US or Maitre de Conferences in France). The position is related to a large group grant in the area between (and including) geometry/topology and mathematical physics, which I'm sure will fit the interests of many readers of this list. For more information, see http://www.math.uu.nl/Positions/ With best regards, Ieke Moerdijk.

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