September 2005 - Categories - categories.org.au

PhD position in Monadic Computational Logics
by Lutz Schroeder 17 Jan '06

17 Jan '06

A 3-year PhD-Position is available in the project "Monadic Computational Logics in HOL" at the University of Bremen. The project is concerned with the implementation and further development of monadic computational logics, including monadic Hoare logic and monadic dynamic logic as well as extensions covering exception handling, as introduced by Till Mossakowski and myself. More detailed information can be found at http://www.informatik.uni-bremen.de/~lschrode/research/projects/HOL-MDL_e.h… Applications or further enquiries may be e-mailed to my address below. -- Lutz -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode(a)informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode -----------------------------------------------------------------------------

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Job: Lectureship
by Alexander Kurz 05 Oct '05

05 Oct '05

Apologies for multiple postings ------------------------------- Please distribute to potential candidates ----------------------------------------- The University of Leicester seeks to appoint two Lecturers (tenure track) who can contribute to existing research in the foundations and applications of algebraic structures and methods in general, and the emerging area of service-oriented computing in particular ... (see the URL below for more) In particular, we are looking for one person who can complement our strengths in theoretical research which is currently represented by Roy Crole, Alexander Kurz, Vincent Schmitt and Fer-Jan de Vries. The official announcement should be available soon at http://www.le.ac.uk/personnel/jobs/a&r.html Closing date is (probably, see URL above) 5 July 2005. If you have any informal enquires please contact one of us. 15-Jun-2005 19:59:14 -0300,7560;000000000000-00000012

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Name of concept?
by Michael Barr 04 Oct '05

04 Oct '05

Is there a name for the following situation: I have a diagram of categories and functors DO(F) DO(A) ------> DO(B) | | | | H | |H | | v F v A ----------> B It does not commute, nor is there even a 2 cell in either direction. What I do have is illustrated below: DO(A) DO(A) /|\ /|\ / | \ / | \ / | \ / | \ H/ | \DO(F) H/ | \DO(F) / P \ / P' \ / | \ / | \ v | v v | v A <== | ==> DO(B) A ==> | <== DO(B) \ | / \ | / \ | / \ | / F\ | /H F\ | /H \ | / \ | / \ | / \ | / vvv vvv B B and, moreover, P -------> HF | | | | | | | | | | v v FH ------> P' commutes.

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ENTCS
by Paul Taylor 28 Sep '05

28 Sep '05

I am sure that I speak for all of us in the category theory community when I say that I am very (relieved and) pleased to hear from Mike Mislove that he and ENTCS are alive and well following the hurricane. On the other hand, I am alarmed by the keenness of Mike (and the many conference organisers who use the services of ENTCS) to give away their colleagues' intellectual property to a company whose sole contribution nowadays to the process of publication is to run a web server. In point of fact, Elsevier DOES NOT require ENTCS authors to transfer copyright. When they were processing my paper for last year's CTCS, they asked me for a copyright transfer, but I refused. Instead, they promptly offered me a license agreement. Although the first version of this was also unacceptable, I succeeded in negotiating another one, which I signed, and they published my paper. You can obtain a LaTeX version of this agreement from www.cs.man.ac.uk/~pt/drafts/Elsevier-licence.tex I urge all of my colleagues to use this, and not give a commercial publisher the monopoly of research in theoretical computer science. I fully understand that younger researchers in particular feel that they have no alternative but to give in to pressure, being in a polically weak position myself. But I have demonstrated that even individuals can have an effect, simply by saying no. If more of us say no then we will succeed in recovering what is properly ours. Paul Taylor PS there is lots of new stuff on the ASD web page since my last posting to "categories" about it.

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Union College Conference
by Susan Niefield 27 Sep '05

27 Sep '05

UNION COLLEGE MATHEMATICS CONFERENCE Saturday and Sunday Schenectady December 3-4, 2005 New York This is a preliminary announcement of the twelfth Union College Mathematics Conference. This year the conference topics will be algebraic topology, category theory, and commutative algebra. In addition to plenary lectures, of interest to the entire conference audience, there will be shorter contributed talks in parallel sessions. Anyone interested in giving such a talk should contact one of the organizers. The deadline for submission of abstracts is November 11th, and for registration is November 18th. PLENARY SPEAKERS John Baez (University of California, Riverside) David Cox (Amherst College) Jesper Grodal (University of Chicago) The meeting will begin with an evening reception on Friday, December 2, and end on Sunday afternoon. For more information about the conference, including registration, submission of abstracts, housing and transportation, please visit our website at www.math.union.edu/~leshk/05Conference/ Union College is centrally located in New York's capital district about 10 miles from the Albany International Airport, easily accessible by train from NYC, and just 3 to 4 hours by car from NYC, Boston, and Montreal. CONFERENCE ORGANIZERS Category Theory Susan Niefield niefiels(a)union.edu Algebraic Topology Brenda Johnson johnsonb(a)union.edu Kathryn Lesh leshk(a)union.edu Commutative Algebra Pedro Teixeira teixeirp(a)union.edu David Vella vellad(a)union.edu We hope to see you in December!

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Computational category theory paper - request for comments
by Shane O'Conchuir 27 Sep '05

27 Sep '05

Hi all, I've written a paper on computational category theory but have for a while been unable to determine whether or not my methods or results are new. I would greatly appreciate any input from the community on this matter. The paper, "Proper Diagrams for Constructing Presheaf-Valued Limits", is about the construction of limits of diagrams of presheaves. The content is not too deep mathematically and is aimed more at computation. The basic idea is that given a functor D from a small category C into a category of presheaves, I construct a second functor D' of the same type where there is a natural transformation D'->D whose components are monomorphisms. The construction has the property that the limit of D is isomorphic to the limit of D'. The intuition is that we remove parts of the presheaves so that the computation of the limit is easier. I like to think of this approach as orthogonal to the use of initial functors to construct limits. Both methods preserve the limit but initial functors reduce the number of objects in the domain of a diagram whereas our transformation reduces the objects in the codomain of the diagram. A concise version of the paper is available at https://www.cs.tcd.ie/Shane.OConchuir/limits/limitspaper.pdf A draft technical report with the missing proofs and some appendices is available at https://www.cs.tcd.ie/Shane.OConchuir/limits/limitstr.pdf The technical report explains how I derived my definition of "spare" element. Any comments, corrections, criticisms, or references are welcome. In particular, I would like to know if any of this seems familiar (apologies in advance!) Also, my choice of terminology ("proper", "inconsistent") probably conflicts with normal use and one of my constructions, 'final proper diagram', may well be called 'initial proper diagram'. Many regards, Shane O'Conchuir Department of Computer Science Trinity College Dublin

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Announcing www.entcs.org
by Michael Mislove 25 Sep '05

25 Sep '05

Dear Colleagues, One of the minor inconveniences that Hurricane Katrina caused has been the temporary failure of the Tulane email servers, both university-wide and within the math department. The latter hosted the ENTCS Macro Home Page, so progress on publishing ENTCS volumes has been hindered since the hurricane. I am happy to announce that ENTCS now has its own, separate web host, which can be found at www.entcs.org Please point your browser at this page, where you will find detailed instructions on how to prepare proposals for publishing material in ENTCS, as well as instructions about how to prepare files both for preliminary, hard copy versions of proceedings for limited distribution at meetings, as well as how to prepare the final versions of papers for publication online at Science at Direct. While the ENTCS production has been hampered over the past month or so, it has been restarted, and publication of ENTCS issues and volumes is now proceeding as usual, with minimal delays. An important point to note is that the procedure for submitting final versions of files for ENTCS has changed. We now ask that the corresponding author of each paper - including the Preface of each volume - submit an electronic copy of the signed Copyright Transfer Form that can be found at the ENTCS Macro Home Page archives. To do this, authors should download the pdf file containing the form, print it out and complete it, including signing it, and then scan it in and include this scanned copy with the files for the final version of their paper that are sent to the Guest Editors of their proceedings. This will expedite the publishing of ENTCS volumes: we now can have final versions online within four weeks of when the files are sent to me for final processing. As usual, if you have any problems or questions about the ENTCS macros, or about ENTCS in general, please let me know. Best regards, Mike Mislove -- Michael Mislove Managing Editor ENTCS

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lax crossed modules
by Ronald Brown 22 Sep '05

22 Sep '05

reply to here Vaughan, David An interesting question! It raises several possible red herrings. 1) What is a lax action of a group (or groupoid) G on a group (or groupoid) A? There is a paper by Brylinski in Cahier on this, with applications to K-theory, if I remember rightly. Another interpretation of this seems to be as a Schreier cocycle (factor set). A relevant paper is 97. (with T. PORTER), ``On the Schreier theory of non-abelian extensions: generalisations and computations''. {\em Proceedings Royal Irish Academy} 96A (1996) 213-227. It is a useful exercise (which I meant to write down, but ...) to translate Brylinski into the terms of a map of a free crossed resolution, and so put this into nonabelian cohomology terms, and potentially allow for calculation using a small free crossed resoution when possible .... This suggests what might be a lax action, but does not complete in an obvious way into a lax crossed module. 2) Another way is to go to 2-crossed modules (Daniel Conduche), which brings in relations with simplicial groups (Tim Porter) and higher Peiffer elements. See also the relations with braided crossed modules and other things in 59. (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism structures for crossed modules'', {\em Proc. London Math. Soc.} (3) 59 (1989) 51-73. 3) There are equivalences of categories crossed modules of groupoids <--> 2-groupoids <--> double groupoids with connections <--> double groupoids with thin elements. I have long found the cubical approach easier to follow and to use than the globular, but it turns out one needs also the globular to define commutative cubes in cubical omega-categories with connections (see a recent paper by Philip Higgins in TAC). This raises the question of "lax cubical omega-categories with connections". What do you laxify, to get an equivalence with one or other notion of weak globular (or other?) omega-category??!! Quite an amusing step, and more do-able, would be to generalise Gray categories to: cubical omega-categories C with an algebra structure C \otimes C \to C, generalising Brown-Gilbert, and using the monoidal closed structure given in 116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118 A nice point about such algebra structures is that they allow for a failure of the interchange law, with a measure of that failure, similar to the way 2-crossed modules give a measure of the failure of the Peiffer law for a crossed module by using a map { , }: P_1 \times P_1 \to P_2. Is this related to Sjoerd Crans' teisi? I have a gut feeling that these strengthened sesquicategories (with a *measure* of the failure of the interchange law) will crop up in a variety of situations, e.g. in rewriting, 2-dimensional holonomy, ...., since the interchange law makes things too abelian, sometimes. This brings in automorphism structures for crossed modules, I guess (Brown-Gilbert again, and of course derived from JHC Whitehead, who first studied such automorphisms). Another thought: the non abelian tensor product of groups derives from properties of the commutator map on groups. Why not develop the corresponding theory for the Peiffer commutator map? Hope that helps Ronnie ------------------------------------------------- *Date:* Mon, 19 Sep 2005 09:41:44 -0700 *From:* Vaughan Pratt <pratt(a)cs.stanford.edu> *To:* Ronnie Brown <mas010(a)bangor.ac.uk> *Reply-to:* pratt(a)cs.stanford.edu *Subject:* [Fwd: categories: Question re lax crossed modules] I'd be interested in knowing this too, in particular what the geometric significance of laxness is. Presumably laxness only enters in the passage from pre-crossed to crossed. Vaughan -------- Original Message -------- Subject: categories: Question re lax crossed modules Date: Mon, 19 Sep 2005 12:44:30 +0930 From: David Roberts <droberts(a)maths.adelaide.edu.au> To: categories(a)mta.ca I have been looking at categorical groups a little and was wondering what a lax crossed module is. A search through various databases has turned up nothing. It would seem that they should be like crossed modules but only satisfy a weakened equivariance property. Any pointers toward a definition would be great. ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts(a)maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts

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Bill on John (and his adequate subcats)
by Peter Freyd 22 Sep '05

22 Sep '05

John Isbell's Adequate Subcategories F. W. Lawvere Submitted to the Topology Atlas (in response to Mel Henriksen's request) For mathematicians of my age, the theory of rings of continuous functions was one of the first exciting research topics we encountered. Many results of that theory have appeared, but its ramifications for category theory are still not fully worked out. A crucial link was provided by John Isbell's contributions around 1960 on the theme of adequate subcategories. Briefly, a subcategory A of a larger category is adequate if every object X of the larger category is canonically the colimit of the category A/X of objects of A equipped with structural maps to X; John's equivalent definition was that the truncated Yoneda embedding of the whole category into the category of set-valued contravariant functors on A is actually a full embedding. The following language is suggestive: (a) The objects of A are figure-types, (b) the objects of A/X are particular figures in X, and (c) the morphisms of A/X (commutative triangles in the big category) are incidence relations between figures. Thus adequacy means that the large category in question consists of objects entirely determined by their A-figures and incidence relations, and that (d) the morphisms in the whole category are nothing but the "geometrically continuous" ones in the sense that they map figures to figures without tearing the incidence relations. For example, if A is the category of countable compact spaces then A is adequate in many large categories constructed in attempts to capture the notion of topological space; in this case a morphism can be identified as a mapping that preserves sequential limits. That example is one of many illustrating that typical large categories of mathematics often have quite small adequate subcategories; it had been studied by Fox in 1945 at the instigation of Hurewicz, who sought a rational notion of function space for use in algebraic topology and functional analysis. In fact, for any A, each space of A-continuous morphisms has its own cohesion, described again by A-figures. The dual notion of a co-adequate subcategory C leads to a contravariant representation of the larger category that can be described in terms of (a) quantity-types, (b) functions, and (c) algebraic operations on functions. The dual of the notion of geometric continuity (that is, a name for naturality of maps of covariant functors instead of contravariant ones) is (d) "algebraic homomorphism" These ideas of John Isbell became fused with the conceptions of Kan, Grothendieck, and Yoneda (emerging in the same period 1958-1960), to form a basic method of analyzing and constructing mathematical categories. That method was used in the early 60's by Freyd, Gabriel, Lawvere, Mitchell, and by Isbell himself, and became as natural as breathing to many algebraists and topologists during the following decades. What does adequacy have to do with rings of continuous functions? The theory of rings of continuous functions springs from a basic philosophical hope to the effect that there should be a near-perfect duality between space and quantity. Such duality questions can be investigated for a great many different categories, but categorical considerations suggest that they need to be brought down to earth in certain respects. John Isbell was also one of the main developers of the theory of locales. This theory revealed that the traditional notion of topological space is algebraic rather than geometric (in the sense of the above analysis) with the infinitary algebras (frames) of open sets playing the dominant role; this merely means that the Sierpinski space, together with "all" its powers, constitutes a coadequate subcategory of the category of sober spaces. John's insistent quest for smallness (as a further requirement on co-adequate subcategories C) brought this analysis qualitatively nearer to real mathematics. If a small co-adequate subcategory is available, it can often be reduced to a single object (for example by taking the product of its objects; or more concretely, the Euclidean plane as a topological object will often serve the purpose of co-adequacy). The endomorphisms of that object then parameterize the unary operations whose preservation by C-homomorphisms serves to exclude ghosts from among detected points and figures. Even among those operations a few may be co-adequate, as the Stone-Weierstrass theorem had shown: addition, multiplication, and conjugation can replace the monoid of all continuous operations for that particular task, and there are many variations on that theme. But apart from such details of presentation, the implicit insight of Czech and Stone, of Hewitt and Nachbin, apparently includes smallness of the algebraic theory C in terms of which spaces are to be (co) analyzed. There was a seeming barrier to the realization of that concrete insight: set theory, in its striving for larger and larger cardinals, had neglected to emphasize that all of the cardinals arising in geometry and analysis in fact satisfy a useful smallness condition: the category of countable sets is co-adequate in the category of all small sets. That is essentially John Isbell's formulation; he proved that it is equivalent to the condition that no small set has the kind of ghost elements called Ulam measures. John knew full well that his formulation for abstract sets would imply that many categories having small adequate subcategories also have small co-adequate subcategories, thus making possible the desired sort of dualities between space and quantity. The ideas of John Isbell contributed to the enlightened understanding of mathematics by lifting some dark clouds of confusion, and they continue to be actively developed and diffused.

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Conference Charles Ehresmann: 100 ans
by Andree Ehresmann 21 Sep '05

21 Sep '05

Conference and papers for Charles Ehresmann's centenary I recall that we organize an International Conference "Charles Ehresmann 100 ans" to commemorate the 100th anniversary of Charles' birth. It will take place at the Universite de Picardie Jules Verne in Amiens, from October 7 to 9, 2005. The first day will consist in general lectures on the life and works of Charles, and their prolongations. The 2 other days, there will be 2 parallel sessions: Category theory, Topology and Geometry, including the 82nd session of the PSSL "Peripatetic Seminar on Sheaves and Logic" and a session of the SIC (Seminaire Itinerant de Categories), Multidisciplinary applications with a Symposium ECHO V. The complete program can be found on the Internet site dedicated to Charles: http://perso.wanadoo.fr/vbm-ehr/ChEh which also contains some photos and documents on Charles, and papers dedicated to him. The abstracts of the lectures presented to the Conference will appear in Volume XLVI-3 of the "Cahiers de Topologie et Geometrie Differentielle Categoriques". The participation to the Conference is free. If you want to participate, please contact me by sending your name and address to ehres(a)u-picardie.fr Same address if you want to send a paper to be posted on the site. Sincerely Andree C. Ehresmann

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