November 1998 - Categories - categories.org.au

Preprint available
by grandis＠dima.unige.it 15 Dec '98

15 Dec '98

The following preprint: M. Grandis, "An intrinsic homotopy theory for simplicial complexes with applications to image processing" is available at: ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ as: Lnk.Nov98.ps *** Abstract. A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets; this structure is mostly viewed as codifying a triangulated space. Here, this structure is used directly to describe "spaces" of interest in various applications, where the associated triangulated space would be misleading. An intrinsic homotopy theory, not based on topological realisation, is introduced. The applications considered here are aimed at metric spaces and digital topology; concretely, at image processing and computer graphics. A metric space X has a structure t_e(X) of simplicial complex at each "resolution" e > 0; the resulting n-homotopy group \pi_n(t_e(X)) detects those singularities which can be captured by an n-dimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. *** Comments would be appreciated. In particular, I am uneasy about a question of terminology. In my opinion, the term "simplicial complex", quite appropriate when the structure is viewed as codifying a triangulated space, is unfit when such objects are treated as "spaces" in themselves (somewhat close to bornological spaces, which have similar axioms on objects and maps). In other words, "simplicial complex" should not refer to the category itself, say C, but to its usual embedding in Top, the simplicial realisation. The two aspects may clash, e.g. with respect to initial or final structures: the coarse C-object on three points (final structure, all parts are distinguished) is realised as a euclidean triangle; a C-subobject is sufficient to produce a topological subspace (a regular subobject in Top), but a C-subspace (a regular subobject in C) is a stronger notion. Moreover, from a more concrete point of view, the simplicial realisation is quite inappropriate in most of the applications considered in this work. The opposition "C-object / simplicial complex" is in part similar to "sequence / series": the second term refers to a more specific view & use of the same data; the clashing of the opposition is particularly evident in the notions of convergence, for a sequence or a series. That's why I am calling a C-object a "combinatorial space". (The term "combinatorial complex" has also been used for simplicial complex; and I wanted a term of the form "attribute + space", to use freely of topological terms like discrete, coarse, subspace...) But of course it is embarassing to propose a new term for a classical notion. Marco Grandis

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TOOLS EUROPE '99 - Call for Submissions
by Karen Ouellette 01 Dec '98

01 Dec '98

[apologies if you receive multiple copies of this announcement] **************************************************************** CALL FOR SUBMISSIONS TOOLS EUROPE '99 Technology of Object-Oriented Languages and Systems "OBJECTS, COMPONENTS, AGENTS" Nancy, France, June 7-10, 1999 http://www.tools.com/europe **************************************************************** TOOLS - Technology of Object-Oriented Languages and Systems - is the major series of international conferences and exhibition entirely devoted to the applications of object-oriented technology. Its emphasis is on the practice of object technology and its applications in industrial environments. TOOLS provides a balanced coverage of the wealth of approaches, trends and variants in the object-oriented community. For anyone interested in OT, the TOOLS Conferences are the best places to learn from the experts and compare experiences with other O-O practitioners. CONFERENCE COMMITTEE ~~~~~~~~~~~~~~~~~~~~~~~~ Program Chair: Richard Mitchell, University of Brighton, UK Tutorials Chair: Alan Cameron Wills, TriReme International Ltd., UK Workshops & Panels Chair: Jan Bosch, University of Karlskrona/Ronneby, Sweden Conference Chair: Jean-Pierre Finance, Henri Poincar� University, France Organising Chair: Martine Gautier, Henri Poincar� University, France Conference Series Chair: Bertrand Meyer, ISE, USA PROGRAMME COMMITTEE ~~~~~~~~~~~~~~~~~~~~~~~ Mehmet Aksit, University of Twente, Netherlands Fran�ois Ban�ilhon, Ardent Software, USA Richard Bielak, CAL FP (US), Inc., USA Gilad Bracha, Sun Microsystems, USA Eduardo Casais, Nokia, Finland Alistair Cockburn, Humans and Technology, USA Dominique Colnet, LORIA - INRIA Lorraine, France Bernard Coulange, Verilog, France Paul Dubois, Lawrence Livermore National Laboratory, USA Jean-Marc Geib, University de Lille, France Yossi Gil, Technion, Israel & IBM Research, USA Petter Graff, InferData, USA Ian Graham, Chase Manhattan Bank, UK Brian Henderson-Sellers, Swinburne University of Technology, Australia Laura Hill, Sun Microsystems, USA Ren� Jacquart, CERT, France Jean-Marc J�z�quel, IRISA/CNRS, France Rachid Guerraoui, Swiss Federal Institute of Technology, Switzerland Sven-Eric Lautemann, University of Frankfurt, Germany Michel Lemoine, ONERA-CERT, France Ian Maung, Platinum Technology, USA Jim McKim, Rensselaer Polytechnic, USA Mira Mezini, University of Siegen, Germany Christine Mingins, Monash University, Australia Frieder Monninger, Object Tools, Germany Meilir Page-Jones, Wayland Systems, USA Jean-Claude Royer, University of Nantes, France Ted Velkoff, Lockheed Martin, USA Kim Wald�n, ENEA, Sweden Tony Wasserman, Software Methods and Tools, USA Wolf Zimmermann, University of Karlsruhe, Germany TECHNICAL PAPERS ~~~~~~~~~~~~~~~~~~ TOOLS EUROPE '99 is now soliciting papers on all aspects of object-oriented technology. All submitted papers will be refereed and judged by the International Programme Committee, not only according to standards of technical quality but also on their usefulness to practitioners and applied researchers. A non-exhaustive list of topics includes: * Production, distribution and revenue collection of industrial-quality reusable components * Extending objects to include, for example, rules * Specification, modeling, and patterns * Architectures and frameworks * Working with a mixture of technologies and languages * Managing software assets * Empirical and field studies * Experience reports: TOOLS Europe '99 is particularly keen on receiving experience reports describing lessons learned from the practical application of objects, components, agents, frameworks, and architectures Submitted papers should be around 12 pages, single or one-and-a-half spaced. Send six copies of your paper to the Programme Chair, to arrive no later than Friday, February 19, 1999. Each submission should be accompanied by a plain-text Notice of Submission, due a couple of days before the paper. Send this notice to <Richard.Mitchell(a)brighton.ac.uk> (with subject line "TOOLS Europe '99 Notice of Submission"), to arrive no later than Friday, February 12, 1999. The proceedings of the 29th TOOLS Conference will be published by IEEE Computer Society Press. Final camera-ready versions of accepted papers will therefore be required to adhere to the IEEE publication format (guidelines available soon), and will contain no more than 12 pages. IMPORTANT DATES: Notice of submission: February 12, 1999 Manuscript submission: February 19, 1999 Notification of acceptance: March 29, 1999 Final copy due at IEEE: April 16, 1999 PROPOSALS SHOULD BE SUBMITTED TO: Richard Mitchell TOOLS EUROPE '99 Programme Chair Faculty of IT University of Brighton Brighton BN2 4GJ UK Phone: +44 1273 642458 Richard.Mitchell(a)brighton.ac.uk TUTORIALS ~~~~~~~~~~ Proposals are sought for high quality tutorials. A tutorial should address a single topic and have a clearly defined tar get audience. A tutorial will usually involve a variety of activities for those attending. Each tutorial is half a day in length. The presenter should have proven experience that qualifies him or her to give the tutorial. Proposals for tutorials should identify the topic, the audience and the relevant experience of the presenter. Please note that you must prepare tutorial notes for the participants. IMPORTANT DATES: Tutorial submission deadline: January 8, 1999 Notification of acceptance: March 29, 1999 PROPOSALS SHOULD BE SUBMITTED TO: Alan Cameron Wills TOOLS EUROPE '99 Tutorials Chair TriReme International Ltd., UK alan(a)trireme.com WORKSHOPS ~~~~~~~~~~~~ Workshops provide a forum for researchers and practitioners to meet and discuss focused issues in an atmosphere that encourages interaction, exchange, and problem solving. A workshop should have a well-defined theme, and a clearly-defined target audience. All topics related to object-oriented technology are potential candidates for workshops. In particular, workshop proposals fall into three categories: * Workshops focusing on a topic in depth. Examples include object-oriented analysis and design methods, object-oriented languages, e.g. Eiffel, etc. * Workshops crossing borders of sub-areas in computer science and/or software engineering. Examples are management and teaching issues. * Workshops addressing the industrial application of object-oriented technology in particular areas, e.g. telecommunications, mobile computing or real-time systems. IMPORTANT DATES: Workshop submission deadline: January 8, 1999 Notification of acceptance: March 29, 1999 PROPOSALS SHOULD BE SUBMITTED TO: Jan Bosch TOOLS EUROPE '99 Workshops & Panels Chair University of Karlskrona/Ronneby, Sweden Jan.Bosch(a)ide.hk-r.se PANELS ~~~~~~~~ The aim of panels is to stimulate discussion about ideas and issues of importance to the object-oriented technology community. They should present interesting and divergent views with solid practical and/or theoretical backing. Panels last about sixty minutes and can be organised in many formats. IMPORTANT DATES: Panel submission deadline: January 8, 1999 Notification of acceptance: March 29, 1999 PROPOSALS SHOULD BE SUBMITTED TO: Jan Bosch TOOLS EUROPE '99 Workshops & Panels Chair University of Karlskrona/Ronneby, Sweden Jan.Bosch(a)ide.hk-r.se ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TOOLS EUROPE '99 IS ORGANISED IN COOPERATION WITH UNIVERSIT� HENRI POINCAR�, NANCY, FRANCE. FOR MORE INFORMATION ON TOOLS USA '99, VISIT OUR WEB SITE AT http://www.tools.com/europe OR CONTACT US AT tools-europe(a)tools.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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commutative monoids
by Tom Leinster 27 Nov '98

27 Nov '98

A commutative monoid is a strict monoidal category with one object, so in this sense one can talk about a category enriched in a commutative monoid. Has anyone looked very hard at such things? Thanks, Tom

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n-categorical miscellany
by john baez 25 Nov '98

25 Nov '98

1) Carlos Simpson writes: >A. Joyal writes: >>In particular, the new theory should help understanding >>homotopy groups of spheres. >I think it is safe to say that the ``n-category crowd'' (myself included) >would love to make some progress on this. The main thing I am wondering >about is: what exactly does one want to know about the homotopy groups of >spheres? for example, is there a concrete question which needs answering? There are a lot of complicated and interesting patterns in the stable homotopy groups of spheres. A lot of these patterns are currently studied using a generalized cohomology theory called complex cobordism theory, together with an intricate mess of spectral sequences: Complex cobordism and stable homotopy groups of spheres / Douglas C. Ravenel. Orlando: Academic Press, 1986. Nilpotence and periodicity in stable homotopy theory / by Douglas C. Ravenel. Princeton, N.J. : Princeton University Press, 1992. It's all far too technical for me to understand, so I don't have much hope of suddenly answering some question that homotopy theorists are stuck on! However, I think that the n-category crowd can give more conceptual explanations of some facts that appear as miracles in the current approach. For example, there's an important relation between complex cobordism theory and formal group laws: the complex cobordism of a point is "the universal formal group law". There should be an elegant conceptual proof of this, but as far as I know, there are just elegant *heuristics* for why it should be true, followed by a grungy computation. 2) Carlos Simpson writes: >On an introduction to n-categories: it would be great if J. Baez could >bundle together his ``This week's finds in math. phys.'' which concern >n-categories. This would make a really nice introduction specially for >friends and family. Especially during the holiday season! I should try to bundle them together more nicely at some point, but right now you can start at: http://math.ucr.edu/home/baez/week73.html , skip the bit about left-right asymmetry in amino acids, read the stuff on n-categories, and then keep clicking on the thing that says: "To continue reading the `Tale of n-Categories', click here." It fizzles out around week100, though I plan to resume it with an issue discussing Carlos Simpson's paper on the stabilization hypothesis and Mark Hovey's book on model categories. 3) Some people have noted that http://xxx.lanl.gov/abs/math.QA/9811139 doesn't give them the paper "HDA4 - 2-Tangles". Sorry! - the delay before the preprint server makes papers publicly available is longer than it used to be. It should be working by tomorrow. If you still have trouble after that (it's a big file and may bust your printer), feel free to email me and ask me to send you a copy.

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Continuity
by Matthieu Amiguet 25 Nov '98

25 Nov '98

Dear categoricians, I'm wondering about the possibility of speaking of trajectories in a category. That is, given a category C, in what sense - if any - can we consider a continuous function R->C (where R are the real numbers). What extra structure is needed on C to consider it as a topological space? what relation with the category structure? I guess there's a link with topos theory, but I can't make it really clear... Subsidiary question: in what case is there a notion of differentiability for the function f? Thank you for direct answers or pointers to (quite easyly understandable) literature. Regards, ---------------------------------- Matthieu Amiguet, doctorant Institut d'Informatique et d'Intelligence Artificielle UniversitÃ© de Neuchatel rue Emile Argand 11 CH - 2000 Neuchatel tel: +41 32 718 27 36 matthieu.amiguet(a)info.unine.ch ----------------------------------

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HDA4: 2-Tangles
by john baez 24 Nov '98

24 Nov '98

The following preprint is now available at the places listed below. Comments and corrections are welcome! -------------------------------------------------------------------- Higher-Dimensional Algebra IV: 2-Tangles John C. Baez, Laurel Langford Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. By this universal property, any unframed self-dual object in a braided monoidal 2-category with duals determines an invariant of 2-tangles in 4 dimensions. --------------------------------------------------------------------- This paper is math.QA/981139 on the mathematics preprint server, so you can get it at: http://xxx.lanl.gov/abs/math.QA/9811139 It's also available as a Postscript file at my website: http://math.ucr.edu/home/baez/hda4.ps If you have any trouble, let me know and I can send you a copy.

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about dimension of morphisms
by Philippe Gaucher 24 Nov '98

24 Nov '98

Bonjour, I think that I have found a small pathological behaviour with dimension of morphisms in omega-categories. This is the following one. Let us first recall that min{m/s_m(x)=x} = min{n/t_n(x)=x} =: dim(x). Because x = s_m(x) = t_m s_m(x) (because m=<m) = t_m(x). Lemma 1 : Let u and v be two composable morphisms of dimension 1 (i.e. t_0(u)=s_0(v)). Suppose that t_0(v)<>s_0(u) (<> means different). Then u o_0 v is of dimension 1. Proof : Suppose that u o_0 v of dimension 0. Then s_0(u) = s_0(u o_0 v) = u o_0 v and t_0(v) = t_0(u o_0 v) = u o_0 v So s_0(u) = t_0(v) : false. Here is now the pathological behaviour : If s_0(u) = t_0(v), here is an example of category with u and v of dimension 1 and u o_0 v of dimension 0 : We take A={alpha,beta,u,v} (four elements) ------------------------------------- alpha beta u v ------------------------------------- s_0 alpha beta alpha beta ------------------------------------- t_0 alpha beta beta alpha ------------------------------------- and for o_0 : ------------------------------------- alpha beta u v ------------------------------------- alpha alpha xxxx u xxxx ------------------------------------- beta xxxx beta xxxx v ------------------------------------- u xxxx u xxxx alpha ------------------------------------- v v xxxx beta xxxx ------------------------------------- (at the intersection of row x and column y, read x o_0 y) All axioms of categories seem to be verified (?). The only composition is a "word" of finite length "... u (beta) v (alpha) u (beta) v..." In (A,s_0,t_0,o_0), u and v are of dimension 1 and u o_0 v of dimension 0 because u o_0 v = alpha. Where am I wrong ? (observe that this behaviour cannot appear with for example a category coming from a composable pasting scheme). pg.

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query
by James Stasheff 21 Nov '98

21 Nov '98

The language of higher category theory in more than analogous to homotopy theory, especially in the cellular version. What is the appropriate reference for a non-categorical reader? Have the A_\infty categories of Smirnov or Fukaya been treated in the categorical literature?? thanks .oooO Jim Stasheff jds(a)math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds

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Jim's query
by T.Porter 21 Nov '98

21 Nov '98

This is not really an adequate reply to Jim's query. The reason is that as I understand it, he is asking for a source that will explore the homotopy theory of the `globular' models of weak infty categories. My approach to the area is in some sense the dual. Starting with the various models for homotopy types or bits of them, try to see what is mirrored in the weak infty category theory by the homotopy structure. This is in some sense the converse to his query but may none-the-less be relevant. My intuition was, and still is, that the Kan condition on simplicial sets gives a composition/pasting up to coherent homotopy. Moreover the `filler' gives the justification for the composite. (Compare the filler structure of the nerve of a category with that in an arbitrary Kan complex.) Accepting that as a starting point, and the idea that the `category' of weak infinity categories should be a weak infinity category, Jean-Marc Cordier and I looked at `locally Kan' simplicially enriched categories (e.g. Trans AMS 349(1997)1-54). With that viewpoint, it becomes clear that the structure of an A_\infty category is needed to make things really `coherent', but that many of the constructions of `ordinary' category theory have A_\infty or homotopy coherent analogues in this setting, which thus serves as a test-bed' for the development of the more general theory. In part this relates to Michael Batanin's paper in the Cahiers where explicit consideration of A_\infty structure is given. That theory looks at the `global' structure to some extent, but simplicially enriched groupoids model all homotopy types, so a corollary of the simplicial to globular type of transition should be that one should be able to construct weak \infty categroies DIRECTLY from the algebra of a simplicially enriched groupoid. The obvious place to look for this is in the Moore complex which carries a hypercrossed complex structure in the sense of Pilar Carrasco and Antonio Cegarra. (This is related to the n-hypergroupoid structures of Jack Duskin.) Exploring the \infty category structure, potentially in their definition, is the aim of another line of research and in low dimensions, this has been attacked by Ali Mutlu and myself, (see very recent articles in TAC or Bangor's preprint list on the web). Getting nearer to Jim's query, any bridge between homotopy theory and higher dimensional category theory should I feel aim to be approachable by algebraic topologists and therefore should start with a recognisable model for homotopy types. Another approach that must be mentioned is that of Tamsamani and Simpson using multisimplicial objects. Presumably this also links in with the cat^n-groupoid approach pioneered some 14 years ago by Loday. This only handles n-types but can be extended to a model that has higher information but in those dimensions above n, the Whitehead products are trivial. (Has anyone looked at the Whitehead and Samelson products from a globular or weak \infty category viewpoint?) The question of simplicial rather than cubical theory is a difficult one. Marco Grandis made a good case for the cubical formulation the other day, and the use of Kan filler conditions in a cubical setting has been for a long time a speciality of Heiner Kamps (see the recent book by him and me!). That theory does not directly address the question of A_\infty category structures, but it does raise a question that deserves some attention. Suppose you want an analogue of a given theorem from homotopy theory but in another non-topological context and you can get a weak composition structure in your setting (typically given by fillers for boxes in a cubical `enrichment'). What fillers/composites do you need for your particular theorem (e.g.Dold's theorem on homotopy equivalence between cofibrations, or long Dold-Puppe type sequences)? This last question also raises that of the motivation for the generalisations. I am convinced these are useful, even important, but perhaps some debate on directions to explore and goals to seek out might help in providing a fuller answer to Jim's question. Tim. ************************************************************ Timothy Porter |tel direct: +44 1248 382492 School of Mathematics |mathematics office: 382475 University of Wales, Bangor | fax: 383663 Dean St. |World Wide Web Bangor |homepage http://www.bangor.ac.uk/~mas013/ Gwynedd LL57 1UT |Mathematics and Knots : exhibition United Kingdom |http://www.bangor.ac.uk/ma/CPM/

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staragain.dvi
by Michael Barr 21 Nov '98

21 Nov '98

A paper titled *-Autonomous categories: once more around the track has just been posted: ftp.math.mcgill.ca/pub/barr/staragain.dvi Basically, it redoes nearly all of the original Lecture Notes volume in about one fifth the space, using the Chu construction and proving a very general theorem. The only example from the original monograph that is not convered by this theorem is the category of Banach balls, the subject of a recent paper by Kleisli and me. That is also thee under the name balls.dvi (if I have remembered it correctly). This paper is to be submitted to tac for the Lambekfestschrift.

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