November 2004 - Categories - categories.org.au

question about fibrations
by Alexey Cherchago 13 Dec '04

13 Dec '04

Dear Cat Community, I have the following question about preservation of properties of fiber categories in the total category: Given a contravariant indexed category F : B^Op -> Cat, and the Grothendieck translation from the indexed category to fibrations Flat(F). Then, the first projection P : Flat(F) -> B forms a (split) fibration, called the flattening of F. Now assume that the fiber categories are complete/cocomplete/adhesive. Would it be the case that the total category Flat(F) of the described (split) fibration is also complete/cocomplete/adhesive? Any references would be appreciated. Thank you kindly. Best regards, Alexey Cherchago 25-Nov-2004 14:18:16 -0400,2139;000000000000-00000000

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smooth spaces
by John Baez 01 Dec '04

01 Dec '04

Dear Categorists - Algebraic topologists like to use not Top but a "convenient category" of topological spaces which has the advantage of being cartesian closed. The most popular seems to be CGHaus, the category of compactly generated Hausdorff spaces, or "Kelly spaces". It's cartesian closed, complete and cocomplete. I'll take these as the requirements for a category being "convenient". Lawvere and others have considered various topoi of "smooth spaces" in their work on synthetic differential geometry. I'm looking for something similar, but apparently a bit different. I want a convenient category of smooth spaces equipped with a forgetful functor to CGHaus: U: Smooth -> CGHaus so I can do differential geometry and the apply it to algebraic topology with the greatest of ease. I want U to be faithful, so that smoothness is just a *property* of continuous maps between smooth spaces. And, I want U to preserve limits and colimits. The topoi used in synthetic differential geometry don't seem to meet these requirements, because the all-important "infinitesimal arrow" D doesn't seem to have a good underlying Hausdorff space. You could say its underlying Hausdorff space is the one-point space, but this would not give a *faithful* functor U. Chen and Mostow have given definitions of "smooth space" that might meet my requirements, and I'm wondering what people think of them: K.T. Chen, Iterated paths integrals of differential forms and loop space homology, Ann. Maths. 97 (1973) 213-237. M. A. Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Diff. Geom. 14 (1979), 255-293. I only have the energy to describe Chen's definition. I'm hoping someone can give me an elegant proof that it gives a category meeting my requirements. I think it does, but I don't know an elegant proof. I'll use "space" to mean an object of CGHaus, though Chen actually uses it to mean an object of Top. Definition: A "smooth space" X is a space equipped with, for each convex open subset U of some R^n, a collection P(U,X) of continuous maps f: U -> X, called "plots". These need to satisfy various properties: 1) If f: U -> X is a plot and g: V -> U is a smooth map with V convex open in some R^m, then fg is a plot. 2) If U is covered by convex open sets U_a and the restriction of f: U -> X to each U_a is a plot, then f is a plot. 3) If U = R^0, every map f: U -> X is a plot. Definition: If X and Y are smooth spaces and g: X -> Y is continuous, g is defined to be "smooth" if for every plot f: U -> X, gf: U -> Y is a plot. We can try to psychoanalyze this definition: Instead of talking about "convex open sets of R^n's" and smooth maps between these, I'd prefer to talk about R^n's and smooth maps between these, because I believe these form an equivalent category. Let's call this category OpenBall. If we do this and then drop all but condition 1), Chen's definition would say that a smooth space is a space X equipped with a presheaf P on OpenBall assigning to each R^n a set of "plots" P(R^n,X). He also requires that P be a subpresheaf of the presheaf that assigns to each R^n the set of all continuous functions from R^n to X. Then, 2) says this presheaf P is actually a sheaf with respect to some Grothendieck topology on OpenBall. Then, 3) is some extra condition that guarantees smooth maps between smooth spaces are determined by what they do to global points. Or if you prefer, a condition that guarantees constant maps are smooth. So, his definition seems like a way of starting with OpenBall equipped with some Grothendieck topology, taking the category of sheaves on this, and then forming a subcategory with the help of the functor OpenBall -> CGHaus. I'm wondering if anyone recognizes this construction as a standard trick for building topoi... or the composite of a couple standard tricks. Or, if there's some similar but better way to meet my requirements! Happy Thanksgiving, jb 27-Nov-2004 13:23:18 -0400,1354;000000000000-00000000

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Contexts for self-similarity
by Tom Leinster 30 Nov '04

30 Nov '04

> i am a bit surprised with tom's line of reasoning: "it's not > topology, so it shouldn't be mentioned" I don't know what I wrote suggesting that I think that; I don't. In two of the papers advertised (first paragraph of the first paper and bottom of p.3 of the second), I listed some other types of self-similarity that I'd like to investigate. Let me expand on the possibilities. 1, 2. Set-theoretic and topological: these are the types of self-similarity that I understand by far the best at present. 3. Type-theoretic: recursive datatypes can be understood as self-similar objects, the best-known example being trees (which can be characterized as a final coalgebra). For instance, given a polynomial p(x) in N[x], you can consider solutions to x = p(x) (in rings, rigs, distributive categories, and rig categories): see the work of Robbie Gates, Peter Hines (TAC vol 6), Marcelo Fiore and I, and probably many others. 4. Conformal/analytic: this is the natural setting for discussing the self-similarity of Julia sets of complex rational functions. 5. Metric: e.g. the Koch snowflake is topologically just a circle, but has interesting metric self-similarity. 6. Measure-theoretic: cf. some of Peter's postings of 1999-2000 concerning integration. 7. Order-theoretic: both Peter's and Dusko's/Vaughan's characterizations of a real interval produce its ordering. (I have some idea of how to handle order in the much more general situation discussed in my papers, but it's early days.) 8. Categorical: e.g. the category of strict omega-categories is the terminal coalgebra for the endofunctor of CAT defined by V |--> V-Cat. (This was an idea of Carlos Simpson.) The same goes for globular sets (omega-graphs), changing V-Cat to V-Graph. 9. Statistical: there are so-called random fractals - e.g. take a black square; divide it into a 3x3 grid and white out each of the 9 subsquares with probability p; do the same to each black subsquare; continue ad infinitum. 10. Algebraic: the Thompson groups are in some sense highly self-similar. (These groups may be best known to readers of this list from Freyd and Heller's work on homotopy idempotents, but are also very tree-y in nature.) Here's a question belonging to (10), to which I don't know the answer. Let C be the category whose objects are triples (V, v_0, v_1) where V is a vector space and v_0 and v_1 are linearly independent vectors in V, and whose maps preserve linear structure and the `basepoints'. There's a `wedge' functor C x C --> C defined by (V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1) where (V + W)/~ is the direct sum with v_1 identified with w_0. (So dim(V wedge W) = dim V + dim W - 1.) There's then an endofunctor G of C given by self-wedging. Question: what, if any, is the terminal G-coalgebra? (I suspect the answer is something to do with measure/integration - again see Peter's previous postings - but really have no idea.) Finally, re citations: I'll stick in a Pavlovic-Pratt reference, as suggested. All the best, Tom 26-Nov-2004 13:34:46 -0400,1447;000000000000-00000000

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fibrations
by Agusti Roig 30 Nov '04

30 Nov '04

****************************************** Alexey Cherchago escribió: > Dear Cat Community, > > I have the following question about preservation of properties of fiber > categories in the total category: > > Given a contravariant indexed category F : B^Op -> Cat, and the > Grothendieck translation from the indexed category to fibrations > Flat(F). Then, the first projection P : Flat(F) -> B forms a (split) > fibration, called the flattening of F. > > Now assume that the fiber categories are complete/cocomplete/adhesive. > Would it be the case that the total category Flat(F) of the described > (split) fibration is also complete/cocomplete/adhesive? > > Any references would be appreciated. > Thank you kindly. > > Best regards, > Alexey Cherchago I'm not an expert on the field and I don't know if for split fibrations you would have a better result, but as far as I know, if you want the total category A of a fibered categoy \pi : A ---> E to be complete, you need the fiber categories A_x to be complete, but also the base category E . If you want also A to be cocomplete, then you need that the fibered category is cofibered and the base and the fibers cocomplete. So, if you have a bifibered category (fibered and cofibered), with base and fibers complete and cocomplete, then the total category is complete and cocomplete (and I don't know anything about "adhesive"). Computations are easy and you can find them, for instance, in section 3 of Model category structures in bifibred categories JPAA 95, (1994), 203 - 223 But, as I said, I'm not an expert: only a user of fibered categories. So I would also appreciate some references for Flat(F) : is this the fibered category associated to F in the sense of SGA1, or am I missing something? Also another question about fibered categories: has someone developped the notion of fibered 2-category? The reason for my question is this: I've encountered the following situation at least four times recently. I have a 2-functor F : E ---> Cat and I am interested in the (co)fibered category associated to F . So I forget the 2-structure and I take it: \pi : A ---> E Then I realize (and I need, at least in one of the examples) that A has also a "natural" (at least in 3 of the four examples I have in mind) structure of 2-category. So my question is: Is there any canonical way to build a fibered 2-category from a 2-functor? I suspect that, in general, the answer should be "no", because in my fourth example the 2-structure of A seems very ad hoc, and has nothing to do with F . So maybe the right questions are: - What should be a "fibered 2-category"? - Which extra conditions do you have to impose on F in order to obtain a natural, canoncial, fibered 2-category from it? -- Agustí Roig Martí Universitat Politècnica de Catalunya Dept. Matemàtica Aplicada I, ETSEIB - FME Diagonal 647 08028 Barcelona

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on Vaughn and Tom
by Peter Freyd 30 Nov '04

30 Nov '04

Vaughn's point is a good one, but not quite apropos of Tom's notion of self-similarity. In the category of ordered sets the final co-algebra of the Pavlovic-Pratt functor is, indeed, the half-open real interval, [0,1). But Tom was in the category of topological spaces. To make [0,1) into a self-similar _space_ one must reinterpret their functor so that the new point of the one-point compactification of each copy of X is identified with the bottom of the next copy. So what interesting space was first described as the final co-algebra of a functor? (I think that locution succeeds in avoiding such things as constant functors.) I would suggest the one-point compactification of the countable discrete set. (The functor, of course, is 1 + X.) Peter 24-Nov-2004 20:03:48 -0400,2672;000000000000-00000000

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Re: Informatics_Theory list
by Bernhard Reus 29 Nov '04

29 Nov '04

Hi Peter, On Monday, November 29, 2004, at 06:05 pm, Peter Selinger wrote: > I recently posted a message to the "categories(a)mta.ca" mailing > list. Apparently your Informatics_Theory list is subscribed to the > categories list, but rejects its messages (I received the below bounce > message). thanks for pointing out to me that these messages bounce back to the authors. I did not know that. This is, of course, not intended. There is a problem that , for reasons yet unknown to me, the categories messages end up on an internal mailing list here in Sussex (of which I happen to be the administrator) that rejects mails from unregistered senders. Cheers, Bernhard

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n-fold categories, n-fold operads
by Stefan Forcey 29 Nov '04

29 Nov '04

Dear category list, Here is yet another operad paper that deals with issues of interest to people in categorical homotopy. The examples mentioned in the abstract actually make up the majority of the text, and come with illustrations. _________________________________________________ Combinatoric n-fold categories and n-fold operads Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras in an iterated monoidal category. We discuss examples of these that live in the previously described categories. Finally we describe the (n-2)-fold monoidal category of n-fold V-operads. _________________________________________________ It is on the arxiv at: http://www.arxiv.org/abs/math.CT/0411561 or from my webpage: http://www.math.vt.edu/people/sforcey/class_home/research.htm Comments and suggestions are welcome indeed! Thanks, Stefan Forcey 30-Nov-2004 08:03:55 -0400,2670;000000000000-00000000

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Open letter: Future of Category Theory Conferences
by F W Lawvere 29 Nov '04

29 Nov '04

Dear categorists, In order to help insure the continuity of the international CT meetings for the advancement of our subject, it has been proposed by Jiri Adamek, Francis Borceux, Dominique Bourn, George Janelidze, Peter Johnstone, Bob Rosebrugh, Jiri Rosicky, Ross Street, and Walter Tholen, that an Advisory Committee be formed with me as chairman. I am willing to accept this position if there are no objections. After January 11th I will name a vice chairman and propose the membership of the committee. Thanking you for your confidence, I send you all best wishes for a happy and productive new year. Bill ************************************************************ 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.buffalo.edu/~wlawvere ************************************************************ 29-Nov-2004 16:05:09 -0400,2108;000000000000-00000000

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Re: Wilson space
by Vaughan Pratt 27 Nov '04

27 Nov '04

> Peter F: > For that we need -- as we called it in Cats and Alligators -- Wilson space. What section number? (Not in the index.) > Actually, > not the space but the linearly ordered set, most easily defined as the > lexicographically ordered subset, W, of sequences with values in > {-1, 0, 1} consisting of all those sequences such that for all n > a(n) = 0 => a(n+1) = 0 (take a finite word on {-1,1} and pad it > out to an infinite sequence by tacking on 0s). (The parenthetical explanation presumably is meant to apply only to those words containing at least one 0, i.e. W should also contain the infinite words on {-1,1} (else it would be countable).) So shouldn't it be called the Wilson chain then, rather than Wilson space? Ditto for the turkey chain I described yesterday (rationals in quadruplicate, vs. triplicate for the Wilson chain). As a topological space, I don't understand what Wilson space is. The order interval topology (generated by the open order intervals, sets of the form {x | p<x<q}) on a chain makes the irrationals Baire space, the reals the continuum, and the-reals-with-each-rational-duplicated Cantor space. If I haven't miscalculated, it makes the Wilson chain simply C+N, the sum of Cantor space and discrete N (natural numbers). (N arises as the middle rational in each triple. Each middle rational appears simultaneously as the greatest element of one open interval and the least element of another; intersect those to get a singleton open.) If Wilson space is not the order interval topology, i.e. not C+N, then surely it must not be Hausdorff. The only natural possibility I could think of in that case is the 1-point compactification of C iterated omega times, i.e. each element of N appears only in cofinite opens, which didn't seem likely. In any of these cases, my analogously defined "turkey space" looks like it ought to be homeomorphic to Wilson space. Vaughan 28-Nov-2004 13:37:05 -0400,5008;000000000000-00000000

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The premier conference in Model-Driven Development - 2005 edition
by Lionel Briand 26 Nov '04

26 Nov '04

<>Apologies if you receive multiple copies of this CFP. ************************************************************************ Call for Papers (scientific, experience, Workshop, tutorials, etc.) MoDELS 2005 ACM/IEEE 8th International Conference on Model Driven Engineering Languages and Systems (formerly the UML series of conferences) October 2 - 7, 2005 Half Moon Resort, Montego Bay, JAMAICA http://www.modelsconference.org/ ************************************************************************ <<MDA>> <<UML>> <<MCI>> and more !!! The growing complexity of computer-based systems has resulted in an awareness of the need to raise the level of abstraction at which such systems are developed. Model-driven development approaches and technologies, in which development is centered round the manipulation of models, address this need. MoDELS 2005 is the premier conference devoted to the topic of model-driven engineering of software-based systems, and covers both languages and development frameworks used to create and evolve complex software systems. This conference is both an expansion and a re-direction of previous Unified Modeling Language (UML) conferences, and replaces that series of conferences for 2005 and beyond. While the UML has played a significant role in the development of software modeling approaches in both academia and industry, there is a need to address modeling issues that go beyond the use of the UML. Initiatives such as the OMG's Model Driven Architecture (MDA) and Model Integrated Computing (MIC) are a reflection of the desire to shift attention to the broader problem of providing cost-effective support for model-driven development. The MoDELS series of conferences is intended to be the premier venue for the exchange of innovative technical ideas and experience related to model-driven development of software-based systems. MoDELS 2005 will include both scientific and experience conference papers, workshops, tutorials, a doctoral symposium, posters and a tool challenge. (See conference web site for further details) Scientific papers: We invite scientific research papers describing innovative research on model-driven development, and innovative research on other aspects of modeling. This also includes well-designed empirical studies and innovative automation solutions and tool architectures. Experience papers: We invite experience papers that focus on reporting project experience with model-driven development. These papers should describe the project context, detail practical lessons learned, and provide insight about how model-driven approaches and technologies can be improved for application in an industrial context. The conference tracks will include sessions on MDA, UML, MIC, and will include papers on the following topics: - Model-driven development methodologies, approaches, and languages - Model transformations - Model Integrated Computing - Unified Modeling Language: What are the next steps? - Empirical studies of modeling and model-driven development - Tool support for any aspect of model-driven development or model use - Models in the development and maintenance process - Model evaluation, formal or heuristics - Metamodeling - Semantics of modeling languages - Domain-specific and concern-oriented modeling IMPORTANT DATES Experience and Scientific Papers: - Hard Deadline for Abstracts: March 21, 2005 - Hard Deadline for Submissions: April 4, 2005 - Notification to Authors: June 6, 2005 - Final Version of Accepted Papers: July 11, 2005 Workshop Proposals: - Deadline for submission: May 6, 2005 - Notification of acceptance: June 24, 2005 Tutorial Proposals - Deadline for submission: June 6, 2005 - Notification of acceptance: July 4, 2005 - Camera-ready tutorial notes: September 5, 2005 Doctoral Symposium,Tools and Exhibits, Posters and Demos: TBA PAPER SUBMISSIONS: Submit your manuscript electronically in Postscript or PDF using the Springer LNCS style: http://www.springer.de/comp/lncs/authors.html Submission web site: http://sql-05.sce.carleton.ca/models2005 Abstract and paper submittal will be available through the web site by January, 2005. Scientific papers should be no longer than 15 pages in length, and experience papers no longer that 10 pages. Papers will undergo a thorough process of review by a program committee comprising leading experts from academia and industry; however, papers that are too long may be rejected without review. Scientific proceedings will be published by Springer-Verlag in the LNCS series, and experience papers will be published in a companion proceedings. All papers must be original, unpublished, and not submitted simultaneously for publication elsewhere. Proposals for advanced workshops, tutorials and posters are requested. See the conference web site http://www.modelsconference.org/ for details. ************************************************************************ General Chair: Stuart Kent, Microsoft, UK Conference Co-Chairs: Geri Georg, Colorado State University, USA, and Ezra Mugisa, The University of the West Indies at Mona, Jamaica Programme Chair: Lionel Briand, Carleton University, Canada Treasurer and Registration Chair: Robert France, Colorado State University, USA Experience Track Chair: Clay Williams, IBM Watson Research Center, USA Workshop Chair: Jean-Michel Bruel, University of Pau, France Tutorials Chair: TBA Panel Chair: Siobhán Clarke, Trinity College, Ireland Doctoral Symposium Chair: Jeff Gray, University of Alabama at Birmingham, USA Tool Exhibition Chair: Gunjan Mansingh, University of the West Indies at Mona, Jamaica Poster Chair: Felix Akinladejo, University of Technology, Jamaica Local Arrangements Chair: Charmaine DeLisser, University of Technology, Jamaica Publicity Chairs: Joao Araujo, University of New Lisbon, Portugal and Emanuel Grant, University of North Dakota, USA Web Chair: Sudipto Ghosh, Colorado State University, USA Program Committee: TBA ************************************************************************ - ******************************************************* Lionel C. Briand, Ph.D., P. Eng. Canada Research Chair (Tier I) in Software Quality Engineering Carleton University Software Quality Engineering Laboratory Department of Systems and Computer Engineering 1125 Colonel By Drive, ME4456 Ottawa, Canada, K1S 5B6 Phone: (613) 520 2600 Ext. 2471 Fax: (613) 520 5727 Email: briand(a)sce.carleton.ca URL: www.sce.carleton.ca/faculty/briand ******************************************************** Do not miss MoDELS /UML 2005: see http://www.modelsconference.org for the 8th International Conference on Model Driven Engineering Languages and Systems (formerly the UML series of conferences) 26-Nov-2004 19:47:23 -0400,2758;000000000000-00000000

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