February 2000 - Categories - categories.org.au

review of `History of Topology'
by Ronnie Brown 06 Mar '00

06 Mar '00

I have put my review of `History of Topology', (ed I M James) (North/Holland Elsevier) for the London Math Soc Newsletter linked from my home page http://www.bangor.ac.uk/~mas010#History This review has some remarks on category theory, groupoids, etc. Comments welcome! Ronnie Brown

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Johnstone review of Clark/Davey
by Peter Freyd 01 Mar '00

01 Mar '00

I am chagrined to find myself as one of two category people in the bibliography of the work reviewed below and -- worse -- Peter says that they ascribed schizophrenic objects "not altogether appropriately" to me. He's right. CMP 1 663 208 (99:06) 18-02 (03G25 06D05 08C15 18A40) Clark, David M.(1-SUNYP); Davey, Brian A.(5-LTRB) Natural dualities for the working algebraist. (English. English summary) Cambridge Studies in Advanced Mathematics, 57. Cambridge University Press, Cambridge, 1998. xii+356 pp. $64.95. ISBN 0-521-45415-8 I began reading this book (conventionally enough) with the first two paragraphs of the Preface: "In 1936 Marshall Stone published a truly novel theorem ... What Stone discovered was a representation for all Boolean algebras which gave algebraists a usable understanding of their structure, using topological spaces to construct the representations ... In fact, Stone proved much more than a representation theorem ... In modern language, he proved that the category of Boolean algebras is dually equivalent to the category of Boolean spaces ...". Finding in these words what I presumed to be a conscious echo of the words: "This book is about a particular theorem -- the Stone representation theorem for Boolean algebras -- and some of the mathematical consequences which have developed from it ... Stone's key idea was the introduction of topology ... this was a really bold idea ... Moreover, [Stone's representation formed] one of the earliest nontrivial examples of an equivalence of categories ..." with which I had begun the Introduction to my own book Stone spaces, published in the same C.U.P. series in 1982 [Cambridge Univ. Press, Cambridge, 1982; MR 85f:54002], I naturally turned next to the Bibliography to check that my book was listed there. After all, though my book dealt with other subjects besides duality, it did have a whole chapter devoted to dualities of the type studied by Clark and Davey, and so it ought to be one of their standard references. My name does not appear in the Bibliography. Indeed, its only occurrence in the entire book is in the list of previous volumes in the series which appears opposite the title page. I mention this at the outset of this review, in order to give the reader a fair chance to make allowance for what he may construe as "sour grapes" on my part. For a related reason, I have departed from Mathematical Reviews tradition by writing the review in the first person; I cannot pretend to have achieved the standards of objectivity and dispassionateness that would be implied by a third-person review. However, it is not only the works of Johnstone which have been overlooked by Clark and Davey. I looked next in the Bibliography for John Isbell's ground-breaking 1972 paper on general functorial semantics [Amer. J. Math. 94 (1972), 535--596; MR 53 #580], and his later work characterizing concrete dualities in terms of commuting subtheories of a ruled theory: neither is present, and Isbell is represented only by a much less important 1980 paper on median algebra. Again, the 1982 paper in which Harold Simmons coined the term "schizophrenic object" [Topology Appl. 13 (1982), no. 2, 201--223; MR 83f:18006], for a set with two commuting algebraic structures, is not there, although Clark and Davey freely use this term in their text. Having discovered this, I began studying the Bibliography more systematically, and soon realized its salient feature: all works written by category-theorists, or making serious use of categorical ideas, are excluded from it -- with the twin exceptions of Saunders Mac Lane's classic Categories for the working mathematician [Springer, New York, 1971; MR 50 #7275] (which, after all, the authors could hardly have left out, given their indebtedness to his title), and of Peter Freyd's 1966 paper on algebra-valued functors [Colloq. Math. 14 (1966), 89--106; MR 33 #4116] (to which I shall return below). This, then, is a book on a categorical subject -- duality -- by authors who are not category theorists, and presumably intended for such a readership. However, the editors of Mathematical Reviews have asked a category-theorist to review it, so the review is written from that point of view. The subject of the book is the construction of dual equivalences, or more generally contravariant adjunctions, between algebraic categories $\scr A$ and categories $\scr X$ whose objects are compact topological spaces equipped with compatible algebraic and/or relational structure. Here one encounters the first restrictive feature of the authors' outlook: for them "algebraic" means "finitary algebraic" (an attitude already debunked by Marshall Stone in 1947), and so the idea of a compact topology as a kind of algebraic structure (indeed, the universal example of a structure which commutes with all finitary structures) cannot be expressed. Nor can the concomitant "unity of opposites" idea that $\scr A$ and $\scr X$ are two categories of the same kind -- they are doomed to remain forever separate, like the lovers on Keats's Grecian urn. Clark and Davey follow the common tradition in universal algebra that the underlying set of an algebra is not allowed to be empty. Having thus amputated the initial objects from many of their categories $\scr A$, they are forced to amputate the terminal objects from the dual categories $\scr X$, leading to unnecessary complications in the descriptions of these categories as quasivarieties. Since they have not amputated the terminal object from $\scr A$, they are forced (in appropriate cases) to allow the empty space as a member of $\scr X$. Thus, for them, a topological algebra does not necessarily have an underlying discrete algebra! As already mentioned, Clark and Davey make heavy use of schizophrenic objects. However, as far as most of the book is concerned, schizophrenic objects are simply a convenient ad hoc way of constructing a contravariant adjunction; the fact that every contravariant adjunction between categories of the type they consider is induced by mapping into a schizophrenic object is not mentioned until page 162, where it is ascribed (not altogether appropriately) to the paper of Freyd mentioned earlier, and stated in such a convoluted way as to be almost unrecognizable. Again, all the schizophrenic objects the authors consider are finite (and topologically discrete); thus, although they refer to Pontryagin duality in their Preface (and Pontryagin appears, along with Birkhoff and Stone, as one of the three mathematicians to whose memory the work is dedicated), they are able to describe only the special case of Pontryagin duality for abelian groups of some fixed finite exponent. All the above criticisms are, in a sense, trivial ones; but their cumulative effect is nontrivial. Category theory is often criticized for consisting entirely of trivialities; but, as Freyd long ago observed, its real function is to demonstrate that the trivial parts of mathematics are trivial for trivial reasons, and that is a valuable service which it performs for the mathematical community. Reading Clark and Davey's book, anyone new to the field would find it difficult or impossible to distinguish between the trivialities and the results with genuine content, since the former are so often presented in ad hoc ways that obscure the underlying pattern. This is not a book that I would recommend a graduate student to read. One might argue that the book's shortcomings are not of major importance in relation to its declared purpose. After all, it contains a wealth of detailed information on particular techniques for establishing duality theorems, and "working algebraists" (read: working universal-algebraists) will undoubtedly find it immensely useful to have all these techniques collected together in one place. I admit the force of that argument; but I also find it seriously worrying, for it carries the implication that universal-algebraists have given up the attempt to engage in dialogue with the rest of the mathematical community. As the authors remark at the end of their Preface, there is much that remains to be done in studying and classifying concrete dualities; but this book is not likely to inspire anyone outside the closed circle of universal-algebraists to take up the task. Reviewed by Peter Johnstone Copyright American Mathematical Society 2000

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Weak algebraic structures
by Tom Leinster 29 Feb '00

29 Feb '00

The following two papers on homotopy-algebraic structures - or "weakened" algebraic structures, if you prefer - are now available. The first one, "Up-to-Homotopy Monoids", is 8 pages long and is essentially a set of notes for the talk I gave at the Louvain-la-Neuve PSSL in October. It can serve as an introduction to the second one, "Homotopy Algebras for Operads" (100 pages, but don't let that scare you: it should be easy for category theorists). Abstracts are below. Tom Leinster * * * "Up-to-Homotopy Monoids" Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper `Homotopy Algebras for Operads', and does not assume any knowledge of operads. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/9912084 * * * "Homotopy Algebras for Operads" We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the morphisms in M have been marked out as `homotopy equivalences'. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any n-fold loop space is an n-fold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A_infinity-spaces, A_infinity-algebras and non-strict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on `change of base', e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we reflect on the advantages and disadvantages of our definition, and on how the definition really ought to be replaced by a more subtle infinity-categorical version. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/0002180

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CFP Category Theory 2000
by walters 28 Feb '00

28 Feb '00

SECOND CALL FOR PAPERS CATEGORY THEORY 2000 (CT 2000) July 16-22, 2000 Centro di Cultura Scientifica "Alessandro Volta" di Villa Olmo Como, Italy The next international summer conference in category theory will be held at Villa Olmo from Sunday 16th July to Saturday 22nd July 2000. We invite submissions in all areas of category theory and its applications, but particularly in the following areas: * algebraic topology and homological algebra, * categorical logic, * categories and computer science, * enriched category theory and 2-dimensional universal algebra, * galois theory and descent, * general category theory. * higher dimensional category theory and quantum algebra, * topos theory and synthetic differential geometry, The web page for the conference is http://www.disi.unige.it/conferences/ct2000/ Electronic submissions are preferred but authors may instead mail 4 copies of an extended abstract, in either case to arrive by 15th March, 2000. Email address for electronic submissions ct2000(a)disi.unige.it Mail address for hardcopy submissions RFC Walters (CT 2000) Dipartimento di Scienze CC., FF., MM., UniversitÃ degli studi dell'Insubria 22100 Como, Italy Important Dates 15 March, 2000: Papers due 15 May, 2000: Notification of acceptance or rejection of papers 16-22 July, 2000: Conference Papers should be submitted in the form of an extended abstract. Papers should begin with the title of the paper, each author's name, affiliation, and e-mail address, followed by a statement as to which of the areas of category theory mentioned above the paper belongs, a succinct statement of the problems and goals that are considered in the paper, the main results achieved, the significance of the work in the context of previous research, and a comparison to past research. The abstract should provide sufficient detail to allow the program committee to evaluate the validity, quality, and relevance of the contribution. The entire extended abstract should not exceed 10 pages. Authors will be notified of acceptance or rejection by 15 May, 2000. Program Committee S.L. Bloom (Stevens Institute, USA) J.M.E. Hyland (Cambridge, U.K.) G. Janalidze (Tbilisi, Georgia) G.M. Kelly (Sydney, Australia) A. Kock (Aarhus, Denmark) I. Moerdijk (Utrecht, The Netherlands) R. ParÃ© (Dalhousie, Canada) R.H. Street (Macquarie, Australia) Organizing Committee A. Carboni (Insubria) G. Rosolini (Genova) R.F.C. Walters (Insubria and Sydney) REGISTRATION Registration for the conference is now possible by consulting the web page http://www.disi.unige.it/conferences/ct2000/. The registration fee is Lit.250000, reduced to Lit.180000 for students. Also consider that the conference dinner will be on Wednesday, July 19, and the cost of a dinner ticket is Lit.70000. 28th February 2000

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preprint: About the globular homology of higher dimensional automata
by Philippe Gaucher 24 Feb '00

24 Feb '00

Bonjour, Here is a short note about the globular homology. Title : About the globular homology of higher dimensional automata Abstract : In this short note, we introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] disappear. Comments : 20 pages Url : http://www-irma.u-strasbg.fr/~gaucher/sglob.ps.gz Cordialement, Philippe Gaucher.

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1970s programmable calculators
by Paul Taylor 24 Feb '00

24 Feb '00

OK, I admit, this has nothing to do with category theory, but maybe one of my mates out there can help me ... Did any of the readers of "categories" own (or use) a programmable calculator in the 1970s? I'm looking for something that would have done about 100 to 1000 instructions per second, with conditional instructions, loops and indexed access to numbered memories. The reason s that I want to set a exam question about elementary ideas of complexity along the lines of "The XY99 calculator could execute this algorithm for a problem size up to nnnn - how big a problem could you solve today with your million-times faster computer?". Info about performance, price, the year it went on the market, etc please, by direct email to me. (OK, I'll summarise the responses I get in another message to "categories".) Paul

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Montreal Seminar announcement list
by Robert Seely 23 Feb '00

23 Feb '00

Due to the recent hard disk failure on triples, I have lost the current electronic mailing list for notices of seminars of the Montreal Category Theory group. The seminar schedule is (as always) available on the triples home page http://www.math.mcgill.ca/triples so anyone interested in what talks are being given at McGill may check there. I will reconstruct the local mailing list, but if anyone wants me to add their names to that list, and they are not presently in Montreal, please email me, and I will do so. This might be a good time to remind people that if emailing folks at McGill, it is always best to use the format user(a)math.mcgill.ca rather than any variant using "triples" in the email address. This allows us to move things around as necessary, when crises strike. It may also be a good time to remind folks that all urls (including ftp) that began "http://triples.math.mcgill.ca/" have been replaced by more generic "http://www.math.mcgill.ca/triples/" (for ftp, the standard url is "ftp://ftp.math.mcgill.ca/pub/"). If you haven't updated your links and bookmarks, please do so, since the old formats are no longer supported. If a McGill link fails to work, check to see if this is the reason. -= rags =- =================================== RAG Seely <rags(a)math.mcgill.ca> <http://www.math.mcgill.ca/rags> ===================================

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TOOLS USA 2000 Call for contributions
by TOOLS Conferences 23 Feb '00

23 Feb '00

Dear Colleague: This is the abbreviated call for contributions for TOOLS USA 2000. The full information is at http://www.tools-conferences.com/usa Please post or forward this information to any other colleague who think might be interested. With best regards, -- TOOLS Conference organization ************************************************************************* TOOLS USA 2000 "Software Serving Society" Santa Barbara, California July 30 - August 3, 2000 http://www.toolsconferences.com/usa CALL FOR CONTRIBUTIONS (deadline 10 March 2000) TOOLS is the major international conference series devoted to applications object technology, component technology and other advanced approaches to software development. TOOLS USA 2000 will be held in Santa Barbara, CA at the Fess Parker Double Tree Resort, one of the most beautiful resorts on the West Coast and will continue the commitment to excellence of earlier TOOLS conferences in Europe, Australia, Asia and the USA since 1989. The proceedings will be published world-wide by the IEEE Computer Society. PAPERS ------ TOOLS USA 2000 is now soliciting papers on all aspects of object and component technology. All submitted papers will be refereed and assessed for technical quality and usefulness to practitioners and applied researchers. TOOLS USA particularly welcomes papers that present general findings based upon industrial experience. Such papers will be judged by the quality of their contribution to industrial best-practice. TUTORIALS, WORKSHOPS AND PANELS ------------------------------- Tutorials, workshops, and panels form an important part of the TOOLS conferences. TOOLS USA 2000 is welcoming proposals for tutorials, workshops and panels on topics related to the theme of the conference. FOR MORE DETAILS AND REGISTRATION INFORMATION PLEASE VISIT THE CONFERENCE WEBSITE AT http://www.tools-conferences.com/usa

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triples down
by Michael Barr 20 Feb '00

20 Feb '00

The disk on triples died yesterday. If you send mail to barr(a)barrs.org or to barr(a)math.mcgill.ca, it should arrive normally. However, any mail sent between 9AM and about 5PM yesterday will have been lost. As will the 199 messages that were waiting for me to act on. Included was at least one referee's report and one journal submission that arrived while I was away and I had left there to deal with when I got back. Which I did Thursday night, but I hadn't done anything by Friday at 9. So if you sent me anything important like that recently, please resend it. We should have triples back in service by Monday or so, but whether we can recover the information on the disk is not clear. Michael

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The Category of all Smooth Manifolds a la Lawvere
by Ellis Cooper 19 Feb '00

19 Feb '00

Hello, In his paper "Qualitative distinctions between some toposes of generalized graphs" (Contemporary Mathematics Volume 92, 1989, pp. 261-299) Lawvere mentions a "powerful theorem" that "justifies bypassing the complicated considerations" usually associated with defining a smooth manifold (charts, atlases, etc.). This is in the context of something called "closed under splitting of idempotents" and the kind of idempotent he is talking about, I think, is what you get if you embed a manifold in a sufficiently high-dimensional space, wrap it inside and out with foam, call that foamy thing an open set, and then the idempotent is the projection of the foam back onto the embedded manifold. What I would like is a carefully written, fully spelled out statement and proof of his theorem. Please advise. In fact, I would be even more delighted by a more standard (motivation, definition, theorem, proof) version of his entire paper, but I suppose that is too much to ask. Please reply directly to me, at Ellis D. Cooper Senior Software Engineer Varian Semiconductor Equipment Associates, Inc. ellis.cooper(a)vsea.com

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