February 2004 - Categories - categories.org.au

preprint: A van Kampen theorem for the homotopy double groupoid of a Hausdorff space
by Ronnie Brown 09 Mar '04

09 Mar '04

Preprint available http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/0… 04.01 : BROWN, R., KAMPS, K.H. & PORTER, T. A van Kampen theorem for the homotopy double groupoid of a Hausdorff space Summary: We show that the homotopy double groupoid of a Hausdorff space defined by the authors in a previous paper satisfies a version of the van Kampen theorem, and so is a suitable tool for non abelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. There is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. Ronnie Brown Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Centre for the Popularisation of Mathematics: http://www.cpm.informatics.bangor.ac.uk/ (reorganised site with new sculpture animations)

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graphics
by jim stasheff 29 Feb '04

29 Feb '04

Thanks to all the suggestions for `flow chart' graphics. I had first asked the alg top list and had no responses whereas here I've received 4 or 5 suggestions gastex, dia (on Linux) xy-pic xfig "The LaTeX Graphics Companion", Goossens, Rahtz, and Mittelbach use the xy-pic and PSTricks packages to draw flowcharts. if anyone has used more than one and cares to comment, I would appreciate that as for example Subject: Re: categories: graphics package Date: Sat, 21 Feb 2004 15:24:28 +0000 (GMT) From: Michael Abbott <michael(a)araneidae.co.uk> To: James Stasheff <stasheff(a)email.unc.edu> References: 1 > Anyone have a recommendation for a software package that makes it easy > to draw the equivalent of flow chart? > with boxes, triangles, circles as junctions?? This can be done, with the dint of much pain, with XyPic directly in LaTeX. The layout tends to be perfect, but learning how to use the tool for anything beyond the simplest diagrams is remarkably painful. On the other hand, xfig produces very nice results with manual layout, and is really quite easy to use (though figuring out how to drop the diagrams into your LaTeX document has its surprises). jim

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The invention of n-categories
by Topos8＠aol.com 28 Feb '04

28 Feb '04

Thanks to all who contributed information on history of the notion of a strict n-category. I'd like to summarize for the benefit of those interested what I have learned. The structure of CAT as a strict 2-category was of course implicit in Eilenberg and MacLane's original 1945 definition of categories, functors and natural transformations. Around 1958 Godement observed the significance of what we now call the exchange law relating horizontal and vertical compostion of natural transformations. In 1960 Charles Ehresmann codified our current notion of a strict 2-category and this notion also appeared in the thesis of his student Jean Benabou shortly thereafter. In 1963 Ehresmann published a defintion of n-tuple categories which include strict n-categories as a special case. However, Ehresmann did not isolate strict n-categories themselves as objects worthy of special attention. At the 1965 La Jolla conference Eilenberg and Kelly presented their paper on closed categories which defined strict n-categories recusively as categories enriched in strict n-1 categories. Around the same time the group surrounding Grothendieck at IHES had informally developed a non-recursive definition of strict n-category similar to what is used today. Finally, at the 1969 Bowdin conference, category theorists recognized that the recursive definition of Eilenberg-Kelly and the non recursive definition of the IHES group were equivalent. As far as I can tell the first significant technical development of the theory of strict n-categories (n > 2 ) appeared in a 1981 Cahiers paper by Ronnie Brown and PJ Higgins and in a 1984 Cahiers paper by Dominique Bourn. Carl Futia

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Who invented n-categories?
by Topos8＠aol.com 27 Feb '04

27 Feb '04

Can anyone offer a reference to the first published work which defined a notion of strict n-category equivalent to that used today? I know that Ehresmann invented n-tuple ( or n-fold ) categories which contain strict n-categories as special cases. If this is the first implicit defintion of strict n-category does anyone know who was the first to isolate our current notion of strict n-category as a particularly interesting special case of an n-tuple category? Carl Futia

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Categorified GNS
by Iain 26 Feb '04

26 Feb '04

I am looking for information on categorifications of C*-algebras. First, I know that there exists a theory for finite dimensional 2-Hilbert spaces due to J.Baez, but is there more recent work on infinite dimensional versions of 2Hilb? Second, has anyone categorified the GNS construction? -I

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On Baez on 'complex Euler characteristics'
by Stephen Schanuel 26 Feb '04

26 Feb '04

Dear colleagues, For those who read the most recent long discursion of John Baez, a few of the errors in the section on distributive categories merit correction: (1) J. B. suggests that Blass published what Lawvere had already worked out. In fact, Lawvere (partly to counteract some incorrect uses of infinite series in analyses of 'data types' in computer science) had worked out the algebra of the rig presented by one generator X and one relation X=1+X^2, roughly by the method in (3) below, and conjectured that this rig could be realized as the isomorphism classes in a distributive (even extensive) category, which conjecture Blass then proved (and a bit more) in "Seven Trees...". (2) The generalization of Blass's theorem to one generator ond one polynomial relation of the 'fixed-point' form X=p(X), where p is a polynomial with natural number coefficients and nonzero constant term is not, as J. B. seems to suggest, due to Fiore and Leinster; it was part of the prize-winning doctoral thesis of Robbie Gates, who (using a calculus of fractions) described explicitly the free distributive category on one object X together with an isomorphism from p(X) to X, proving that this category is extensive and that its rig of isomorphism classes satisfies no further relations, i.e. is the rig R presented by one generator and the one relation above. (3) If p is as in (2) and of degree at least 2, the algebra of the rig R is made by J. B. to seem mysterious. It is more easily understood in the way the X=2X+1 case was treated in my "Negative Sets..." paper; just show that the Euler and dimension homomorphisms, tensoring with Z and with 2 (the rig true/false) respectively, are jointly injective. In this case the dimension rig has only three elements, which explains why the Euler characteristic captures almost, but not quite, everything. Greetings to all, Steve Schanuel

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Eilenberg and Kelly invented n-categories?
by Topos8＠aol.com 26 Feb '04

26 Feb '04

I have received many helpful responses to my question asking who invented n-categories. However, a short while ago I was rereading the 1965 La Jolla paper by Eilenberg and Kelly (Closed categories, Proc. Conf.on Cat. Alg, Springer, 1965). On page 552 of the conference proceedings, chapter 4, section 2 of their paper, the authors write: [S]o one may define an n-category with morphisms of every type i [between 1 and n], a morphism of type i connecting two of type i - 1...". This is a very explicit reference to the notion of a strict n-category. The definition of an n-category given is that of a category enriched in n-1 categories. Eilenberg and Kelly also offer a reference to to Ehresmann's 1963 paper on structured categories but I can only find Ehresman's definition of n-tuple categories in his paper. Carl Futia

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TAC: new class file feature
by TAC 25 Feb '04

25 Feb '04

A new version of the style file for Theory and Applications of Categories has been posted. Notably it contains new possibilities for references associated with proclamations. The corresponding addition to the TAC author information page follows, but this is also a reminder that articles prepared for TAC using its style are processed much more quickly... Best wishes, Bob Rosebrugh The new section of TAC Author Information: A theorem (or any theorem-like environment), whether introduced by \theorem or by \begin{theorem} may have an optional argument that is material to be set in upright font and enclosed in square brackets. This is normally used to give credit, but may also be used for naming the theorem. For example, \theorem[Fermat's last theorem] For any integers ...\endtheorem will result in THEOREM [Fermat's last theorem] For any integers ... and \theorem[Fermat's last theorem \cite{w94}] For any integers ...\endtheorem might result in THEOREM [Fermat's last theorem [Wiles, 1994]] For any integers ... A few points should be noted. If the parenthetical material itself contains brackets, then it must be enclosed in braces. For example, \theorem[Fermat's last theorem {\cite[Theorem 4.7]{w94}}] For any integers ...\endtheorem will result in only one pair of brackets in the output, not two. Nothing (in particular no \label) may intervene between \theorem or \begin{theorem} and this parenthetical remark. A \label can follow this argument with no change in functionality. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Robert Rosebrugh, Managing Editor http://www.tac.mta.ca/tac Theory and Applications of Categories tac(a)mta.ca Dept of Math, Mt. Allison University 67 York Street Sackville, NB E4L 1E6 Canada +1-506-364-2530 fax: -364-2583 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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Equivalences and psuedo-equivalences (two items)
by Topos8＠aol.com 25 Feb '04

25 Feb '04

The answer to this question must be in the literature somewhere but I haven't been able to track it down or figure it out for myself. Let A and B be bicategories. By a functor from A to B I shall mean a morphism of underlying reflexive globular sets that preserves all operations and identities on the nose and satisfies the standard coherence conditions. A psuedo-functor is a morphism that preserves operations and identities only up to 1 cells that are equivalences in B ( a 1-cell f is an equivalence if there is a 1-cell g such that fg and gf are both defined and isomorphic to the respective identitity 1-cells). (These 1-cells must of course satisfy additional, standard coherence conditions.) An equivalence from A to B is a functor that is essentially surjective on 0-cells (every 0-cell of B is equivalent in B to a 0-cell in the image of F) and which induces an equivalence of 1-categories between A( x,y ) and B ( Fx, Fy ) for all 0-cells x, y in A. A psuedo-equivalence from A to B is a psuedo-functor that has these same properties. Question 1 : If F is a psuedo-equivalence from A to B does there exist an equivalence G from A to B? (references/counterexamples?) Question 2: Same as question 1 but with A and B strict 2-categories. Qestion 3: If the answer to question 1 is yes, then can G be chosen to be equivalent to F in the bicategory whose 0-cells are psuedo-functors from A to B? I shall be very grateful for any guidance the list can provide on these questions. Carl Futia ------------------------------------------------------ SECOND ITEM: Subject: Correction to my last query I have to apologize for a silly error in my last request for help. The problem lies with the definition of psuedo functor. I should have said that a psuedo functor preserves operations up to a 2-cell isomorphism, not a 1-cell equivalence. Sorry! Carl Futia

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Workshop on Quantum Programming Languages
by Peter Selinger 24 Feb '04

24 Feb '04

CALL FOR PAPERS 2nd International Workshop on Quantum Programming Languages (QPL2004) July 12-13, 2004, Turku, Finland Affiliated with LICS 2004 http://quasar.mathstat.uottawa.ca/~selinger/qpl2004/ * * * The goal of this workshop is to bring together researchers working on mathematical formalisms and programming languages for quantum computing. In the last few years, there has been a growing interest in logical tools, languages, and semantical methods for analyzing quantum computation. These foundational approaches complement the more mainstream research in quantum computation which emphasizes algorithms and complexity theory. Possible topics include the syntax and semantics of quantum programming languages, new paradigms for quantum programming, specification of quantum algorithms, higher-order quantum computation, quantum data types, reversible computation, axiomatic approaches to quantum computation, concurrent and distributed quantum computation, compilation of quantum programs, semantical methods in quantum information theory, and categorical models for quantum computation. The first workshop in this series was held June 15-16, 2003, in Ottawa, Canada. SUBMISSION PROCEDURES: The workshop will be a 1.5-day workshop, with some invited speakers and some contributed presentations. Those who wish to give a contributed presentation should submit a one-page abstract (or a full paper up to 12 pages, if available) by April 25 to selinger(a)mathstat.uottawa.ca (please put "workshop submission" in the subject line). Authors of accepted abstracts or papers are invited to submit a full version (subject to a page limit, details to be announced) for inclusion in the workshop proceedings (to be distributed at the workshop) by June 7. If there is a sufficient number of papers, we will also arrange for them to be published as a volume of ENTCS, probably in association with other LICS or ICALP associated workshops (details to be announced). REGISTRATION: Registration and local arrangements will be handled through the LICS 2004 main conference (http://www.dcs.ed.ac.uk/home/als/lics/lics04/) There will be a small fee for attending the workshop, which will cover lunch, coffee, and proceedings. IMPORTANT DATES/DEADLINES: Submission of abstracts: April 25, 2004 Notification of acceptance: May 20, 2004 Paper for inclusion in proceedings: June 7, 2004 Workshop: July 12-13, 2004 CONTACT INFORMATION: Organizer: Peter Selinger Department of Mathematics and Statistics University of Ottawa, Canada Email: selinger(a)mathstat.uottawa.ca

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