Date: Tue, 30 Apr 1996 23:53:57 -0700 From: Vaughan Pratt <pratt@cs.Stanford.EDU> Boole.Stanford.EDU, in whose /pub directory one can find the Structures Directory, /pub/structdir, is also where you can find /pub/ABSTRACTS, listing the abstracts of some fifty papers written by members of my group over the past ten years, in chronological order. About a third of these are on Chu spaces. Their chronological order is very little help for people hoping to learn more about Chu spaces from them. I've therefore reorganized the abstracts of those papers in the form of a web-browsable guide, accessible as http://boole.stanford.edu/chuguide.html The abstracts are grouped into Introductory (2), Applications (10), Technical Notes (5), and Prehistory (5). Each abstract includes a link to the postscript file, which you can view and/or print if your web browser knows about gzipped files (.gz) and Postscript (.ps). I've only listed papers from Stanford, for related papers from elsewhere see the bibliographies of the introductory papers. Please let me know of any difficulties you encounter with this guide. Vaughan Pratt Date: Wed, 1 May 1996 09:27:58 -0300 (ADT) Subject: Re: question Date: Wed, 1 May 96 10:13 BST From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk> Fred Linton's comment re Cantor--Bernstein is of course correct, provided you assume classical logic. However, Barbosa's question makes sense in any topos with a natural number object, and it's well known that Cantor--Bernstein can fail in non-Boolean toposes. Is there always an isomorphism A* x A* =~ (A x 2)* in such a topos? I suspect the answer is yes, but it may take some ingenuity to construct it. Peter Johnstone Date: Wed, 1 May 1996 09:29:11 -0300 (ADT) Subject: Re: question Date: Wed, 1 May 1996 12:24:09 +0100 From: Tim Heap <timh@dcs.ed.ac.uk> categories writes: Luis> Date: Tue, 30 Apr 1996 19:07:19 +0000 From: Luis Soares Barbosa Luis> <lsb@di.uminho.pt> Luis> This is probably a trivial question, but I would appreciate any Luis> pointer to a solution: Luis> Let A be a denumerable set and A* be the set of finite sequences Luis> of A. It is easy to show (by defining two injections) that the Luis> set (A* x A*) is isomorphic (as a set) to (A x 2)* -- where 2 is Luis> a two element set and x the Cartesian product. The question is Luis> how to exhibit the isomorphism (actually the 2 obvious Luis> injections are not mutually inverse). There's almost certainly a constructive proof of the Cantor-Schroeder-Bernstein theorem in the circumstances that you describe, which would necessarily contain a construction of a bijection given two appropriate injections --- it wouldn't be hard to makes this as explicit as you like. If you really want to exhibit this isomorphism explicitly, this suggests to me that your looking for something like an algorithm: If this is not the case, then even a non-constructive proof would do if you're prepared to a allow primitive corresponding to the law of the excluded middle in the recipe for your isomorphism --- it's just not always obvious how to interpret this in any computational sense. Alternatively, you could simply introduce some mutant lexicographic-type ordering on each of the two sets (using the ubiquitous `diagonal' trick (that's not diagonalisation) if necessary, i.e. if A is infinite). Neither of these suggestions is perceptibly natural or canonical i'm afraid, but i rather suspect that there is no particularly distinguished candidate with respect to the level of structure that you describe. t!m Date: Wed, 1 May 1996 09:30:12 -0300 (ADT) Subject: (A x 2)* = A* x A* Date: Wed, 1 May 1996 07:49:59 -0400 From: Peter Freyd <pjf@saul.cis.upenn.edu> There's a good category question here. View (A x 2)* and A* x A* as functors on the category of non-empty sets and Show that they are naturally equivalent. It's not hard if you switch to the category of pointed sets (whereon each is naturally equivalent to A*). Date: Wed, 1 May 1996 14:00:49 -0300 (ADT) Subject: Re: question Date: Wed, 1 May 1996 09:07:43 -0400 (EDT) From: Andre Joyal <joyal.andre@uqam.ca> On Tue, 30 Apr 1996 Luis Soares Barbosa wrote
Let A be a denumerable set and A* be the set of finite sequences of A. It is easy to show (by defining two injections) that the set (A* x A*) is isomorphic (as a set) to (A x 2)* -- where 2 is a two element set and x the Cartesian product. The question is how to exhibit the isomorphism (actually the 2 obvious injections are not mutually inverse).
If we put F(A)=(A* x A*) and G(A)= (A x 2)* for any set A we obtain two functors F,G:Sets -->Sets. Are they naturally isomorphic? This is not the original question but a related one. Observe that both functors are analytic, which means that they have a power series expansion: F(A)= sum_n n A^n and G(A)= sum_n 2^n A^n The two power series are different and it follows that F is not isomorphic to G. Andre Joyal PS: For the theory of analytic functors one can read my paper '' Foncteurs Analytiques et Especes de Structures'' Combinatoire Enumerative Springer Lecture Notes 1234 (1985). Date: Wed, 1 May 1996 19:51:10 -0300 (ADT) Subject: Re: (A x 2)* = A* x A* Date: Wed, 1 May 1996 16:03:35 -0400 From: Peter Freyd <pjf@saul.cis.upenn.edu> Andre's analytic functors do it all. As he points out, they answer, negatively, the question I put (answering it, note, before he could have read my question). And they provide a much nicer proof than the one I had in mind for the category of strict maps between pointed sets. Well, yeah, I didn't notice that my "not hard" proof was only good for what the CS people call "strict maps." What it comes to is that if you first apply the functor \A. 1+A and then Andre's functors F , G you get isomorphic functors: ((1+A) x 2)* = (1+A)* x (1+A)* because in each case you get sum_n Nx(A^n). Date: Thu, 2 May 1996 23:32:52 -0300 (ADT) Subject: Barbosa's question Date: Thu, 2 May 96 10:42 BST From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk> Andre Joyal's neat argument not only shoots down Peter Freyd's conjectured extension of Barbosa's question, it also shoots down mine (i.e. does the isomorphism (A* x A*) =~ (A x 2)* hold for A an object of a non-Boolean topos with NNO?) Take E to be the object classifier (i.e. the functor category [Set_f,Set]) and A to be the generic object of this topos (i.e. the inclusion functor Set_f --> Set). Then (A* x A*) and (A x 2)* are simply the restrictions to Set_f of Andre's functors F and G, and these are already non-isomorphic (since both functors are finitary). Peter Johnstone Date: Thu, 2 May 1996 23:35:27 -0300 (ADT) Subject: Re: question Date: Thu, 2 May 1996 11:30:14 +0100 From: Ralph Loader <loader@oban.ox.ac.uk> cat-dist@mta.ca writes:
Date: Wed, 1 May 96 10:13 BST From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>
Fred Linton's comment re Cantor--Bernstein is of course correct, provided you assume classical logic. However, Barbosa's question makes sense in any topos with a natural number object, and it's well known that Cantor--Bernstein can fail in non-Boolean toposes. Is there always an isomorphism A* x A* =~ (A x 2)* in such a topos?
If we have a 1-1 correspondance between A and the set N of natural numbers, then this reduces to N* x N* =~ (N x 2)*, and constructive proofs of N* =~ N and N x N =~ N are easy, so presumably this will work in any topos with a NNO. On the other hand, A* x A* =~ (A x 2)* for all A implies excluded middle. An informal argument follows: Suppose A* x A* =~ (A x 2)* for all A. Let phi be a proposition. We show phi or not phi. Take a and b such that a=b iff phi; it suffices to show a=b or not a=b. Let A be the set {a,b}. Let f : (A x 2)* --> A* x A* and g : (A* x A*) --> (A x 2)* be inverse. We take 2 = {0,1} for convenience. First look at f applied to x in S = {<(x,0),(y,0),(z,0)> | x,y,z in A} The images of each of these is in the form (<c_1,...,c_m>,<d_1,...,d_n>). If either m or n depends on x in S, then f is non-constant, S is not a singleton, and so not a=b. Alternatively m, n don't depend on x in S (we're using excluded middle for equality on natural numbers). Let T = {(<x_1,...,x_m>,<y_1,...,y_n>) | x,y in A}. Consider the g(w) for w in T. If these are not all in the form (x,0),(y,0),(z,0), then we have two distinct members of T, so not a=b. Otherwise, g and f restrict to a bijection between S and T. If not m+n = 3, then we have A^i =~ A^j with not i=j and it follows that a=b. Otherwise, we are left with m+n=3. We now repeat with this with S = {<(x,i),(y,j),(z,k)>|x,y,z in A}, for each of the other seven possibilities of i,j,k in {0,1}. There are two possibilities: either eventually we obtain (a=b or not a=b), or we find for all 8 triples (i,j,k), some m and n such that m+n=3. But the latter case is impossible: as f and g are bijections, different triples (i,j,k) must give different pairs (m,n), a contradiction as there are eight triples, but only four pairs. [We've implicitly used the following fact: If n is a natural number, and psi(x) is decideable for all x in A^n, then psi(x) for all x in A^n, is also decidable] Ralph. P.S. The headers on this message probably won't survive the list distribution. My email address is loader@maths.ox.ac.uk, not loader@oban.ox.ac.uk. Date: Thu, 2 May 1996 23:36:46 -0300 (ADT) Subject: coherent toposes Date: Thu, 2 May 1996 15:03:07 +0200 (MET DST) From: Paul Johnson <johnson@math.muni.cz> In all the resources I have available, Deligne's Theorem (axiom) is proven to be a consequence of axiom of choice. (That is, in TTT, Johnstone's Topos Theory, Maclane/Moerdijk). Am I wrong to think it is actually a consequence of only prime ideal theorem??? The best possible result I can imagine is that ***constructively*** every coherent topos E admits a surjection sh(X) ---> E (ie epimorphism in the category of geometric morphisms) where X is a coherent locale. But I can't prove it. Paul. Date: Thu, 2 May 1996 23:38:08 -0300 (ADT) Subject: Re: (A x 2)* = A* x A* Date: Thu, 2 May 1996 10:59:58 -0400 (EDT) From: Andre Joyal <joyal.andre@uqam.ca> I would like to correct an error in my power series expansion of A* x A*. You might have already noticed it. The correct expansion is A* x A* = sum_n (n+1) x A^n Of course, the error does not affect the proof that A* x A* is not isomorphic to (2 x A)* as a functor from Sets to Sets. Peter is right about the isomorphism between the functors F(X+1) and G(X+1) where F(A)=A* x A* and G(A)=(2 x A)*. It might be worth to notice that if we substitute the category Sets, of sets and all maps, by the category Bi of sets and bijections then F and G become isomorphic as functors Bi -->Sets. In this case Cantor's argument applies since the category of functors Bi-->Sets is a Boolean topos (and since there are natural injections F>-->G and G>-->F in this category). However, if we substitute the category Sets by the category In of sets and injections then it can be shown F and G are not isomorphic as functors In -->Sets. Andre J. Date: Thu, 2 May 1996 23:39:47 -0300 (ADT) Subject: re: question Date: Thu, 2 May 1996 18:33:58 -0400 From: Todd Wilson <twilson@CS.Cornell.EDU> ----------------------------------------------------------------
Let A be a denumerable set and A* be the set of finite sequences of A. It is easy to show (by defining two injections) that the set (A* x A*) is isomorphic (as a set) to (A x 2)* -- where 2 is a two element set and x the Cartesian product. The question is how to exhibit the isomorphism (actually the 2 obvious injections are not mutually inverse).
Thanks. L. S. Barbosa
Fred Linton's comment re Cantor--Bernstein is of course correct, provided you assume classical logic. However, Barbosa's question makes sense in any topos with a natural number object, and it's well known that Cantor--Bernstein can fail in non-Boolean toposes. Is there always an isomorphism A* x A* =~ (A x 2)* in such a topos? I suspect the answer is yes, but it may take some ingenuity to construct it.
Peter Johnstone
Given total enumerations of A and B (in other words bijections N -> A and N -> B), there are standard ("canonical") constructions that produce total enumerations of A*, A x 2, A x B, and so on, using primitive recursive functions, which are thus available in any topos with natural numbers object. For example, the Cantor Pairing Function <x,y> = ((x+y)^2 + 3x + y)/2 is a bijection N x N -> N. (In fact, by the Fueter-Polya theorem, it is the unique such bijection up to swapping x and y that is given by a quadratic polynomial with real coefficients.) Similarly, the function mapping N* to N given by the dyadic coding <a0,...,an> |-> 2^a0 + 2^{a0+a1+1} + ... + 2^{a0+...+an + n} (the empty sequence is coded by 0) is a "primitive recursive" bijection in the extended sense where lists are a recursive data type. By appropriately composing these enumerations and their inverses, one gets canonical bijections between any two sets built from totally enumerated sets using x, *, and so on, by going from one of the sets back to N and then from N to the other set. --Todd Wilson Date: Fri, 10 May 1996 15:10:30 -0300 (ADT) Subject: Master Class - Utrecht Date: Fri, 10 May 1996 11:10:22 +0200 From: I. Moerdijk <moerdijk@math.ruu.nl> Dear colleagues, We have a one year preliminary program at Utrecht for which we accept a number of fully funded students. The courses this year relate to sheaves, knots, etc, so will interest some category theorists. I would be grateful if you could bring this to the attention of your students. Ieke Moerdijk. ---------------------------------- Master Class 1996-1997 Mathematical Research Institute The Mathematical Research Institute (MRI) in The Netherlands has been set up jointly by the mathematics departments of the universities of Groningen, Nijmegen, Twente and Utrecht. Its aim is to promote research, organise graduate courses and seminars, and stimulate international contacts and exchange. The MRI is one of the main mathematical institutes in The Netherlands. Master Class The MRI organises a Master Class (MC). Every year three parallel MC programmes are offered, each with its own topic. These are 1-year programmes of lectures, seminars and a test problem for bright undergraduate students and beginning graduate students of any nationality. The MC student follows one of these programmes. The language for all MC activities is English. Students who have successfully completed a Master Class programme can apply for admission to the Graduate Studies Programme (GSP) of the MRI. Studies in the GSP can take 4 years and aim at a doctoral thesis, based on original mathematical research. Admission to the GSP is subject to a severe selection procedure. Topics for the Master Class 1996-97 # Numerical Analysis and Supercomputing # Stochastics and Operations Research # Complex Geometry and Topology Each programme includes # two full days of lectures and seminars per week, # a test problem: individual work on a mathematical problem under the guidance of a tutor. The programme runs from September 1st until June 30th. There are two blocks of courses of about thirteen weeks each. Participants are expected to become actively involved in seminars, workshops, discussions. Time will also be reserved for preparing for the examinations and for work on the test problem. In 1996-97 the following courses will be offered: NUMERICAL ANALYSIS AND SUPERCOMPUTING # Seminar: Numerical solution of nonlinear problems (O. Axelsson) # Finite element and boundary element methods (B. Polman) # Splines and wavelets (C. Traas) # Numerical treatment of surfacewaves (P. Zandbergen) # Subspace methods for linear problems and eigenproblems (H. van der Vorst/G. Sleijpen) # Seminar: Preconditioning techniques (H. van der Vorst/G. Sleijpen) # Elliptic boundary value problems (J. Duistermaat) # either: Numerical programming (R. Bisseling) or: Supercomputers and numerical linear algebra (O. Axelsson) STOCHASTICS AND OPERATIONS RESEARCH # Algorithmic methods in mathematical operations research (U. Faigle, W. Kern) # Seminar: Stochastic models in applied operations research (W. Nawijn) # Stochastic processes (M. van Zuijlen, E. van Doorn) # Stochastic analysis (F. den Hollander, H. Maassen) # Non-parametric statistics (B. Levit) # Seminar: Quantum stochastics (R. Gill) # Time series analysis (H. Dehling, T. Mikosch) # Seminar: Spatial stochastics (H. Dehling, A. van Enter, J. Roerdink) COMPLEX GEOMETRY AND TOPOLOGY # Sheaves and algebraic topology (W. van der Kallen, I. Moerdijk) # Complex geometry (E. Looijenga) # Riemann-Hilbert correspondence on P^1 (F. Beukers, G. Heckman) # Introduction to singularities (D. Siersma, J. Steenbrink) # D-modules (J. Steenbrink, R. van Doorn) # Intersection homology (W. van der Kallen, I. Moerdijk) # Arrangements of hyperplanes (M. Hazewinkel, D. Siersma) # Seminar on hypergeometric functions (G. Heckman, E. Opdam, J. Stienstra) Examination and certificate Each participant is required to take examinations on the courses and to present a written account of his/her work on the test problem. When the examinations and the test problem are successfully passed a certificate will be awarded. Expenses and fellowships The fee for admission to the MC, including tuition fees, is Dfl. 5,000. Since the MC activities take place in different cities the participants should expect expenses up to Dfl. 1,500 per year for travelling in the Netherlands. Moreover, there will be expenses for visa and for travelling to and from the Netherlands. The other normal costs of living (including housing and insurance) can be estimated at Dfl. 1,200 per month. The MRI can only offer a very limited number of fellowships for participation in the MC. Foreign students are therefore requested to apply for scholarships or funding in their home country. The MRI also encourages participation by way of student exchange programmes. Students are requested to check if their home university is involved in exchange programmes with one of the MRI universities (Groningen, Nijmegen, Twente, Utrecht). The MRI can offer a few full fellowships, intended for students with no possibility of funding in their home country and a few partial fellowships, intended for students with limited funding. In some cases the admission fee will be waived. Housing The MC activities will be spread over the four participating universities. The MRI will help candidates to find suitable housing accommodation near one of the participating universities. Usually one day per week the student will travel to another university. Travelling can take up to two and a half hours each way. Application Application is open to all those who are now in the final year of their undergraduate studies in Mathematical Sciences or have started their graduate studies. Women are especially encouraged to apply. To apply one should complete (a copy of) the application form attached to this folder and send it together with the following documents to Jean Arthur: # Curriculum vitae. # Academic record: list of subjects/classes taken at the home university; subjects for degree examination; photocopy of diploma (if available). # At least one letter of recommendation from a member of the academic staff of the home university. # A summary of the applicant's financial circumstances (if a fellowship is being applied for). N.B.: incomplete applications will not be considered; clearly indicate the topic of your choice; there are no special application forms for fellowships or housing. If you require further information or have any questions, do not hesitate to contact: Ms. Jean Arthur (secretary MRI) University of Utrecht, Mathematical Research Institute P.O. Box 80.010 3508 TA Utrecht The Netherlands (tel: 31-30-2531472, fax: 31-30-2518394, e-mail: mri@math.ruu.nl). Information on WWW We plan to have the contents of the folder and the latest information about the Master Class 1996-1997 on World Wide Web at the following address: http://www.math.ruu.nl/mri Deadlines for application Applicants seeking admission to the MC in September 1996 are requested to contact the secretariat as soon as possible. Deadline for applications: March 1st, 1996. Candidates will hear at the beginning of May 1996, whether or not their application has been successful. For students not requiring an MRI-fellowship (or a visa to enter The Netherlands) the deadline for application is June 1st, 1996. Prof.dr. D. Siersma (director MRI) Dr. J. Stienstra (coordinator MC) Application form MASTER CLASS 1996-1997 Name Address Postal Code & City Country Telephone number E-Mail / Fax / Telex I am particularly interested in the class about: Numerical Analysis and Supercomputing Stochastics and Operations Research Complex Geometry and Topology In order to participate, I would need no support partial support full support Date: Fri, 10 May 1996 15:32:54 -0300 (ADT) Subject: Announcing Hypatia Electronic Library Date: Fri, 10 May 1996 18:36:06 +0100 (BST) From: Hypatia Electronic Library <hypatia@doc.ic.ac.uk> Dear Colleagues, This is to tell you about the new electronic library, Hypatia, which has been implemented by my colleagues at QMW. (Apologies for duplicate copies of this message.) HH HH ii HH HH tt HHHHHHH yy yy p ppp aaaa tttt ii aaaa HH HH yy yy pp pp aa aa tt ii aa aa HH HH yy yy pp pp aa aa tt tt ii aa aa HH HH yy ppppp aaa aa tttt ii aaa aa yy pp yyyy pp http://hypatia.dcs.qmw.ac.uk Hypatia is at Queen Mary and Westfield College in London's East End. Hypatia is a directory of research workers in computer science and mathematics, and a library of their papers. It contains material which has been published by placing it on a publicly accessible web or ftp site, and a certain amount of "public domain" information about authors (name, affiliation, email address and phone number, etc). It also assembles bibliographic information. Hypatia is not a web crawler, but a mirrored archive with a web interface. The archive is indexed by author and by research group, and is equipped with a search engine. Hypatia works by having a database of registered authors. Many of the readers of this message will already be on the database. You can find out if you are on our database simply by searching for yourself in Hypatia. If you are not, then you can register and ensure that your papers are lodged in Hypatia simply by filling in an electronic form, which you can obtain via the Hypatia home page. If you are already registered, then you can use the same form to correct any incorrect or out of date information we have about you. (Inevitably, and despite our best efforts, not all the information on Hypatia's database will be correct.) You can use the same form on behalf of your colleagues to help expand Hypatia's coverage. Hypatia would also like to encourage you to compile a BibTeX-format database of your own papers. We hope that this will help people to cite accurately the definitive versions of your work. On-line help for this is available. Hypatia has been implemented by Mark Dawson, who set up and ran the popular mirrored archive at theory,doc.ic.ac.uk. Mark has benefited from discussions with Paul Taylor. Hypatia is still under development and welcomes your comments, suggestions, requests for help, and corrections to <hypatia@dcs.qmw.ac.uk>. Edmund Robinson, Department of Computer Science, Queen Mary and Westfield College, University of London, London E1 4NS Date: Tue, 14 May 1996 16:06:35 -0300 (ADT) Subject: one day workshop Date: Tue, 14 May 1996 16:13:06 +0200 From: Giuseppe Rosolini <rosolini@athena.disi.unige.it> SEMINARIO DI TEORIA ED APPLICAZIONI DELLE CATEGORIE -- TAC We organize a one-day workshop on theory and applications of categories. It will take place on Wednesday, May 29, starting at 11.00, at DIMA/DISI in Genoa. Non local participants include Marcelo Fiore, Michael Makkai, Jiri Rosicky, Dana Scott. We shall circulate a detailed programme for the workshop a few days before. It will be distributed to the usual list of the TAC seminar, but NOT to the other lists this message is sent to (apologies if you are reading a copy of it). Please inform G. Rosolini if you want to receive further announcements. Information about reaching Genoa and accomodation is available on the net at the URL below, but please do not hesitate to contact any of the organizers if you may not link to it. The organizers Aurelio Carboni Marco Grandis Eugenio Moggi Giuseppe Rosolini --------------------- For more information, please look at http://www.disi.unige.it/seminars/tac/ or write to rosolini@disi.unige.it G.Rosolini, Dip.di Matematica via Dodecaneso 35 16146 Genova ITALY tel +39 (0)10 3536630 fax +39 (0)10 3536699 Date: Tue, 14 May 1996 16:17:30 -0300 (ADT) Subject: *-Autonomous categories ? Date: Mon, 13 May 1996 12:50:28 +0100 From: Justin Pearson <justin@dcs.rhbnc.ac.uk> Dear All, Given a *-autonomous categories C, we have an isomorphism of Hom sets: Hom(A,B) iso HOM(B^\perp , A^\perp) where B^\perp is the dual of B etc. The question is, if you there is another isomorphism of Hom sets Hom(A,B) iso HOM(B,A) Does this force the duality to become trivial, i.e. A^\perp iso A for all A? I suspect the answer is no, but something tells me it might be yes. But my intuition and attempts to resolve the matter have, so far, failed me. Regards Justin Pearson Computer Science Royal Holloway University of London Egham Surrey TW20 0EX U.K. Tel: +44(0)1784 443912 Email: justin@dcs.rhbnc.ac.uk Date: Wed, 15 May 1996 08:54:59 -0300 (ADT) Subject: Re: *-Autonomous categories ? Date: Wed, 15 May 96 09:55 BST From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk> The answer is no. Take a closed symmetric monoidal category C which happens to be self-dual (e.g. finite-dimensional vector spaces), and perform (the trivial case of) the Chu construction on it -- i.e. take C^op x C with the duality (A,B)^\perp = (B,A). Peter Johnstone Date: Wed, 15 May 1996 14:21:48 -0300 (ADT) Subject: Re: *-Autonomous categories ? Date: Wed, 15 May 1996 09:46:34 -0400 From: Michael Barr <barr@triples.math.mcgill.ca> There answer is no. One point to remember is that the category could even be discrete, that is the only arrows are identities. An easy example is this. Let G be a group (and assume not every element is of order dividing 2). If you want a symmetric example, assume G is commutative. Now make a category whose objects are the elements of G and arrows are only identities, so there is an arrow a --> b iff a = b and then there is only one. So Hom(a,b) = Hom(b,a). The monoidal structure is the group multiplication, a --o b = a\inv b (and b o-- a = b a\inv) and a* = a\inv. Here is a less trivial example. Take a CMC with finite products in which Hom(a,b) \iso Hom(b,a) (finite dimensional vector spaces, say) and form Chu(C,1) (1 is terminal). An object is a pair (a,a') where a and a' are arbitrary objects of C and Hom((a,a'),(b,b'))= Hom(a,b) x Hom(b',a') \iso Hom(b,a) x Hom(a',b') = Hom((b,b'),(a,a')), while (a,a')* = (a',a). Michael Barr Date: Fri, 17 May 1996 08:51:57 -0300 (ADT) Subject: BRICS positions Date: Wed, 15 May 1996 14:37:25 +0200 From: Uffe Henrik Engberg <engberg@brics.dk> Postdoctoral positions at BRICS There are several postdoctoral positions at BRICS for a period of one to two years starting next year, 1997. Applications by researchers are welcome in the areas of logic, semantics, algorithms and complexity theory. Applications for positions should preferably be sent by e-mail and include curriculum vitae and two or three names of referees for recommendations as well as the referees' regular mail addresses and, if possible, e-mail addresses (see below). BRICS, a Centre for Basic Research in Computer Science, is funded by the Danish National Research Foundation for the period 1994-1998. Its aim is to establish in Denmark important areas of basic research in the mathematical foundations of Computer Science, notably Algorithmics (including Complexity Theory) and Mathematical Logic. The Centre is to develop these areas as a joint effort between the theoretical-computer- science groups at Aarhus University and Aalborg University. The research plan is based on a committment to develop Algorithms and Complexity Theory, and Logic integrated with existing strong activities in Semantics of Computation, using a combination of long-term efforts and a number of short-term, intensive programmes, within carefully chosen scientific themes. Organizationally, BRICS is an autonomous centre with its own management, and yet with its activities strongly integrated in the existing infrastructure and student environments at the two universities. The scientific planning is the responsibility of the following committee: Glynn Winskel, Professor (Aarhus), Director Mogens Nielsen, Associate Professor (Aarhus), Codirector Erik Meineche Schmidt, Associate Professor (Aarhus), Codirector Uffe Engberg, (Aarhus), Project Manager Kim Guldstrand Larsen, Professor (Aalborg) Peter D.Mosses, Associate Professor (Aarhus) Michael Schwartzbach, Associate Professor (Aarhus) Arne Skou, Associate Professor (Aalborg) Sven Skyum, Associate Professor, Reader (Aarhus) Further information on BRICS can be accessed through World Wide Web (WWW) and anonymous FTP. To connect to the BRICS WWW entry, open the URL: http://www.brics.dk/ The BRICS WWW entry contains updated information about activities, courses and researchers as well as access to electronic copies of information material and reports of the BRICS Series (look under Publications). To access the information via anonymous FTP do the following: ftp ftp.brics.dk cd pub/BRICS get README Addresses: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK - 8000 Aarhus C Denmark. Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: BRICS@brics.dk How to apply for a position at BRICS ------------------------------------ Applications for positions should preferably be sent by e-mail and include ^^^^^^ - curriculum vitae and - two or three names of referees for recommendations with the referees' + regular mail addresses and, if possible, + e-mail addresses, as well as - an URL to your WWW home directory if available. The various parts of the application (application letter, CV, etc.) can be sent by e-mail as e.g. uuencoded PostScript, clear ASCII text or if it causes troubles, just as an URL in which case we will try to load the files. Date: Tue, 28 May 1996 12:12:08 -0300 (ADT) Subject: revised `From Horn clause to Makkai sketch resolution' Date: Mon, 27 May 1996 17:14:57 -0400 From: James Otto <otto@triples.math.mcgill.ca> Dear people, The May 27, 1996 revision of `From Horn clause to Makkai sketch resolution' is now at (linked to, indirectly linked to) ftp://triples.math.mcgill.ca/pub/otto/res ftp://triples.math.mcgill.ca/pub/otto/otto.html ftp://triples.math.mcgill.ca/ctrc.html 1. Axiom templates are gone. This simplifies the main result --- lifting (for l.f.p. logic programming) --- and eliminates a section. 2. The examples are greatly improved. Even and + are added and head consolidation is gone. 3. Some terminology is improved and more motivation is added. Regards, Jim Otto