From rrosebru@mta.ca Tue Mar 2 12:13:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Mar 2004 12:13:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AyCU0-00016d-00 for categories-list@mta.ca; Tue, 02 Mar 2004 12:11:00 -0400 X-Authentication-Warning: mercury.comlab: jg owned process doing -bs Date: Sun, 29 Feb 2004 11:49:11 +0000 (GMT) From: Jeremy Gibbons To: Subject: categories: Re: graphics In-Reply-To: <200402250526.i1P5Q8gA007761@coraki.Stanford.EDU> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 1 On Tue, 24 Feb 2004, Vaughan Pratt wrote: > If you're a Picasso in such things then > starting off with xfig isn't so bad, but if you're more of a Mondrian you > might prefer to work in Latex's picture environment from the get-go. > ... > The only other widely supported language I know that's expressive enough to > write a complete macro library in 20 lines for capabilities like flowcharts, > digital or analog circuits, and lattice diagrams and that interoperates > smoothly with Latex is PostScript. I'm surprised no-one seems to have mentioned John Hobby's METAPOST language. This is a mostly declarative language, with the capability of solving linear equations. (So you can, for example, express that one item appears halfway between two others, and that relationship will be maintained as either endpoint moves.) It also integrates beautifully with LaTeX. If you're neither a Picasso nor a Mondriaan, but you like writing programs, I would have said this was the tool for you. Should come with any decent TeX distribution. Jeremy From rrosebru@mta.ca Tue Mar 2 12:13:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Mar 2004 12:13:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AyCVb-0001Dz-00 for categories-list@mta.ca; Tue, 02 Mar 2004 12:12:39 -0400 Message-Id: <200403010541.i215fXm23694@math-cl-n03.ucr.edu> Subject: categories: golden objects To: categories@mta.ca (categories) Date: Sun, 29 Feb 2004 21:41:33 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 2 Dear Categorists - I've attempted to improve my web version of "week202" based on Steve Schanuel's comments. Here's the next issue, which contains two different constructions by Robin Houston, both of which give a "golden object" - i.e. an object G in some rig category, equipped with an isomorphism between G and G^2 + 1. Best, jb ....................................................................... Also available at http://math.ucr.edu/home/baez/week203.html February 24, 2004 This Week's Finds in Mathematical Physics - Week 203 John Baez Last week I posed this puzzle: to find a "Golden Object". A couple days ago I got a wonderful solution from Robin Houston, a computer science grad student at the University of Manchester. So, I want to say a bit more about the golden number, then describe his solution, and then describe how he found it. Supposedly the Greeks thought the most beautiful rectangle was one such that when you chop a square off one end, you're left with a rectangle of the same shape. If your original rectangle was 1 unit across and G units long, after you chop a 1-by-1 square off the end you're left with a rectangle that's G-1 units across and 1 unit long: G ......................... . . . . . . . . . . . . 1 . . . 1 . . . . . . . 1 . G-1 . ......................... So, to make the proportions of the little rectangle the same as those of the big one, you want "1 is to G as G-1 is to 1" or in other words: 1/G = G - 1 or after a little algebra, G^2 = G + 1 so that G = (1 + sqrt(5))/2 = 1.618033988749894848204586834365... while 1/G = 0.618033988749894848204586834365... and G^2 = 2.618033988749894848204586834365... (At this point I usually tell my undergraduates that the pattern continues like this, with G^3 = 3.618... and so on - just to see if they'll believe anything I say.) These days, the number G is called the Golden Number, the Golden Ratio, or the Golden Section. It's often denoted by the Greek letter Phi, after the Greek sculptor Phidias. Phidias helped design the Parthenon - and supposedly packed it full of golden rectangles, to make it as beautiful as possible. The golden number is a great favorite among amateur mathematicians, because it has a flashy sort of charm. You can find it all over the place if you look hard enough - and if you look too hard, you'll find it even in places where it's not. It's the ratio of the diagonal to the side of a regular pentagon! If you like the number 5, you'll be glad to know that 5 + sqrt(5) G = sqrt[-------------] 5 - sqrt(5) If you don't, maybe you'd prefer this: G = exp(arcsinh(1/2)) My favorite formulas for the golden number are G = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ... and the continued fraction: 1 G = 1 + --------- 1 + 1 -------- 1 + 1 ------- 1 + 1 ------ 1 + 1 ---- 1 + 1 ---- . . . These follow from the equations G^2 = G + 1 and G = 1 + 1/G, respectively. If you chop off the continued fraction for G at any point, you'll see that G is also the limit of the ratios of successive Fibonacci numbers. See "week190" for a very different proof of this fact. However, don't be fooled! The charm of the golden number tends to attract kooks and the gullible - hence the term "fool's gold". You have to be careful about anything you read about this number. In particular, if you think ancient Greeks ran around in togas philosophizing about the "golden ratio" and calling it "Phi", you're wrong. This number was named Phi after Phidias only in 1914, in a book called _The Curves of Life_ by the artist Theodore Cook. And, it was Cook who first started calling 1.618... the golden ratio. Before him, 0.618... was called the golden ratio! Cook dubbed this number "phi", the lower-case baby brother of Phi. In fact, the whole "golden" terminology can only be traced back to 1826, when it showed up in a footnote to a book by one Martin Ohm, brother of Georg Ohm, the guy with the law about resistors. Before then, a lot of people called 1/G the "Divine Proportion". And the guy who started *that* was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid's Elements. In 1509, Pacioli published a 3-volume text entitled Divina Proportione, advertising the virtues of this number. Some people think da Vinci used the divine proportion in the composition of his paintings. If so, perhaps he got the idea from Pacioli. Maybe Pacioli is to blame for the modern fascination with the golden ratio; it seems hard to trace it back to Greece. These days you can buy books and magazines about "Elliot Wave Theory", a method for making money on the stock market using patterns related to the golden number. Or, if you're more spiritually inclined, you can go to workshops on "Sacred Geometry" featuring talks about the healing powers of the golden ratio. But Greek texts seem remarkably quiet about this number. The first recorded hint of it is Proposition 11 in Book II of Euclid's "Elements". It also shows up elsewhere in Euclid, especially Proposition 30 of Book VI, where the task is "to cut a given finite straight line in extreme and mean ratio", meaning a ratio A:B such that A:B::(A+B):A (i.e., "A is to B as A+B is to A") This is later used in Proposition 17 of Book XIII to construct the pentagonal face of a regular dodecahedron. Of course, Euclid wasn't the first to do all these things; he just wrote them up in a nice textbook. By now it's impossible to tell who discovered the golden ratio and just what the Greeks thought about it. For a sane and detailed look at the history of the golden ratio, try this: 1) J. J. O'Connor and E. F. Robertson, The Golden Ratio, http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html While I'm at it, I should point out that you that Theodore Cook's book introducing the notation "Phi" is still in print: 2) The Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art: with Special Reference to the Manuscripts of Leonardo da Vinci, Dover Publications, New York, 1979. If you want to see what Euclid said about the golden ratio, you can also pick up a cheap copy of the Elements from Dover - but it's probably quicker to go online. There are a number of good places to find Euclid's Elements online these days. Topologists know David Joyce as the inventor of the "quandle" - an algebraic structure that captures most of the information in a knot. Now he's writing a high-tech edition of Euclid, complete with Java applets: 3) David E. Joyce's edition of Euclid's Elements, http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Joyce is carrying on a noble tradition: back in 1847, Oliver Byrne did a wonderful edition of Euclid complete with lots of beautiful color pictures and even some pop-up models. You can see this online at the Digital Mathematics Archive: 4) Oliver Byrne's edition of Euclid's Elements, online at the Digital Mathematics Archive, http://www.sunsite.ubc.ca/DigitalMathArchive/ The most famous scholarly English translation of Euclid was done by Sir Thomas Heath in 1908. You can find it together with an edition in Greek and a nearly infinite supply of other classical texts at the Perseus Digital Library: 5) Thomas L. Heath's edition of Euclid's Elements, online at The Perseus Digital Library, http://www.perseus.tufts.edu/ But I'm digressing! My main point was that while G = (1 + sqrt(5))/2 is a neat number, it's a lot easier to find nuts raving about it on the net than to find truly interesting mathematics associated with it - or even interesting references to it in Greek mathematics! The cynic might conclude that the charm of this number is purely superficial. However, that would be premature. First of all, there's a certain sense in which G is "the most irrational number". To get the best rational approximations to a number you use its continued fraction expansion. For G, this converges as slowly as possible, since it's made of all 1's: 1 G = 1 + --------- 1 + 1 -------- 1 + 1 ------- 1 + 1 ------ 1 + 1 ---- 1 + 1 ---- . . . We can make this more precise. For any number x there's a constant c(x) that says how hard it is to approximate x by rational numbers, given by lim inf |x - p/q| = c(x)/q^2 q -> infinity where q ranges over integers, and p is an integer chosen to minimize |x - p/q|. This constant is as big as possible when x is the golden ratio! It'd be ironic if the famously "rational" Greeks, who according to legend even drowned the guy who proved sqrt(2) was irrational, chose the most irrational number as the proportions of their most beautiful rectangle! But, it wouldn't be a coincidence. Their obsession with ratios and proportions led them to ponder the situation where A:B::(A+B):A, and this proportion instantly implies that A and B are incommensurable, since if you assume A and B are integers and try to find their greatest common divisor using Euclid's algorithm, you get stuck in an infinite loop. Euclid even mentions this idea in Proposition 2 of Book X: If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable. He doesn't explicitly come out and apply it to what we now call the golden ratio - but how could he not have made the connection? For more info on the Greek use of continued fractions and the Euclidean algorithm, check out the chapter on "antihyphairetic ratio theory" in this book: 6) D. H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction, Oxford U. Press, Oxford, 1987. Anyway, it's actually important in physics that the golden number is so poorly approximated by rationals. This fact shows up in the Kolmogorov- Arnold-Moser theorem, or "KAM theorem", which deals with small perturbations of completely integrable Hamiltonian systems. Crudely speaking, these are classical mechanics problems that have as many conserved quantities as possible. These are the ones that tend to show up in textbooks, like the harmonic oscillator and the gravitational 2-body problem. The reason is that you can solve such problems if you can do a bunch of integrals - hence the term "completely integrable". The cool thing about a completely integrable system is that time evolution carries states of the system along paths that wrap around tori. Suppose it takes n numbers to describe the position of your system. Then it also takes n numbers to describe its momentum, so the space of states is 2n-dimensional. But if the system has n different conserved quantities - that's basically the maximum allowed - the space of states will be foliated by n-dimensional tori. Any state that starts on one of these tori will stay on it forever! It will march round and round, tracing out a kind of spiral path that may or may not ever get back to where it started. Things are pretty simple when n = 1, since a 1-dimensional torus is a circle, so the state *has* to loop around to where it started. For example, when you have a pendulum swinging back and forth, its position and momentum trace out a loop as time passes. When n is bigger, things get trickier. For example, when you have n pendulums swinging back and forth, their motion is periodic if the ratios of their frequencies are rational numbers. This is how it works for any completely integrable system. For any torus, there's an n-tuple of numbers describing the frequency with which paths on this torus wind around in each of the n directions. If the ratios of these frequencies are all rational, paths on this torus trace out periodic orbits. Otherwise, they don't! KAM theory says what happens when you perturb such a system a little. It won't usually be completely integrable anymore. Interestingly, the tori with rational frequency ratios tend to fall apart due to resonance effects. Instead of periodic orbits, we get chaotic motions instead. But the "irrational" tori are more stable. And, the "more irrational" the frequency ratios for a torus are, the bigger a perturbation it takes to disrupt it! Thus, the most stable tori tend to have frequency ratios involving the golden number. As we increase the perturbation, the last torus to die is called a "golden torus". You can actually *watch* tori breaking into chaotic dust if you check out the applet illustrating the "standard map" on this website: 7) Takashi Kanamaru and J. Michael T. Thompson, Introduction to Chaos and Nonlinear Dynamics, http://www.sekine-lab.ei.tuat.ac.jp/~kanamaru/Chaos/e/Standard/ The "standard map" is a certain dynamical system that's good for illustrating this effect. You won't actually see 2d tori, just 1d cross-sections of them - but it's pretty fun. For more details, try this: 8) M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley, New York, 1989. In short, the golden number is the best frequency ratio for avoiding resonance! Some audiophiles even say this means the best shaped room for listening to music is one with proportions 1:G:G^2. I leave it to you to find the flaw in this claim. For more dubious claims, check out the ad for expensive speaker cables at the end of this article. KAM theory is definitely cool, but we shouldn't rest content with this when skeptics ask if the golden number is all it's cracked up to be. I figure it's part of our job as mathematicians to keep on discovering mind-blowing facts about the golden number. A small part, but part: we shouldn't give up the field to amateurs! Penrose has done his share. His "Penrose tiles" take crucial advantage of the self-similarity embodied by the golden number to create nonperiodic tilings of the plane. This helped spawn a nice little industry, the study of "quasicrystals" with 5-fold symmetry. Here's a good introduction for mathematicians: 9) Andre Katz, A short introduction to quasicrystallography, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 496-537. By the way, this same book has some nice stuff on the role of the golden number in KAM theory and the theory of iterated maps from the circle to itself: 10) Predrag Cvitanovic, Circle maps: irrationally winding, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 631-658. 11) Jean-Christophe Yoccoz, Introduction to small divisors problems, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 659-679. Conway and Sloane are also pulling their weight. Starting from the relation between the golden ratio, the isosahedron, and the 4-dimensional big brother of the icosahedron (the "600-cell"), they've described how to construct the coolest lattices in 8 and 24 dimensions using "icosians" - which are certain quaternions built using the golden ratio. I discussed this circle of ideas in "week20", "week59" and "week155". But if you want some really scary formulas involving the golden ratio, Ramanujan is the one to go to. Check these out: 1 -------------- 1 + exp(-2pi) ------------- 1 + exp(-4pi) = exp(2pi/5) [sqrt(G sqrt(5)) - G] ------------ 1 + exp(-6pi) ----------- 1 + exp(-8pi) --------- . . . and 1 + exp(-2pi sqrt(5)) ------------------- 1 + exp(-4pi sqrt(5)) ----------------- 1 + exp(-6pi sqrt(5)) ------------------ 1 + exp(-8pi sqrt(5)) ------------------ . . . sqrt(5) = exp(2pi/5) [ ------------------------------------- - G] 1 + [5^{3/4} (G - 1)^{5/2} - 1]^{1/5} These are special cases of a monstrosity called the Rogers-Ramanujan continued fraction, which is a kind of "q-deformation" of the continued fraction for the golden ratio. For details, start here: 12) Eric W. Weisstein, Rogers-Ramanujan Continued Fraction, http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html The golden number also shows up in the theory of quantum groups. I talked about this in "week22" so I won't explain it again here. But, I can't resist mentioning that Freedman, Larsen and Wang have subsequently shown that a certain topological quantum field theory called Chern-Simons theory, built using the quantum group SU_q(2), can serve as a universal quantum computer when the parameter q is a fifth root of unity. And, this is exactly the case where the spin-1/2 representation of SU_q(2) has quantum dimension equal to the golden number! 13) Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation, available at quant-ph/0001108. But don't get the wrong idea: it's not that some magic feature of the golden number is required to build a universal quantum computer! It's just that the 5 seems to be the *smallest* number n such that SU_q(2) Chern-Simons theory is computationally universal when q is an nth root of 1. That's pretty much everything I know about the golden number. So now, what about this "Golden Object" puzzle? Basically, the problem was to find an object that acts like the golden number. The golden number has G = G^2 + 1, so we want to find a object G equipped with a nice isomorphism between G and G^2 + 1. If G is just a set, this means we want a nice one-to-one correspondence between pairs of elements of G, and elements of G together with one other It doesn't matter what that other thing is, so let's call it "@". (You may be wondering about the word "nice". The point is, the problem is too easy if we don't demand that the solution be nice in some way - some way that I don't feel like making precise.) Here's Robin Houston's answer: Define a "bit" to be either 0 or 1. Define a "golden tree" to be a (planar) binary tree with leaves labelled by 0, 1, or *, where every node has at most one bit-child. For example: /\ is a golden tree, but /\ is not. /\ 1 /\ * 0 * 0 1 Let G be the set of golden trees. We define an isomorphism f: G^2 -> G + {@} as follows. First we define f(X, Y) when both X and Y are golden trees with just one node, this node being labelled by a bit. We can identify such a tree with a bit, and doing this we set f(0, 0) = 0 f(0, 1) = 1 f(1, 0) = * f(1, 1) = @ In the remaining case, where the golden trees X and Y are not just bits, we set f(X, Y) = /\ X Y There are different ways to show this function f is a one-to-one correspondence, but the best way is to see how Houston came up with this answer! He didn't just pull it out of a hat; he tackled the problem systematically, and that's why his solution counts as "nice". It's easy to find a set S equipped with an isomorphism S = P(S) where P is some polynomial with natural number coefficients. You just use the fixed-point principle described in "week108". Namely, you start with the empty set, keep hitting it with P forever, and take a kind of limit. This is how I built the set of binary trees last week, as a solution of T = T^2 + 1. The problem is that the isomorphism we seek now: G^2 = G + 1 (1) is not of this form. So, what Houston does is to make a substitution: G = H + 2 Given this, we'd get (1) if we had H^2 + 4H + 4 = H + 3 (2) and we'd get (2) if we had H^2 + 4H + 1 = H (3) which is of the desired form. We can rewrite (3) as H = 1 + H^2 + 2H + H2 and in English this says "an element of H is either a *, or a pair consisting of two guys that are either bits or elements of H - but not both bits". So, a guy in H is a golden tree! But, if it has just one node, that node can only be labelled by a *, not a 0 or 1. This means there are precisely 2 golden trees not in H. So, G = H + 2 is the set of all golden trees, and our calculation above gives an isomorphism G^2 = G + 1. Voila! Note that to derive (3) from (1) we need to subtract, which in general is not allowed in this game. Here we are subtracting constants, and Houston says that's allowed by the "Garsia-Milne involution theorem". I don't know this theorem, so I'll make a note to myself to learn it. But luckily, we don't really need it here: we only need to derive (1) from (3), and that involves addition, so it's fine. Part of what makes Houston's solution "nice" is that it suggests a general method for turning polynomial equations into recursive definitions of the form S = P(S). Another nice thing is that his trick delivers a structure type G(X) that reduces to G when X = 1. To get this, first use the fixed-point method to construct a structure type H(X) with an isomorphism H(X) = (H(X) + X)^2 + 2H(X) Then, define G(X) = H(X) + X + 1 and note that this gives G(X)^2 = G(X) + X which reduces to G^2 = G + 1 when X = 1. As if this weren't enough, Houston also gave another solution to the puzzle. He showed that James Propp's proposed Golden Object, described last week, really is a Golden Object! Maybe Propp already knew this, but I sure didn't. The idea of the proof is pretty general. Suppose we've got a category that's a "2-rig" in the sense of "week191". And, suppose we've got an object X equipped with an isomorphism X = 1 + 2X (4) so that X acts like "-1". For example, following Schanuel and Propp, we can take the category of "sigma-polytopes" and let X be the open interval: then isomorphism (4) says (0,1) = (0,1/2) + {1/2} + (1/2,1) Or, following Houston, we can take the category of sets and let X be the set of finite bit-strings. Then (4) says "a finite bit-string is either the empty bit-string, or a bit followed by a finite bit-string". The relation between these two examples is puzzling to me - if anyone understands it, let me know! But anyway, either one works. Now let G be the object of "binary trees with X-labelled leaves": G = X + X^2 + 2X^3 + 5X^4 + 14X^5 + 42X^6 + ... where the coefficients are Catalan numbers. Let's show that G is a Golden Object. To do this, we'll use (4) and this isomorphism: G = G^2 + X (5) which says "a binary tree with X-labelled leaves is a pair of such trees, or a degenerate tree with just one X-labelled node". The formula for G involving Catalan numbers is really just the fixed-point solution to this! Here is Houston's fiendishly clever argument. Suppose Z is any type equipped with an isomorphism Z = Z' + X for some Z'. Then Z + X + 1 = Z' + 2X + 1 = Z' + X = Z This applies to Z = G^2, since G^2 = (X + G^2)^2 = (2X + 1 + G^2)^2 has a X term in it when you multiply it out, so it's of the form Z' + X. Therefore we have an isomorphism G^2 = G^2 + X + 1 But we also have an isomorphism G + 1 = G^2 + X + 1 by (5). Composing these, we get our isomorphism G^2 = G + 1. Golden! I'll stop here. Quote of the week: "As a high-end cable manufacturer, Cardas Audio strives to address every detail of cable and conductor construction, no matter how small. An elegant solution deals with quality, not quantity. Cable geometry problems are resolved in the cable's design, not after the fact with filters. George Cardas received U.S. Patent Number 4,628,151 for creating Golden Section Stranding Audio Cable. It is truly unique. George introduced the concept of Golden Section Stranding to high-end audio, but the Golden Ratio, 1.6180339887... : 1, is as old as nature itself. The Golden Ratio is the mathematical proportion nature uses to shape leaves and sea shells, insects and people, hurricanes and galaxies, and the heart of musical scales and chords. "Discovered" by the Greeks, but used by the Egyptians in the Great Pyramid centuries before, man has employed the Golden Ratio to create his most beautiful and naturally pleasing works of art and architecture." - Cardas Audio speaker cable advertisement ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html From rrosebru@mta.ca Fri Mar 5 10:20:06 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 05 Mar 2004 10:20:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzG77-0003rg-00 for categories-list@mta.ca; Fri, 05 Mar 2004 10:15:45 -0400 Message-ID: <4045ED10.4040502@elet.polimi.it> Date: Wed, 03 Mar 2004 15:34:56 +0100 From: Pietro Braione User-Agent: Mozilla Thunderbird 0.5 (Windows/20040207) X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Question on universal arrows Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Virus-Scanned: by amavisd-new-20030616-p7 at elet.polimi.it Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 3 Greetings to all the readers. I am studying category theory on my own and I did not succeed in finding an answer to the following question. Colimits can be defined in terms of universal arrows: the colimit of a functor F: S -> C is the universal arrow, in the category C^S, from F to the diagonal functor D: C -> C^S. Can we also define universal arrows in terms of colimits? If yes, how? Thank you very much Pietro Braione From rrosebru@mta.ca Fri Mar 5 10:20:06 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 05 Mar 2004 10:20:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzG7V-0003tg-00 for categories-list@mta.ca; Fri, 05 Mar 2004 10:16:09 -0400 Date: Wed, 3 Mar 2004 17:28:26 -0400 (AST) From: TAC To: categories@mta.ca Subject: categories: TAC Reprints: announcement Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 4 The Editors are in the process of preparing a number of TAC Reprints that will consist of a number of papers by Peter Freyd that were circulated but never published. The papers will be individually refereed and when ready this number will appear. In the meantime, as papers are approved they will be posted temporarily at ftp.math.mcgill.ca/pub/barr/abcat/ The first such paper has now been posted under the name homotopy.pdf. It is the one that shows that the homotopy category is not concrete. best wishes, Bob Rosebrugh =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Robert Rosebrugh, Managing Editor http://www.tac.mta.ca/tac Theory and Applications of Categories tac@mta.ca Dept of Math, Mt. Allison University 67 York Street Sackville, NB E4L 1E6 Canada +1-506-364-2530 fax: -364-2583 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= From rrosebru@mta.ca Sun Mar 7 10:45:05 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Mar 2004 10:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzzUR-00062i-00 for categories-list@mta.ca; Sun, 07 Mar 2004 10:42:51 -0400 From: David Yetter To: Subject: categories: Re: mystification and categorification Date: Fri, 5 Mar 2004 10:55:26 -0600 User-Agent: KMail/1.5.4 References: <002a01c401ab$cd50b370$1767eb44@grassmann> In-Reply-To: <002a01c401ab$cd50b370$1767eb44@grassmann> MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Content-Disposition: inline Message-Id: <200403051055.26794.dyetter@math.ksu.edu> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 5 Categorification is a bit like quantization: it isn't a construction so much as a desideratum for a relationship between one thing and another (in the case of categorification an (n+1)-categorical structure and an n-categorical structure; in the case of quantization a quantum mechanical system and a classical mechanical system). Categorification wants to find a higher-dimensional categorical structure corresponding to a lower-dimensional one by weakening equations to natural isomorphisms and imposing new, sensible, coherence conditions. In general, for the original purpose for which it was proposed--constructions of TQFT's and models of quantum gravity--one wants the highest categorical level to have a linear structure (hence Baez wanting tensor product and a sum it distributes over, rather than cartesian product and coproduct). Specific lower-dimensional categories with structure are 'categorified' by finding a higher-dimensional category with the new structure which 'lies over' the lower dimensional one in the way an additive monoidal category lies over its Grothendieck rig. For instance any (k-linear) monoidal category with monoid of isomorphism classes M is a categorification of M, and more generally (k-linear) monoidal categories are a categorification of monoids. A simple example shows why it is not a construction: commutative monoids (as rather special categories with one object) admit two different categorifications: symmetric monoidal categories and braided monoidal categories (each regarded as a kind of bicategory with one object). There is a good argument for regarding braided monoidal categories as the 'correct' categorification: the Eckmann-Hilton theorem ('a group in GROUPS is an abelian group' or, really as the proof shows, 'a monoid in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category in MONCAT is a braided monoidal category. From rrosebru@mta.ca Sun Mar 7 10:45:05 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Mar 2004 10:45:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzzSU-0005xk-00 for categories-list@mta.ca; Sun, 07 Mar 2004 10:40:50 -0400 Message-ID: <4048AFE9.903@math.tulane.edu> Date: Fri, 05 Mar 2004 10:50:49 -0600 From: Michael Mislove Reply-To: mwm@linus.math.tulane.edu Organization: Tulane University User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.6b) Gecko/20031205 Thunderbird/0.4 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: MFPS 20 Second Announcement and Call for Participation Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 6 Apologies to those who receive multiple copies of this announcement Dear Colleagues, This is the Second Announcement of the Twentieth Workshop on the Mathematical Foundations of Programming Semantics and a Call for Participation at the meeting. MFPS 20 will be held on the campus of Carnegie Mellon University from May 23 through May 26 of this year. The meeting will be co-located with the Annual Meeting of the Association of Symbolic Logic, which will take place at CMU from May 19 through midday, May 23. MFPS 20 will feature invited lectures by: o Christel Baier (Bonn) o Radha Jagadeesan (DePaul) o Pat Lincoln (SRI) o Luke Ong (Oxford) o Dana Scott (CMU) o Alex Simpson (Edinbrgh) Each invited talk will have an accompanying special session focusing on the topic of the plenary talk, and featuring talks by a number of leading researchers in the area. Noting that this is the 20th anniversary of MFPS, we are planning a panel discussion for one evening of the meeting. Details about this will be forthcoming later. Slots for contributed talks are available for those who wish to contribute a talk These slots will be allocated on a first come, first serve basis. If you are interested in giving a talk at the meeting, send an email to mfps@math.tulane.edu with the title and abstract. Also indicate if the talk would be appropriate for one of the special sessions. The deadline for submitting a title and abstract is April 1, 2004. We anticipate support from the US Office of Naval Research, and a portion of that support is reserved to help defray the expenses of women, minorities and graduate students to attend the meeting. If you need some support to attend the meeting and are a member of one of these groups, please send an email to mfps@math.tulane.edu describing your needs. Because these funds are limited, we are unable to underwrite the full cost of participation at the meeting, so we expect requests that address a portion of the costs. To access more details about the meeting, please point your browser at http://www.math.tulane.edu/~mfps/mfps20.htm where you also will find a link to the Registration Page for the conference. The latter contains information about hotel accommodations in addition to a registration form that can be submitted online. MFPS 20 is collaborating with ASL to encourage participants to attend both meetings. In particular, participants at MFPS 20 will have access to the ASL meetings on Saturday, May 22. A link the to the ASL meeting information is accessible on the MFPS Home Page. MFPS is organized by Achim Jung (Birmingham), Stephen Brookes (CMU), Catherine Meadows (NRL), Michael Mislove (Tulane) and Prakash Panangaden (McGill). Best regards, Mike Mislove =============================================== Professor Michael Mislove Phone: +1 504 862-3441 Department of Mathematics FAX: +1 504 865-5063 Tulane University URL: http://www.math.tulane.edu/~mwm New Orleans, LA 70118 USA =============================================== From rrosebru@mta.ca Sun Mar 7 10:47:27 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Mar 2004 10:47:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1AzzXX-0006BI-00 for categories-list@mta.ca; Sun, 07 Mar 2004 10:46:03 -0400 Message-Id: <200403060649.i266nuaG014947@coraki.Stanford.EDU> X-Mailer: exmh version 2.6.3 04/04/2003 with nmh-1.0.4 To: categories@mta.ca Subject: categories: Re: mystification and categorification In-Reply-To: Message from "Stephen Schanuel" of "Thu, 04 Mar 2004 00:44:46 EST." <002a01c401ab$cd50b370$1767eb44@grassmann> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Date: Fri, 05 Mar 2004 22:49:56 -0800 From: Vaughan Pratt Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 7 >While I'm airing my confusions, can anyone tell me what >'categorification' means? I don't know any such process; the simplest >exanple, 'categorifying' natural numbers to get finite sets, seems to me >rather 'remembering the finite sets and maps which gave rise to natural >numbers by the abstraction of passing to isomorphism classes'. A fair question. I attended John's Coimbra lectures on this stuff in 1999 but a lot of it leaked out afterwards. If I had to guess I'd say he was categorifying the free monoid on one generator to make it a monoidal category, but then how did the monoid end up as coproduct and the generator as the final object? One suspects some free association there---John, how *do* you make that connection? With regard to categorification in general, sets seem to play a central role in at least one development of category theory. The homobjects of "ordinary" categories are homsets. (In that sense I guess "ordinary" must entail "locally small.") 2-categories are what you get if instead you let them be homcats, suitably elaborated. Going in the other direction, if you take homsets to be vacuous, not in the sense that they are empty but rather that they are all the same, then you get sets. One more step in that direction makes everything look the same, which may have something to do with the Maharishi Yogi hiring category theorists for the math dept. of his university in Fairfield, Iowa. (When I spoke last with the MY's "Minister of World Health," an MD who like Ross Street was a classmate of mine but eight years earlier starting in 1957, the entire conversation seemed to be largely a skirting of the minefield of the sameness of everything, which may subconsciously have been behind my obscure reply to Peter Freyd's posting a while back about unique existence going back to Descartes, where I tried to one-up him by claiming it went *much* further back.) Categorification isn't the only way to get to 2-categories, which can be understood instead in terms of the interchange law as a two-dimensional associativity principle. However John has got a lot of mileage out of the categorification approach, which one can't begrudge in an era where mileage and minutes are as integral to a balanced life as one's checkbook. (Q: How many minutes in a month? A: Depends on your plan.) >Since in the category of sets, any nasty old infinite set satisfies >the golden equation and many others, I have taken the liberty of >interpreting 'nice' to mean at least 'satisfying no unexpected >equations'. Quite right. I would add to this "and satisfying the expected equations." The "nasty sets" of which Steve speaks fail to satisy such expected equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra, for heaven's sake. Why on earth forget that structure prior to taking the second exponentiation? Set theorists seem to think that they can simply forget structure without paying for it, but in the real world it costs kT/2 joules per element of X to forget that structure. If set theorists aren't willing to pay real-world prices in their modeling, why should we taxpayers pay them real-world salaries? Large cardinals are a figment of their overactive imaginations, and the solution to consistency concerns is not to go there.) Vaughan Pratt From rrosebru@mta.ca Fri Mar 5 10:21:43 2004 -0400 Message-ID: <002a01c401ab$cd50b370$1767eb44@grassmann> From: "Stephen Schanuel" To: Subject: mystification and categorification Date: Thu, 4 Mar 2004 00:44:46 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Status: RO X-Status: X-Keywords: X-UID: 9 I was unable to understand John Baez' golden object problem, nor his description of the solutions. He refuses to tell us what 'nice' means, but let me at least propose that to be 'tolerable' a solution must be an object in a category, and John doesn't tell us what category is involved in either of the solutions; at least I couldn't find a specification of the objects, nor the maps, so I found the descriptions 'intolerable', in the technical sense defined above. He is very generous, allowing one to use a category with both plus and times as extra monoidal structures. (Does anyone know an example of interest in which the plus is not coproduct?) This freedom is unnecessary; a little algebra plus Robbie Gates' theorem provides a solution G to G^2=G+1 which satisfies no additional equations, in an extensive category (with coproduct as plus, cartesian product as times). Briefly, here it is. A primitive fifth root of unity z is a root of the polynomial 1+X+X^2+X^3+X^4, hence satisfies 1+z+z^2+z^3+z^4+z=z, which is of the 'fixed point' form p(z)=z with p in N[X] and p(0) not 0. Gates' theorem then says that the free distributive category containing an object Z and an isomorphism from p(Z) to Z is extensive, and its Burnside rig B (of isomorphism classes of objects) is, as one would hope, N[X]/(p(X)=X); that is, Z satisfies no unexpected equations. Since the degree of p is greater than 1, an easy general theorem tells us (from the joint injectivity of the Euler and dimension homomorphisms) that two polynomials agree at the object Z if and only if either they are the same polynomial or both are non-constant and they agree at the number z.Now the 'algebra': the golden number is 1+z+z^4. So G=1+Z+Z^4 satisfies G^2=G+1, as desired. It satisfies no unexpected equations, because the relation X^2=X+1 reduces any polynomial in N[X] to a linear polynomial, and these reduced forms have distinct Euler characteristics, i.e. differ at z. Thus the homomorphism from N[X]/(X^2=X+1) to B (sending X to G) is injective, and that's all I wanted. Since in the category of sets, any nasty old infinite set satisfies the golden equation and many others, I have taken the liberty of interpreting 'nice' to mean at least 'satisfying no unexpected equations'. One could ask for more; the construction above has produced a distributive, but not extensive, category whose Burnside rig is N[X]/(X^2=X+1), the full subcategory with objects polynomials in G. (If it were extensive, it would be closed under taking summands, but every object in the larger category is a summand of G.) I don't know whether there is an extensive category with N[X]/(X^2=X+1) as its full Burnside rig; perhaps one or both of the examples John mentioned would do, if I knew what they were. While I'm airing my confusions, can anyone tell me what 'categorification' means? I don't know any such process; the simplest exanple, 'categorifying' natural numbers to get finite sets, seems to me rather 'remembering the finite sets and maps which gave rise to natural numbers by the abstraction of passing to isomorphism classes'. Finally, a note to John: While you're trying to give your audience some feeling for the virtues of n-categories, couldn't you give them a little help with n=1, by being a little more precise about objects and maps? Greetings to all, and thanks for your patience while I got this stuff off my chest, Steve Schanuel From rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Rcb-0007a8-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:45:09 -0400 Subject: categories: Re: mystification and categorification From: Tom Leinster To: categories@mta.ca In-Reply-To: <002a01c401ab$cd50b370$1767eb44@grassmann> References: <002a01c401ab$cd50b370$1767eb44@grassmann> Content-Type: text/plain Organization: Message-Id: <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk> Mime-Version: 1.0 X-Mailer: Ximian Evolution 1.2.2 (1.2.2-5) Date: 07 Mar 2004 19:43:05 +0000 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 Steve Schanuel wrote: > a category with both plus and times as extra monoidal structures. > (Does anyone know an example of interest in which the plus is not > coproduct?) Here are two examples that I've come across previously of rig categories in which the plus is not coproduct: (i) the category of finite sets and bijections, with + and x inherited from the category of sets; (ii) discrete rig categories, which are of course the same thing as rigs. > This freedom is unnecessary; a little algebra plus Robbie > Gates' theorem provides a solution G to G^2=G+1 which satisfies no > additional equations, in an extensive category (with coproduct as plus, > cartesian product as times). If you *do* allow yourself the freedom to use any rig category then an even simpler solution exists, also satisfying no additional equations: just take the rig freely generated by an element G satisfying G^2 = G + 1 and regard it as a discrete rig category. > Since in the category of sets, any nasty old infinite set satisfies > the golden equation and many others, I have taken the liberty of > interpreting 'nice' to mean at least 'satisfying no unexpected > equations'. I'd interpret "nice" differently. (Apart from anything else, the trivial example in my previous paragraph would otherwise make the golden object problem uninteresting.) "Nice" as I understand it is not a precise term - at least, I don't know how to make it precise. Maybe I can explain my interpretation by analogy with the equation T = 1 + T^2. A nice solution T would be the set of finite, binary, planar trees together with the usual isomorphism T -~-> 1 + T^2; a nasty solution would be a random infinite set T with a random isomorphism to 1 + T^2. (Both these solutions are in the rig category Set with its standard + and x.) I regard the first solution as nice because I can see some combinatorial content to it (and maybe, at the back of my mind, because it has a constructive feel), and the second as nasty because I can't. I'm not certain what I think of the solution given by the set of not-necessarily-finite binary planar trees (nice?), or by the set of binary planar trees of cardinality at most aleph_5 (probably nasty). Maybe the finding of a "nice" solution is similar in spirit to the finding of a "concrete interpretation" of a combinatorial identity. As an extremely simple example, consider the identity saying that each entry in Pascal's triangle is the sum of the two above it, (n+1 choose r) = (n choose r-1) + (n choose r). This is a doddle to prove, but all the same you'd be missing something if you didn't know the standard "concrete interpretation": choosing r objects out of n+1 objects amounts to EITHER choosing the first one and then choosing r-1 of the remaining n OR ... . Even if the challenge of finding a "nice solution" or "concrete interpretation" isn't made precise, I think there is a shared sense of what would count as an answer, and finding an answer is in general not straightforward. Best wishes, Tom From rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Rbf-0007VD-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:44:11 -0400 Subject: categories: jobs galore in Glasgow From: Tom Leinster To: categories@mta.ca Content-Type: text/plain Organization: Message-Id: <1078680962.20775.76.camel@tl-linux.maths.gla.ac.uk> Mime-Version: 1.0 X-Mailer: Ximian Evolution 1.2.2 (1.2.2-5) Date: 07 Mar 2004 17:36:02 +0000 Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 11 The University of Glasgow is hiring four Lecturers in Mathematics. Some of these appointments may be at the Senior Lecturer level. Details are at http://www.jobs.ac.uk/jobfiles/PK506.html As I only started here myself in October, maybe I can say some things that will be useful to people thinking of applying. I've found the department a very pleasant place to be. As well as Richard Steiner and me in category theory, there are a lot of algebraists (one of the largest algebra groups in the UK), a good bunch of topologists and geometers, and (best of all) a lot of interaction between the different groups. For example, last term there was a study group/"working seminar" on model categories run by the algebraic topologists. There are plenty of people around with whom a category theorist can have a meaningful mathematical conversation, and I've yet to meet anyone here who makes me feel like an alien (as sometimes happens). So despite there not being a large category theory group as such (yet), I've felt no sense of intellectual isolation. And as you can see from the rate we're hiring, there's a lot of fresh blood coming in. Glasgow is the fourth-oldest university in the UK, after St Andrew's and two places whose names I forget. Glasgow was 1990 European city of culture, 1999 UK city of architecture and design, and 2003 European capital of sport. I am told that we also have the tallest cinema in Europe. If you want any further information then feel free to get in touch. Tom From rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0RfN-0000Ce-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:48:01 -0400 Message-Id: <200403072050.i27KoTB12570@math-cl-n03.ucr.edu> Subject: categories: golden objects To: categories@mta.ca (categories) Date: Sun, 7 Mar 2004 12:50:29 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 12 Dear Categorists - Sorry to take a while to respond. People at UCR have been unable to receive posts on the category theory mailing list, due to problems with our internet connection. I'd asked for some nice examples of an object G in a rig category equipped with an isomorphism from G to G^2 + 1. Steve Schanuel replied: >I was unable to understand John Baez' golden object problem, nor his >description of the solutions. He refuses to tell us what 'nice' means, [...] The problem was deliberately open-ended, but you seem to have understood it perfectly, since you've given a nice solution, including a precise specification of what you consider "nice". Let me repeat the two solutions given by Robin Houston: 1) The first solution works in any rig category having an object H equipped with an isomorphism to H^2 + 4H + 1. The solution is to take G = H + 2. I described how Houston uses the isomorphism H -> H^2 + 4H + 1 to construct an isomorphism G^2 -> G + 1. What's nice about this is that it reduces a problem that's not obviously of fixed-point form to one that is. 2) Houston's second solution works in any monoidal cocomplete category, tensor product distributing over colimits, that contains an object X equipped with an isomorphism to 2X + 1. The solution is to let G be the object of "binary planar rooted trees with X-labelled leaves", i.e. G = X + X^2 + 2X^3 + 5X^4 + 14X^5 + 42X^6 + ... where the coefficients are Catalan numbers. He uses the obvious isomorphism G -> G^2 + X to construct an isomorphism G^2 -> G + 1. What's nice about this is that it shows Propp's originally proposed golden object really is one: just take the category of sigma-polytopes with its cartesian product, and let X be the open interval! And, it makes precise the sense in which the alternating sum of Catalan numbers equals the golden ratio. Steve writes: >I don't know whether there is an extensive category with N[X]/(X^2=X+1) >as its full Burnside rig; perhaps one or both of the examples John >mentioned would do, if I knew what they were. I think example 1) does the job if we take the free distributive category on an object H equipped with an isomorphism to H^2 + 4H + 1. Right? Steve also writes: >He is very generous, allowing one to use a category with both plus >and times as extra monoidal structures. (Does anyone know an example >of interest in which the plus is not coproduct?) This freedom is >unnecessary [...] It's unnecessary, but handy: I think there's also an golden object in the rig category of reps of quantum SU(2) at a suitable value of q. Here the tensor product is not cartesian. In the lingo of quantum group theory, this object has "quantum dimension" equal to the golden number. It's interesting how such nonintegral but algebraic "dimensions" show up naturally in quantum group theory, just as nonintegral but algebraic "cardinalities" show up in the theory of distributive categories. I don't know any golden objects in rig categories where the plus is not coproduct, and I agree that such rig categories arise less often than those where times is not product. But, if you use the obvious way of making the groupoid of finite sets into a rig category, + isn't coproduct, nor is x product. > While I'm airing my confusions, can anyone tell me what > 'categorification' means? I don't know any such process; the simplest > exanple, 'categorifying' natural numbers to get finite sets, seems to me > rather 'remembering the finite sets and maps which gave rise to natural > numbers by the abstraction of passing to isomorphism classes'. You're right: categorification is not a systematic process! I explained this idea back in "week121", and also in my paper "Categorification", at http://www.arXiv.org/abs/math.QA/9802029. Here's what I said. If one studies categorification one soon discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of facts we learned in high school! There is a good reason for this. All along, we have been unwittingly `decategorifying' mathematics by pretending that categories are just sets. We `decategorify' a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects. To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and `count' it, setting up an isomorphism between it and some set of `numbers', which were nonsense words like `one, two, three, ...' specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented. According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification. While the historical reality is far more complicated, categorification really has led to tremendous progress in mathematics during the 20th century. For example, Noether revolutionized algebraic topology by emphasizing the importance of homology groups. Previous work had focused on Betti numbers, which are just the dimensions of the rational homology groups. As with taking the cardinality of a set, taking the dimension of a vector space is a process of decategorification, since two vector spaces are isomorphic if and only if they have the same dimension. Noether noted that if we work with homology groups rather than Betti numbers, we can solve more problems, because we obtain invariants not only of spaces, but also of maps. In modern language, the nth rational homology is a *functor* defined on the *category* of topological spaces, while the nth Betti number is a mere *function*, defined on the *set* of isomorphism classes of topological spaces. Of course, this way of stating Noether's insight is anachronistic, since it came before category theory. Indeed, it was in Eilenberg and Mac Lane's subsequent work on homology that category theory was born! Decategorification is a straightforward process which typically destroys information about the situation at hand. Categorification, being an attempt to recover this lost information, is inevitably fraught with difficulties. >Finally, a note to John: While you're trying to give your audience >some feeling for the virtues of n-categories, couldn't you give them a >little help with n=1, by being a little more precise about objects and >maps? I hope it's clearer now. Best, jb From rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Ree-00002p-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:47:16 -0400 Message-Id: <200403072100.i27L0MH12680@math-cl-n03.ucr.edu> Subject: categories: golden objects - typo To: categories@mta.ca (categories) Date: Sun, 7 Mar 2004 13:00:22 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 13 A typo: >I'd asked for some nice examples of an object G in a rig category >equipped with an isomorphism from G to G^2 + 1. ^^^^^^^^^^^^ It should have been "from G^2 to G + 1". From rrosebru@mta.ca Mon Mar 8 16:49:21 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:21 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Re9-0007ir-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:46:45 -0400 Message-ID: <404B8E44.3030701@unt.edu> Date: Sun, 07 Mar 2004 15:04:04 -0600 From: Mike Oliver Organization: University of North Texas User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.4) Gecko/20030630 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: mystification and categorification References: <200403060649.i266nuaG014947@coraki.Stanford.EDU> In-Reply-To: <200403060649.i266nuaG014947@coraki.Stanford.EDU> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Authentication-Info: Submitted using SMTP AUTH at out005.verizon.net from [4.10.156.194] at Sun, 7 Mar 2004 15:04:03 -0600 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 14 Vaughan Pratt wrote: > Quite right. I would add to this "and satisfying the expected equations." > The "nasty sets" of which Steve speaks fail to satisy such expected > equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra, > for heaven's sake. Why on earth forget that structure prior to taking the > second exponentiation? Set theorists seem to think that they can simply > forget structure without paying for it, but in the real world it costs > kT/2 joules per element of X to forget that structure. If set theorists > aren't willing to pay real-world prices in their modeling, why should we > taxpayers pay them real-world salaries? Large cardinals are a figment of > their overactive imaginations, and the solution to consistency concerns is > not to go there.) I will answer you in a Popperian key: Large cardinals are falsifiable, and are not yet falsified. They may in fact be figments of our imaginations, but then why do they keep on *working*? Could be just a coincidence -- but so could all other observation; that way lies the nullification of science in general. It's an illusion, by the way, to think that you can be rid of concerns about consistency by dumping large cardinals, that you can thus achieve a priori justification for apodictic certainty. That doesn't exist even for the natural numbers; Ed Nelson is quite right on this point. As to the question of taxpayer funding, I will not attempt to justify it (I'm a libertarian in politics), but will merely note that many taxpayers probably feel that way about *all* pure mathematics. From rrosebru@mta.ca Mon Mar 8 16:49:52 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:49:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0Rfr-0000HF-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:48:31 -0400 Message-ID: <404C39CA.60809@mcs.le.ac.uk> Date: Mon, 08 Mar 2004 09:15:54 +0000 From: "V. Schmitt" User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.2) Gecko/20031007 Debian/1.0.2-3 X-Accept-Language: en MIME-Version: 1.0 To: categories Subject: categories: RA position Leicester Pure math Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 15 Dear All, There is currently a research assistantship position being advertised in Leicester, in pure mathematics (details given below). Interviews are likely to be late April, with the RA starting by 1st June. Please can you forward this to anyone who might be interested. Best wishes, Robert -------------- University of Leicester Research Associate in Pure Mathematics Department of Mathematics R&AIA 18,265 to 27,339 pa (lower end of scale) Available for 3 years, to start by 1st June 2004. Ref: R0646 The main aim of this EPSRC-funded project will be to study the actions of braid groups on derived categories by generalising recent examples from the representations of associative algebras, of simple Lie algebras (category O), and quantum groups. The project will also study the representations of the braid group induced by such actions. These developments will be used in order to develop a more general theory of braid group actions on derived categories. Possession of a PhD in mathematics, or nearing completion of a relevant PhD, is essential. Informal enquiries can be made to Dr Robert Marsh, Tel +44 (0) 116 252 5107 or email rjm25@mcs.le.ac.uk, or Professor Steffen Koenig, Tel. +44 (0) 116 252 5670 or email sck5@mcs.le.ac.uk. Downloadable application forms and further particulars are available by following the link below, or in hardcopy from the Personnel Office, tel: 0116 252 5114, fax: 0116 252 5140, email: personnel@le.ac.uk, www.le.ac.uk/personnel/jobs. Please note that CVs will only be accepted in support of a fully completed application form. Closing date: 19 March 2004. ---------------------------------------------------------------------- Steffen Koenig Department of Mathematics University of Leicester University Road Leicester, LE1 7RH UK Office F24, Phone 0044 (0)116 252 5670 http://www.mcs.le.ac.uk/~skoenig/ ---------------------------------------------------------------------- From rrosebru@mta.ca Mon Mar 8 16:50:15 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:50:15 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0RgU-0000OL-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:49:11 -0400 Date: Mon, 08 Mar 2004 10:20:23 +0000 From: Steve Vickers User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:1.3) Gecko/20030312 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: mystification and categorification References: <200403060649.i266nuaG014947@coraki.Stanford.EDU> In-Reply-To: <200403060649.i266nuaG014947@coraki.Stanford.EDU> Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 16 Vaughan Pratt wrote: >(The power set of a set is a Boolean algebra, >for heaven's sake. Why on earth forget that structure prior to taking the >second exponentiation? Set theorists seem to think that they can simply >forget structure without paying for it, but in the real world it costs >kT/2 joules per element of X to forget that structure. If set theorists >aren't willing to pay real-world prices in their modeling, why should we >taxpayers pay them real-world salaries? Large cardinals are a figment of >their overactive imaginations, and the solution to consistency concerns is >not to go there.) > >Vaughan Pratt > Dear Vaughan, I like it! But there's still the question of just what structure the power set has. Constructively it's not a Boolean algebra in general, though it is a frame. And is it even a set? You can in fact only say that by removing the structure, which is exactly what you told the set-theorists not to do. And in this instance it's arguable. Topos theorists say it is a set, predicative type theorists say it isn't. Part of the structure of the power "set" is topological - the Scott topology, with the inclusion order as its specialization order. But to formalize it as topological space, point-set + topological structure, you again have to forget structure in order to get a point-set. Taking this seriously generally brings you to point-free topology in some form or other. Steve Vickers. From rrosebru@mta.ca Mon Mar 8 16:51:20 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Mar 2004 16:51:20 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0RhV-0000WH-00 for categories-list@mta.ca; Mon, 08 Mar 2004 16:50:13 -0400 Message-ID: <404C9AA0.5000509@cs.bham.ac.uk> Date: Mon, 08 Mar 2004 16:09:04 +0000 From: Steve Vickers User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:1.3) Gecko/20030312 X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories Subject: categories: Papers on Lawvere's generalized metric spaces References: <4006C74B.15F9C427@bangor.ac.uk> In-Reply-To: <4006C74B.15F9C427@bangor.ac.uk> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 I have just put on the web two papers that explore constructive localic (and also topos) aspects of Lawvere's generalized metric spaces (see TAC Reprints 1, "Metric spaces, generalized logic and closed categories"). My papers are: "Localic completion of generalized metric spaces I" "Localic completion of generalized metric spaces II: Powerlocales" They can both be found through http://www.cs.bham.ac.uk/~sjv/README.html Lawvere describes generalized metric spaces as categories enriched over the extended half line [0, infinity]. I replace this enrichment category by a locale with the same points, and show how to obtain a completion as a locale. Topos aspects include a proof that the generic localic completion is an opfibration in the 2-category of grothendieck toposes. My second paper shows how the powerlocales of the localic completions are themselves also localic completions, and uses this to analyse questions of compactness. Steve Vickers. From rrosebru@mta.ca Tue Mar 9 19:50:55 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Mar 2004 19:50:55 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0qyN-0000IZ-00 for categories-list@mta.ca; Tue, 09 Mar 2004 19:49:19 -0400 Message-Id: Date: Tue, 9 Mar 2004 13:23 +0100 From: femke@few.vu.nl (Femke van Raamsdonk) To: categories@mta.ca Subject: categories: HOR'04: last call for abstracts Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 18 ******************************** * * * HOR'04 CALL FOR ABSTRACTS * * * ******************************** 2nd international Workshop on Higher-Order Rewriting http://www-i2.informatik.rwth-aachen.de/HOR04/ IMPORTANT DATES: Mar 17 2004 : deadline electronic submission of paper Apr 9 2004 : notification of acceptance of papers Apr 23 2004 : deadline for final version of accepted papers The aim of HOR is to provide an informal and friendly setting to discuss recent work and work in progress concerning higher-order rewriting. INVITED TALKS: Mariangola Dezani Torino Mark-Oliver Stehr Hamburg TOPICS of interest include (but are not limited to): APPLICATIONS: proof checking, theorem proving, generic programming, declarative programming, program transformation. FOUNDATIONS: pattern matching, unification, strategies, narrowing, termination, syntactic properties, type theory. FRAMEWORKS: term rewriting, conditional rewriting, graph rewriting, net rewriting, comparisons of different frameworks. IMPLEMENTATION: explicit substitution, rewriting tools, compilation techniques. SEMANTICS: semantics of higher-order rewriting, higher-order abstract syntax PROGRAM/ORGANIZING COMMITTEE: Delia Kesner Paris kesner@pps.jussieu.fr Femke van Raamsdonk Amsterdam femke@cs.vu.nl Joe Wells Edinburgh jbw@macs.hw.ac.uk HOR'04 SUBMISSIONS: Abstracts between 2 and 5 pages. As HOR is meant to be a platform to discuss ongoing research we are also interested in abstract describing work in progress, or problems in higher-order rewriting. PUBLICATION: The proceedings of HOR 2004 will be published as a technical report of the Computer Science Department of RWTH Aachen. LOCAL ARRANGEMENTS: Juergen Giesl RWTH Aachen, Germany giesl@informatik.rwth-aachen.de From rrosebru@mta.ca Tue Mar 9 19:50:55 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Mar 2004 19:50:55 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0qwt-0000CM-00 for categories-list@mta.ca; Tue, 09 Mar 2004 19:47:47 -0400 In-Reply-To: <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk> References: <002a01c401ab$cd50b370$1767eb44@grassmann> <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk> Mime-Version: 1.0 (Apple Message framework v612) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <13EDA32B-71B8-11D8-BE9C-000A95A85E4A@brics.dk> Content-Transfer-Encoding: 7bit To: categories@mta.ca From: Pawel Sobocinski Subject: categories: Re: mystification and categorification Date: Tue, 9 Mar 2004 10:54:02 +0000 X-Mailer: Apple Mail (2.612) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 19 On 7 Mar 2004, at 19:43, Tom Leinster wrote: > I'd interpret "nice" differently. (Apart from anything else, the > trivial example in my previous paragraph would otherwise make the > golden > object problem uninteresting.) "Nice" as I understand it is not a > precise term - at least, I don't know how to make it precise. Maybe I > can explain my interpretation by analogy with the equation T = 1 + T^2. > A nice solution T would be the set of finite, binary, planar trees > together with the usual isomorphism T -~-> 1 + T^2; a nasty solution > would be a random infinite set T with a random isomorphism to 1 + T^2. > (Both these solutions are in the rig category Set with its standard + > and x.) I regard the first solution as nice because I can see some > combinatorial content to it (and maybe, at the back of my mind, because > it has a constructive feel), and the second as nasty because I can't. > I'm not certain what I think of the solution given by the set of > not-necessarily-finite binary planar trees (nice?), or by the set of > binary planar trees of cardinality at most aleph_5 (probably nasty). From a computer science point of view, both the first "nice" solution (finite binary trees) and the second "nice?" solution (possibly non-finite binary trees) are canonical, in the sense that the first is the carrier of the initial algebra for the endofunctor 1+X^2 on Set, while the second is the carrier of its final coalgebra. All the best, Pawel. From rrosebru@mta.ca Tue Mar 9 19:50:55 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Mar 2004 19:50:55 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B0qxu-0000Gv-00 for categories-list@mta.ca; Tue, 09 Mar 2004 19:48:50 -0400 Message-ID: <404DAA18.8030403@itu.dk> Date: Tue, 09 Mar 2004 12:27:20 +0100 From: Thomas Hildebrandt User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.2) Gecko/20030208 Netscape/7.02 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: CTCS 2004 - 3rd Call for Papers - Deadline April 9th Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Virus-Scanned: by amavisd-new at itu.dk Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 20 [Apologies if you receive this more than once] ----------------------------------------------------------------- (APPSEM-II event) 3rd CALL FOR PAPERS 10th conference on Category Theory and Computer Science (CTCS'04) August 12-14, 2004 IT University of Copenhagen, Denmark www.itu.dk/research/theory/ctcs2004/ Submission Deadline: April 9th, 2004 ----------------------------------------------------------------- Invited Speakers: Francois Bergeron (Quebec) Martin Hyland (Cambridge) Robin Milner (Cambridge) Andrew Pitts (Cambridge) Thomas Streicher (Darmstadt) ----------------------------------------------------------------- The purpose of the CTCS conference series is the advancement of the foundations of computing using the tools of category theory. Previous meetings have been held in Guildford (Surrey), Edinburgh (twice), Manchester, Paris, Amsterdam, Cambridge, S. Margherita Ligure (Genova), and Ottawa. The emphasis is upon applications of category theory, but it is recognized that the area is highly interdisciplinary. Typical topics of interest include, but are not limited to, category-theoretic aspects of the following: * Coalgebras and computing * Concurrent and distributed systems * Constructive mathematics * Declarative programming and term rewriting * Domain theory and topology * Foundations of computer security * Linear logic * Modal and temporal logics * Models of computation * Program logics, data refinement, and specification * Programming language semantics * Type theory The proceedings of the conference will be published as a special issue of ENTCS (Electronic Notes in Theoretical Computer Science). ----------------------------------------------------------------- PROGRAMME COMMITTEE Lars Birkedal, Chair (IT University of Copenhagen) Marcelo Fiore (University of Cambridge) Masahito Hasegawa (Kyoto University) Bart Jacobs (University of Nijmegen) Ugo Montanari (University of Pisa) Valeria de Paiva (Palo Alto Research Center) Dusko Pavlovic (Kestrel Institute) John Power (University of Edinburgh) Edmund Robinson (Queen Mary, University of London) Peter Selinger (University of Ottawa) ----------------------------------------------------------------- ORGANIZING COMMITTEE E. Moggi, Chair, (Genova) S. Abramsky (Oxford) P. Dybjer (Chalmers) B. Jay (Sydney) A. Pitts (Cambridge) Local organizing committee: C. Butz T. Hildebrandt A.L. Moerk ----------------------------------------------------------------- SUBMISSION OF PAPERS Papers should be submitted, preferably in electronic form, to ctcs04@itu.dk. Papers are limited to 15 pages, and must be submitted in dvi, postscript, or pdf format, possibly gzipped and/or uuencoded, or sent as a standard email attachment. All submissions must be received by April 9th, 2004. If you cannot submit your paper electronically, please contact the program chair at ctcs04@itu.dk. ----------------------------------------------------------------- IMPORTANT DATES April 1st, 2004: Student Grants (see below) April 9th, 2004: Submission deadline June 1st, 2004: Notification of authors of accepted papers July 1st, 2004: Registration deadline, and Revised papers due ----------------------------------------------------------------- Associated events: FIRST GRADUATE STUDENT SUMMER SCHOOL, August 9-11. Inspired by the success of the graduate student preconference of CTCS'02 in Ottawa, the CTCS of this year will have a graduate student summer school from August 9-11, sponsored by the FIRST graduate school (www.first.dk). The goal is to prepare students (with basic knowledge of category theory) for CTCS, through mini-courses in the basic areas underlying some of the fields of the conference. The school will offer the following mini-courses (5 lectures each): * Coalgebras, Modal Logic and Stone Duality (Lecturer: Alexander Kurz) * Game Semantics (Lecturer: Guy McCusker) * Operational Semantics (Lecturer: Pawel Sobocinski) * Categorical Models for Concurrency (Lecturer: Thomas Hildebrandt) Registration Deadline: July 1st. CMCIM WORKSHOP, August 11. In between the summer school and the CTCS conference, August 11th, there will be a half-day workshop on Categorical Methods in Concurrency, Interaction and Mobility. The workshop has previously been held in connection with CONCUR 2002 and CONCUR 2003. We invite submissions of extended abstracts (less than 5 pages), presenting recent results, challenges or work in progress. There will be no formal proceedings of the workshop, informal proceedings will be distributed at the workshop. Thus, accepted material may be published elsewhere at a later date. Workshop participation is *free*, but requires registration. Registration is done by sending an email to hilde@itu.dk, containing `CMCIM2004-registration' in the subject, and your full name and institution in the body. Submissions should be sent as PostScript files to: hilde@itu.dk, containing `CMCIM-submission' in the subject, and in the body the full names of the author(s), title, and a text-only abstract. Workshop Organizers: Thomas Hildebrandt Alexander Kurz CMCIM Workshop Registration Deadline: July 1st. CMCIM Workshop Submission Deadline: June 21th. ----------------------------------------------------------------- STUDENT GRANTS (application deadline: April 1st!!) A limited number of grants is available for funding the stay of international students at a nearby youth hostel during the summer school and conference. See http://www.itu.dk/research/theory/ctcs2004/accomodation.html ----------------------------------------------------------------- CONFERENCE, SUMMER SCHOOL and WORKSHOP HOMEPAGE Updated information is available from http://www.itu.dk/research/theory/ctcs2004/ ----------------------------------------------------------------- SPONSORSHIP The CTCS conference and summer school are APPSEM-II events, sponsored by the FIRST graduate school (www.first.dk) and the Theory Department at the IT University of Copenhagen (www.itu.dk/English/research/theory/) ----------------------------------------------------------------- POSTER http://www.itu.dk/research/theory/ctcs2004/plakata4.pdf From rrosebru@mta.ca Thu Mar 11 17:16:52 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Mar 2004 17:16:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B1XTL-0002QQ-00 for categories-list@mta.ca; Thu, 11 Mar 2004 17:12:07 -0400 To: categories@mta.ca Subject: categories: Picture of Kan? X-Mailer: mh-e 6.1; nmh 1.0.4+dev; Emacs 21.4 Date: Thu, 11 Mar 2004 11:21:36 +0000 From: N Ghani Message-Id: <20040311112136.56C632F7BA@pc38.mcs.le.ac.uk> Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 21 I've got to do some publicity for funding agencies on a grant I have that uses Kan extensions. I've been told that they like pictures (as opposed to diagrams). So, does anyone have a picture of Daniel Kan? I tried his web page but it is rather minimal and without an email address Neil From rrosebru@mta.ca Fri Mar 12 13:18:27 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Mar 2004 13:18:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B1qG8-0004yW-00 for categories-list@mta.ca; Fri, 12 Mar 2004 13:15:44 -0400 To: categories@mta.ca Subject: categories: Re: Picture of Kan? From: Dan Christensen In-Reply-To: <20040311112136.56C632F7BA@pc38.mcs.le.ac.uk> (N. Ghani's message of "Thu, 11 Mar 2004 11:21:36 +0000") References: <20040311112136.56C632F7BA@pc38.mcs.le.ac.uk> Date: Thu, 11 Mar 2004 18:21:35 -0500 Message-ID: <87brn37yww.fsf@uwo.ca> User-Agent: Gnus/5.110002 (No Gnus v0.2) Emacs/21.3 (gnu/linux) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Spam-Score: 0.496 () MAILTO_TO_SPAM_ADDR X-Scanned-By: MIMEDefang 2.39 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 23 N Ghani writes: > I've got to do some publicity for funding agencies on a grant I have > that uses Kan extensions. I've been told that they like pictures (as > opposed to diagrams). So, does anyone have a picture of Daniel Kan? The hopf archive has a couple: This one has (left to right) Bill Singer, George Whitehead, Dan Kan and someone else that looks familiar. It's from 1980 at Cornell. http://hopf.math.purdue.edu/pictures/kan.gif This one has Charles Rezk and Dan Kan. 1995 or 1996, MIT: http://hopf.math.purdue.edu/pictures-95-96/mit/resk-kan.jpg It you write to Clarence Wilkerson (the maintainer of hopf, wilker at math purdue edu) or to Phil Hirschhorn (close friend of Kan's, sees him daily, psh at math mit edu) you might be able to dig up better pictures. Dan From rrosebru@mta.ca Fri Mar 12 13:18:27 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Mar 2004 13:18:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B1qHX-000597-00 for categories-list@mta.ca; Fri, 12 Mar 2004 13:17:11 -0400 Message-ID: <005901c40839$0faef9a0$05e5fea9@gateway.2wire.net> Reply-To: "Vidhyanath Rao" From: "Vidhyanath Rao" To: References: <002a01c401ab$cd50b370$1767eb44@grassmann> <1078688585.20775.171.camel@tl-linux.maths.gla.ac.uk> Subject: categories: Quillen model structure of category of toposes/locales? Date: Fri, 12 Mar 2004 08:50:46 -0500 MIME-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: 7bit X-Priority: 3 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 24 I would like some references to model structures on the category of toposes/locales (thinking of them as generalized spaces), perhaps even the category of internal toposes of a given (boolean?) topos. What I know of is about model structures on the category of simplicial objects in a topos. Along the same lines, does it make sense to ask about ``internally simplicial objects'' in a topos with an NNO (i.e., do such toposes have an internal category that looks like the category of internally ``finite sets with linear order and order preserving maps'')? Nath Rao From rrosebru@mta.ca Fri Mar 12 13:18:45 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Mar 2004 13:18:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B1qIz-0005Gk-00 for categories-list@mta.ca; Fri, 12 Mar 2004 13:18:41 -0400 Message-ID: <4051D920.7010601@doc.ic.ac.uk> Date: Fri, 12 Mar 2004 15:37:04 +0000 From: Concur 2004 MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Concur 2004: Call for Papers Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 25 Call for Papers for CONCUR 2004 CONCUR 2004, the 15th International Conference on Concurrency Theory, takes place at the Royal Society, London, Tuesday 31 August - Friday 3 September 2004. As well as the main event, we will also hold many workshops on Monday 30th August and Saturday 4th February, as detailed below. Further information is available at the conference's web site http://www.doc.ic.ac.uk/concur2004 Important dates Submission: Friday 9 April 2004 (submission is now open) Notification: Monday 31 May 2004 Final version: Tuesday 15 June 2004 The purpose of the CONCUR conferences is to bring together researchers, developers and students in order to advance the theory of concurrency, and promote its applications. Interest in this topic is continuously growing, as a consequence of the importance and ubiquity of concurrent systems and their applications, and of the scientific relevance of their foundations. Submissions are solicited in all areas of semantics, logics and verification techniques for concurrent systems. Principal topics include (but are not limited to): - Basic models and logics of concurrent and distributed computation (such as process algebras, Petri nets, domain theoretic or game theoretic models, modal and temporal logics). - Specialized or enriched models (such as circuits, synchronous systems, real time and hybrid systems, stochastic systems, data bases, mobile and migrating systems, parametric protocols, security protocols). - Related verification techniques and tools (such as state-space exploration, model-checking, synthesis, abstraction, automated deduction, testing). - Related programming models (such as distributed, constraints or object oriented, graph rewriting, as well as associated type systems, static analyses, abstract machines, and environments). Authors are invited to submit an extended abstract not exceeding 15 pages electronically via the web submission form at the conference's web site. Submissions will be evaluated by the program committee for inclusion in the proceedings, which will be published by Springer-Verlag in the Lecture Notes in Computer Science series. Papers must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. Simultaneous submissions to other conferences are not allowed. There will be a special issue of the journal Theoretical Computer Science associated with Concur 2004. Invited speakers * Sriram K. Rajamani (Microsoft Research): Invited Talk on Zing (a model-checking tool based on ideas from process algebra) * Peter O'Hearn (Queen Mary University of London) and Steve Brookes (Carnegie-Mellon): Tutorial on Local Reasoning about Concurrent Imperative Programs * Bengt Jonsson (Uppsala): Tutorial on Regular Model Checking Organisers * General chair: Philippa Gardner * Programme Committee co-chairs: Philippa Gardner, Nobuko Yoshida * Workshops organisers: Vladimiro Sassone, Julian Rathke * Local organisers: Iain Phillips, Sergio Maffeis, Alex Ahern Program committee Philippa Gardner, co-chair, UK Nobuko Yoshida, co-chair, UK Luca Aceto, Denmark Bruno Blanchet, Germany/France Steve Brookes, USA Luca De Alfaro, USA Paul Gastin, France Petr Jancar, Czech Republic Joost-Pieter Katoen, Netherlands Dietrich Kuske, Germany Cosimo Laneve, Italy Michael Mendler, Germany Ugo Montanari, Italy Catuscia Palamidessi, France Vladimiro Sassone, UK PS Thiagarajan, Singapore Antti Valmari, Finland Wang Yi, Sweden Concur 2004 gratefully acknowledges sponsorship from Microsoft Research. Monday workshops - Workshop on Structural Operational Semantics, SOS 2004 URL = http://www.cs.auc.dk/~luca/SOS-WORKSHOP Contact = Luca Aceto - 11th International Workshop in Expressiveness in Concurrency, EXPRESS 2004 URL = http://www.win.tue.nl/express04 Contact = Flavio Corradini - II Workshop on Object-Oriented Developments, WOOD 2004 URL = http://www.dsi.unive.it/wood2004 Contact = Viviana Bono - 3rd International Workshop on Foundations of Coordination Languages and Software Architectures, FOCLASA 2004 URL = http://www.info.fundp.ac.be/~jmj/Foclasa04 Contact = Jean-Marie JACQUET - 2nd International Workshop on Security Issues in Coordination Models, Languages and Systems, SECCO 2004 URL = http://www.cs.unibo.it/secco04/ Contact = Gianluigi Zavattaro - BIOCONCUR 2004 URL = http://www.imm.dtu.dk/bioconcur04 Contact = Anna Ingolfsdottir Friday afternoon + Saturday - Global Ubiquitous Computing URL = http://www.cogs.susx.ac.uk/fgc04 Contact = Julian Rathke Saturday workshops - 3rd International Workshop on Parallel and Distributed Methods in VerifiCation, PDMC 2004 URL = http://www.fi.muni.cz/~brim/PDMC04/ Contact = Martin Leucker - First International Workshop on Practical Applications of Stochastic Modelling, PASM 2004 URL = http://www.doc.ic.ac.uk/pasm2004 Contact = Jeremy Bradley - Infinity URL = http://www.lfcs.ed.ac.uk/Infinity04 Contact = Julian Bradfield - Fourth International Workshop on Automated Verification of Critical Systems, AVoCS 2004 URL = http://www.doc.ic.ac.uk/~mrh/AVoCS04.html Contact = Michael Huth From rrosebru@mta.ca Mon Mar 15 14:12:37 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Mar 2004 14:12:37 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B2wWZ-0005Y8-00 for categories-list@mta.ca; Mon, 15 Mar 2004 14:09:15 -0400 Message-Id: <5.2.0.9.0.20040315075042.01c306e8@pop.cwru.edu> X-Sender: cxm7@pop.cwru.edu (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.0.9 Date: Mon, 15 Mar 2004 09:23:11 -0500 To: categories@mta.ca From: Colin McLarty Subject: categories: Topos cohomology, context and technical questions Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 26 Thanks to Christopher Townsend and Carsten Butz for help on cohomology in an elementary topos. It seems the general theory is not much advanced beyond what it was in Johnstone 1977. My question came out of a conversation with algebraic geometers several years ago, which I have taken up again lately. Deligne, for example, describes toposes as one of Grothendieck's great ideas (one of Grothendieck's four "idees maitresses"). But for him and many other geometers their value lies in organizing cohomology. Insofar as Grothendieck toposes support a simple general theory of cohomology, and elementary toposes do not, these people find only Grothendieck toposes interesting. Certainly there is a lot to say for elementary topos theory even from that perspective: The elementary topos axioms organize the theory of Grothendieck toposes. Elementary toposes have some cohomology theory though not so simple and general. And elementary toposes have other roles. What interests me, now, is how far elementary topos theory helps with cohomology per se. One approach is to notice: The elementary theory of "a topos whose Abelian groups have enough injectives" supports a considerable general theory of cohomology via injective resolutions. But I have not worked out how far it really goes. (People with foundational interests will notice the exact result depends on whether and how this theory works with infinite complexes. There are various approaches depending on what you mean by "elementary".) This raises my first technical question: SGA 4 proves inverse image functors preserve flat modules, but the transparent proof assumes enough points (Exp. V Prop. 1.7). Deligne gives a far from transparent proof, for all (Grothendieck) toposes, in an appendix on "local inductive limits". He urges the reader "to avoid, as a matter of principle, reading this appendix". Is the result proved more simply somewhere? Do "local inductive limits" survive today in some form? In short, can we follow Deligne's advice on not reading this appendix, and still prove his result? I have made no progress on the appendix yet, as the opening definition is full of typos. If there is a cleaner exposition I'd rather start with that. The second question: The IHES version of SGA 4 gives a faulty proof that, in every (Grothendieck) topos, rings admit a standard kind of resolution over any cover by tensoring with a resolution of the integers. This is Prop. 1.4 of Expose V. The Springer-Verlag version corrects the mistake by proving the result only when the topos has enough points (Prop 1.11 Exp. V). Johnstone 1977 recovers the theorem for the case of a presheaf topos (Lemma 8.2) which is the case of interest and easily extends to any topos with enough points. Is that version optimal, in some easy to prove sense? Is there an easy example of a ring in a Grothendieck topos where the resolution fails? Is it known to be optimal in any sense? best, Colin From rrosebru@mta.ca Thu Mar 18 14:14:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Mar 2004 14:14:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B41zi-0001Ng-00 for categories-list@mta.ca; Thu, 18 Mar 2004 14:11:50 -0400 Message-ID: <40578302.FFF898F4@math.upenn.edu> Date: Tue, 16 Mar 2004 17:43:14 -0500 From: jim stasheff To: "categories@mta.ca" Subject: categories: [Fwd: 2 questions] Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 27 You might also post your quesiton to the category analog of Don's list: Date: Tue, 16 Mar 2004 15:58:18 -0500 From: Don Davis Subject: 2 questions Two postings, both questions. Also, I announce that Martin Bendersky and I have organized a special session in homotopy theory at the AMS meeting in Lawrenceville, NJ, on April 17-18. This session, and several others, honor Bill Browder on his 70th birthday. There will be a banquet in his honor. The program of this special session can be found at http://www.ams.org/amsmtgs/2102_program_ss3.html#title or on the conference page of this discussion group, http://www.lehigh.edu/~dmd1/conf.html...........DMD ____________________________________________________________ Subject: Question for the discussion list. Date: Tue, 16 Mar 2004 09:31:11 -0800 (PST) From: Keir Lockridge Is the full subcategory of finite objects in the stable category equivalent, as a triangulated category, to a full subcategory of the derived category of an abelian category? Can the abelian category be chosen to be symmetric monoidal and so that S^0 corresponds to a resolution of the unit object? Keir Lockridge University of Washington Department of Mathematics lockridg@math.washington.edu ___________________________________________________ Subject: Question Date: Tue, 16 Mar 2004 20:47:40 +0100 From: "boccellari" I am looking for any kind of result concerning the following question. Consider a space X and an element x \in H^*(X;Z_2). Take M(X,K;x) path component of x in the space of maps M(X,K) where K is a suitable generalized Eilenberg-MacLane space. Question: Is the evaluation map ev : M(X,K;x) \times X \rightarrow K either a fibration or a cofibration? What about it when X is a Thom space, x its thom class and K = K(Z_2,n)? Thank you for your help. Yours faithfully Tommaso Boccellari From rrosebru@mta.ca Thu Mar 18 14:14:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Mar 2004 14:14:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B421k-0001kL-00 for categories-list@mta.ca; Thu, 18 Mar 2004 14:13:56 -0400 Date: Wed, 17 Mar 2004 16:48:31 -0700 (MST) From: robin@cpsc.ucalgary.ca Reply-To: robin@cpsc.ucalgary.ca Subject: categories: FMCS 2004 (Calgary) To: categories@mta.ca MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 28 A preliminary announcement for FMCS 2004: __________________________________________________________ FFFFFFF M M CCC SSS F MM MM C S FFFFF M MM M C SSS F M M C S F M M CCC SSS 2004 (CALGARY) _________________________________________________________ http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_04/index.html _________________________________________________________ This year's meeting on Foundational Method in Computer Science (FMCS 2004) will be hosted by the Programming Languages Group at the University of Calgary, June 4th -- June 6th and will be held at the Kananaskis Field Station. Next years meeting will be in Vancouver at UBC ... __________________________________________________________ FOUNDATIONAL METHODS IN COMPUTER SCIENCE The workshop is an informal meeting to bring together researchers in mathematics and computer science with a focus on the applications of category theory in computer science. It is a three day meeting, which starts with a day of tutorials (Friday 4th June) aimed at students and newcomers to category theory, followed by a day and a half of research talks (Saturday, 5st June and Sunday, 6th June). There will be a series of invited presentations (TBA). The remaining research talks are solicited from the participants. If you wish to give a talk please let us know by sending an abstract and title to Robin Cockett or Craig Pastro (see e-mail address below) before the 21st May. Student participation at FMCS is particularly encouraged. There are limited funds to provide support for students who wish to attend the workshop (see below). __________________________________________________________ Support for graduate students: __________________________________________________________ We particularly encourage graduate students to attend FMCS and to present their work. If you wish to give a presentation at FMCS you should send a title and a brief abstract to Robin Cockett (robin@cpsc.ucalgary.ca). We would like to receive these abstracts before beginning of May so that we can arrange the schedule before the meeting itself. Some limited funding is available to support graduate students who wish to attend FMCS. To apply for this money, you should contact Robin Cockett with the following information: 1) A one-page email letter stating your background as well as why you are interested in attending. 2) The letter should also state whether you have access to any other funding to attend. 3) An email letter of reference from your supervisor or an appropriate other person. Applications should be made as soon as possible but before Friday 7th May. __________________________________________________________ Organizer: Robin Cockett (robin@cpsc.ucalgary.ca) Local organizer: Craig Pastro (pastroc@cpsc.ucalgary.ca) From rrosebru@mta.ca Thu Mar 18 14:14:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Mar 2004 14:14:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B420I-0001Vg-00 for categories-list@mta.ca; Thu, 18 Mar 2004 14:12:26 -0400 Date: Wed, 17 Mar 2004 02:47:18 +0000 (GMT) From: Dr Eugenia Cheng To: categories@mta.ca Subject: categories: 80th PSSL - third announcement Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 29 PSSL 80 - Cambridge, UK Third announcement: registration deadline Dear All, This is the third announcement for the 80th Peripatetic Seminar on Sheaves and Logic, which will be held on the weekend of 3rd and 4th April in Cambridge, UK. If you would like to attend the PSSL please send an e-mail to Eugenia Cheng, using the form attached below. Please register by the end of this week (ie by Sunday 21st March) in order to be guaranteed accommodation at Newnham College. Please note that the list of talks is already very full and it will be hard to squeeze much more in! For further information including list of registered participants and talks, see: http://www.dpmms.cam.ac.uk/~elgc2/pssl80 The full text of the main announcement can be found at: http://www.dpmms.cam.ac.uk/~elgc2/pssl80/announce2.html We look forward to seeing you in April. With best regards, Eugenia Cheng ----------------------------------------------------------------------- REGISTRATION FORM (please delete as appropriate) I, __________________________________________, would like to attend the 80th PSSL. I will not be giving a talk / I would like to give a talk entitled ________________________________________________________________________ My affiliation is _____________________________________ (University etc) I will not require accommodation in Newnham College / I would like accommodation in Newnham College for the nights of _____________ April I will not be attending the dinner on Saturday / I would like to attend the dinner on Saturday for 15 pounds. I need the following non-standard combination of the above options: ____________________________________________________________________ I have the following special dietary requirements: ____________________________________________________________________ I expect that the amount payable will be: Residential + dinner 117 pounds Residential without dinner 102 pounds Non-residential + dinner 41 pounds Non-residential without dinner 26 pounds + ____ extra nights @ 32 pounds TOTAL _________________ I will pay in cash upon registration / I would like to pay online ------------------------------------------------------------------- From rrosebru@mta.ca Thu Mar 18 14:14:56 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Mar 2004 14:14:56 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B4211-0001cB-00 for categories-list@mta.ca; Thu, 18 Mar 2004 14:13:11 -0400 Date: Wed, 17 Mar 2004 10:29:56 -0500 To: categories@mta.ca From: Colin McLarty Subject: categories: Deligne on Govorov-Lazard, proof and history Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 30 Deligne's proof that inverse image functors preserve flat modules is to internalize the Govorov-Lazard proof that a module is flat iff it is a projective colimit of finite free modules, and to show inverse image functors preserve these internalized colimits (SGA 4, Springer-Verlag edition, Exp. V appendix). The classical proof (say, Eisenbud COMMUTATIVE ALGEBRA appendix 6, theorem A6.6) looks good for topos logic though I admit I have not worked through internalizing it. Also, I have still not fought through the definitions in Deligne's appendix to SGA 4. I would still appreciate any reference to a cleaner treatment if anyone knows one. It seems likely that Deligne's "local inductive limits" are internal colimits in the modern sense but I have not read the thing closely enough yet to really say that. This seems to me a strikingly sophisticated lifting of a theorem, from SET to any topos, for 1972. The current methods for routine lifting were just being created over here in elementary topos theory at that time. I believe Deligne did not know of that. Or am I making too much of this proof? I'd appreciate any historical reflections on relations between such lifting in the Grothendieck school, and methods of elementary topos theory. Colin From rrosebru@mta.ca Sun Mar 21 11:58:58 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Mar 2004 11:58:58 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B55Hj-0007O3-00 for categories-list@mta.ca; Sun, 21 Mar 2004 11:54:47 -0400 Date: Wed, 17 Mar 2004 12:37:05 +0000 (GMT) From: Paul B Levy To: categories@mta.ca Subject: categories: new book: Call-By-Push-Value Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 31 Dear category theorists, My book "Call-By-Push-Value" has been published by Kluwer in the "Semantic Structures in Computation" bookseries http://www.wkap.nl/prod/b/1-4020-1730-8 - and I hope it will be of interest to many people on this list. Although it is based on my PhD thesis, subsequent research has led to a great deal of simplification. In particular, the categorical structure of the programming language - an adjunction between values and stacks - is now completely apparent. Here's the blurb: Call-By-Push-Value A Functional/Imperative Synthesis "Call-by-push-value is a programming language paradigm that, surprisingly, breaks down the call-by-value and call-by-name paradigms into simple primitives. This monograph, written for graduate students and researchers, exposes the call-by-push-value structure underlying a remarkable range of semantics, including operational semantics, domains, possible worlds, continuations and games. "After introducing basic ideas using domain semantics and a stack machine, the book is layered to appeal to readers in a variety of fields. One strand treats semantics of store, culminating in a possible world model for general storage cells. Another "implements" call-by-push-value by translating it into the Jump-With-Argument continuation language, enabling an account of pointer game semantics that explains its arenas, pointers and question/answer labelling in concrete computational terms. Yet another gives a categorical picture of call-by-push-value: an adjunction between values and stacks. "Incorporating recent simplifications, this is a key text for anyone interested in lambda-calculus, programming language foundations or applications of category theory." Please feel free to email me, either about the book or about the subject! Regards Paul http://www.cs.bham.ac.uk/~pbl From rrosebru@mta.ca Mon Mar 22 15:07:33 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 22 Mar 2004 15:07:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5Uio-0001wb-00 for categories-list@mta.ca; Mon, 22 Mar 2004 15:04:26 -0400 Date: Mon, 22 Mar 2004 09:51:19 +0100 From: Thomas Hildebrandt To: categories@mta.ca Subject: categories: CTCS 2004 - 3rd Call for Papers - Deadline April 9th Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Virus-Scanned: by amavisd-new at itu.dk Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 32 [Apologies if you receive this more than once] ----------------------------------------------------------------- (APPSEM-II event) 3rd CALL FOR PAPERS 10th conference on Category Theory and Computer Science (CTCS'04) August 12-14, 2004 IT University of Copenhagen, Denmark www.itu.dk/research/theory/ctcs2004/ Submission Deadline: April 9th, 2004 ----------------------------------------------------------------- Invited Speakers: Francois Bergeron (Quebec) Martin Hyland (Cambridge) Robin Milner (Cambridge) Andrew Pitts (Cambridge) Thomas Streicher (Darmstadt) ----------------------------------------------------------------- The purpose of the CTCS conference series is the advancement of the foundations of computing using the tools of category theory. Previous meetings have been held in Guildford (Surrey), Edinburgh (twice), Manchester, Paris, Amsterdam, Cambridge, S. Margherita Ligure (Genova), and Ottawa. The emphasis is upon applications of category theory, but it is recognized that the area is highly interdisciplinary. Typical topics of interest include, but are not limited to, category-theoretic aspects of the following: * Coalgebras and computing * Concurrent and distributed systems * Constructive mathematics * Declarative programming and term rewriting * Domain theory and topology * Foundations of computer security * Linear logic * Modal and temporal logics * Models of computation * Program logics, data refinement, and specification * Programming language semantics * Type theory The proceedings of the conference will be published as a special issue of ENTCS (Electronic Notes in Theoretical Computer Science). ----------------------------------------------------------------- PROGRAMME COMMITTEE Lars Birkedal, Chair (IT University of Copenhagen) Marcelo Fiore (University of Cambridge) Masahito Hasegawa (Kyoto University) Bart Jacobs (University of Nijmegen) Ugo Montanari (University of Pisa) Valeria de Paiva (Palo Alto Research Center) Dusko Pavlovic (Kestrel Institute) John Power (University of Edinburgh) Edmund Robinson (Queen Mary, University of London) Peter Selinger (University of Ottawa) ----------------------------------------------------------------- ORGANIZING COMMITTEE E. Moggi, Chair, (Genova) S. Abramsky (Oxford) P. Dybjer (Chalmers) B. Jay (Sydney) A. Pitts (Cambridge) Local organizing committee: C. Butz T. Hildebrandt A.L. Moerk ----------------------------------------------------------------- SUBMISSION OF PAPERS Papers should be submitted, preferably in electronic form, to ctcs04@itu.dk. Papers are limited to 15 pages, and must be submitted in dvi, postscript, or pdf format, possibly gzipped and/or uuencoded, or sent as a standard email attachment. All submissions must be received by April 9th, 2004. If you cannot submit your paper electronically, please contact the program chair at ctcs04@itu.dk. ----------------------------------------------------------------- IMPORTANT DATES April 1st, 2004: Student Grants (see below) April 9th, 2004: Submission deadline June 1st, 2004: Notification of authors of accepted papers July 1st, 2004: Registration deadline, and Revised papers due ----------------------------------------------------------------- Associated events: FIRST GRADUATE STUDENT SUMMER SCHOOL, August 9-11. Inspired by the success of the graduate student preconference of CTCS'02 in Ottawa, the CTCS of this year will have a graduate student summer school from August 9-11, sponsored by the FIRST graduate school (www.first.dk). The goal is to prepare students (with basic knowledge of category theory) for CTCS, through mini-courses in the basic areas underlying some of the fields of the conference. The school will offer the following mini-courses (5 lectures each): * Coalgebras, Modal Logic and Stone Duality (Lecturer: Alexander Kurz) * Game Semantics (Lecturer: Guy McCusker) * Operational Semantics (Lecturer: Pawel Sobocinski) * Categorical Models for Concurrency (Lecturer: Thomas Hildebrandt) Registration Deadline: July 1st. CMCIM WORKSHOP, August 11. In between the summer school and the CTCS conference, August 11th, there will be a half-day workshop on Categorical Methods in Concurrency, Interaction and Mobility. The workshop has previously been held in connection with CONCUR 2002 and CONCUR 2003. We invite submissions of extended abstracts (less than 5 pages), presenting recent results, challenges or work in progress. There will be no formal proceedings of the workshop, informal proceedings will be distributed at the workshop. Thus, accepted material may be published elsewhere at a later date. Workshop participation is *free*, but requires registration. Registration is done by sending an email to hilde@itu.dk, containing `CMCIM2004-registration' in the subject, and your full name and institution in the body. Submissions should be sent as PostScript files to: hilde@itu.dk, containing `CMCIM-submission' in the subject, and in the body the full names of the author(s), title, and a text-only abstract. Workshop Organizers: Thomas Hildebrandt Alexander Kurz CMCIM Workshop Registration Deadline: July 1st. CMCIM Workshop Submission Deadline: June 21th. ----------------------------------------------------------------- STUDENT GRANTS (application deadline: April 1st!!) A limited number of grants is available for funding the stay of international students at a nearby youth hostel during the summer school and conference. See http://www.itu.dk/research/theory/ctcs2004/accomodation.html ----------------------------------------------------------------- CONFERENCE, SUMMER SCHOOL and WORKSHOP HOMEPAGE Updated information is available from http://www.itu.dk/research/theory/ctcs2004/ ----------------------------------------------------------------- SPONSORSHIP The CTCS conference and summer school are APPSEM-II events, sponsored by the FIRST graduate school (www.first.dk) and the Theory Department at the IT University of Copenhagen (www.itu.dk/English/research/theory/) ----------------------------------------------------------------- POSTER http://www.itu.dk/research/theory/ctcs2004/plakata4.pdf From rrosebru@mta.ca Tue Mar 23 14:39:17 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Mar 2004 14:39:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5qku-0001Vi-00 for categories-list@mta.ca; Tue, 23 Mar 2004 14:36:04 -0400 From: Peter Selinger Message-Id: <200403221911.i2MJBIB15322@quasar.mathstat.uottawa.ca> Subject: categories: Update: Workshop on Quantum Programming Languages To: categories@mta.ca (Categories List) Date: Mon, 22 Mar 2004 14:11:18 -0500 (EST) X-Mailer: ELM [version 2.5 PL3] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 33 Dear colleagues, relative to what I sent last month, the following announcement has been updated in three places, marked "*** NEW ***" below. Best wishes, -- Peter SECOND 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. INVITED SPEAKER: *** NEW *** Richard Jozsa (Bristol) PROGRAM COMMITTEE: *** NEW *** Samson Abramsky (Oxford) Prakash Panangaden (McGill) Peter Selinger (Ottawa) SUBMISSION PROCEDURES: The workshop will be a 1.5-day workshop. Those who wish to give a talk should submit a one-page abstract (or a full paper up to 12 pages, if available) by April 25 to selinger@mathstat.uottawa.ca (please put "workshop submission" in the subject line). Authors of accepted abstracts or papers will be encouraged to submit a full version (subject to a page limit, details to be announced) for inclusion in the informal workshop proceedings (to be distributed at the workshop) by June 7. There will also be a special issue of the journal MSCS devoted to the workshop, see below. SPECIAL PROCEEDINGS ISSUE OF MSCS: *** NEW *** There will be a special issue of the journal Mathematical Structures in Computer Science (MSCS, Cambridge University Press) devoted to this workshop and areas represented at the workshop. There will be a period of at least 4 months for revising and submitting the papers after the workshop. 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 the informal proceedings. IMPORTANT DATES/DEADLINES: Submission of abstracts: April 25, 2004 Notification of acceptance: May 20, 2004 Paper for informal proceedings: June 7, 2004 Workshop: July 12-13, 2004 Journal paper due: November 21, 2004 CONTACT INFORMATION: Organizer: Peter Selinger Department of Mathematics and Statistics University of Ottawa, Canada Email: selinger@mathstat.uottawa.ca (revised March 22, 2004) From rrosebru@mta.ca Tue Mar 23 14:39:17 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Mar 2004 14:39:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5qmJ-0001d0-00 for categories-list@mta.ca; Tue, 23 Mar 2004 14:37:31 -0400 Date: Tue, 23 Mar 2004 14:17:37 +0000 From: Peter McBurney User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.0.2) Gecko/20030708 X-Accept-Language: en-us, en MIME-Version: 1.0 To: Subject: categories: CFP: Knowledge and Games Content-Type: text/plain; charset=iso-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-MIME-Autoconverted: from 8bit to quoted-printable by tethys.server.csc.liv.ac.uk id i2NEHbP15385 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 34 WITH APOLOGIES FOR MULTIPLE POSTINGS Call for participation A Workshop on Knowledge and Games Liverpool, UK 10th + 11th July 2004 Organising committee + Peter McBurney (University of Liverpool) + Wiebe van der Hoek (University of Liverpool) + Sieuwert van Otterloo (University of Liverpool) + Michael Wooldridge (University of Liverpool) Programme Committee: Johan van Benthem (University of Amsterdam) Valentin Goranko (Rand Afrikaans University) Wilfrid Hodges (Queen Mary University of London) Barteld Kooi (University of Groningen) Alessio Lomuscio (King's College London) Peter McBurney (University of Liverpool) John-Jules Meyer (Utrecht University) Marc Pauly (IRIT Toulouse) Karl Tuyls (Vrije Universiteit Brussel) Mike Wooldridge (University of Liverpool) Ann Now=E9 (Vrije Universiteit Brussel) The aim of this workshop is to bring together researchers from the=20 multi-agent systems, logic, and game theory communities in order to discuss work that combines, in some way, formal=20 theories of knowledge and games. The workshop will take place immediately after the European Agent Systems Summer=20 School, which will also be held in Liverpool. This two day workshop is intended as an informal meeting, with plenty of=20 opportunities for discussion. We invite participants to send in short papers or abstracts of ongoing research or=20 previously published results, which will be informally reviewed. The authors of accepted papers will be invited to=20 present their work at the workshop. Informal proceedings will be distributed during the workshop. In addition, all=20 authors of previously unpublished papers are offered the opportunity to submit a full paper to "Knowledge,=20 Rationality and Action", which will be reviewed by the workshop PC." Important dates (all in 2004) Paper deadline: May 1st Acceptance notification May 14th Early registration deadline May 28th Workshop: Saturday 10 and Sunday 11th of July (directly after EASSS summer school) Topics of interest include, but are not restricted to: + epistemic logic + coalition/cooperation logics + game semantics of logic + logics for games + formal theories of knowledge and action + connections between process models and game models + knowledge games + epistemic model checking + epistemic properties of solution concepts For more information, contact: Sieuwert van Otterloo sieuwert _ at _ csc.liv.ac.uk www.csc.liv.ac.uk/~sieuwert/knowledgegames =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D From rrosebru@mta.ca Tue Mar 23 14:39:17 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Mar 2004 14:39:17 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5qlO-0001Y7-00 for categories-list@mta.ca; Tue, 23 Mar 2004 14:36:34 -0400 From: "M.M. Bonsangue" Date: Tue, 23 Mar 2004 10:30:11 +0100 Message-Id: <200403230930.i2N9UB802412@pc157aa.liacs.nl> To: categories@mta.ca Subject: categories: Preliminary call for Participation: FMCO 2004 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 **************** PRELIMINARY CALL FOR PARTICIPATION ******************** Third International Symposium on Formal Methods for Components and Objects (FMCO 2004) The objective of this symposium is to bring together top researchers in the area of software engineering to discuss the state-of-the-art and future applications of formal methods in the development of large component-based and object-oriented software systems. DATES 2 - 5 November 2004 PLACE Lorentz Center, Leiden University, Leiden, The Netherlands URL http://fmco.liacs.nl/fmco04.html Participation is limited to about 80 people, based on a first-in first-served policy. For more information about participation and registration see the FMCO site at http://fmco.liacs.nl/fmco04.html or consult either F.S. de Boer (frb@cwi.nl) or M.M. Bonsangue (marcello@liacs.nl). PRELIMINARY PROGRAM TUESDAY 2nd, November 2004 8:45 - 9:00 Welcome 9:00 - 10:00 Keynote: Robin Milner (Cambridge University, UK) 10:00 - 10:30 Break 10:30 - 11:15 Rocco de Nicola (University of Firenze, IT) 11:15 - 12:00 Eugenio Moggi (Genova University, IT) 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Kim Bruce (Williams College, USA) 14:30 - 15:00 Break 15:00 - 15:45 Julian Rathke (Sussex University, UK) 15:45 - 16:00 Break 16:00 - 16:45 Martin Steffen (Kiel University, DE) 16:45 - 17:30 Marcello Bonsangue (LIACS, NL) WEDNESDAY 3rd, November 2004 9:00 - 10:00 Keynote: Tom Henzinger (University of California, Berkeley, USA) 10:00 - 10:30 Break 10:30 - 11:15 Susanne Graf (Verimag, FR) 11:15 - 12:00 Wang Yi (Uppsala University, SE) 12:00 - 13:15 Lunch break 13:15 - 14:15 Keynote: Thomas Ball (Microsoft Research at Redmond, USA) 14:15 - 14:30 Break 14:30 - 15:15 Frits Vaandrager (Nijmegen University, NL) 15:15 - 16:00 Tobias Nipkow (Munchen University, DE) 17:00 - 19:15 Social Event 19:30 - Dinner THURSDAY 4th, November 2004 9:00 - 10:00 Keynote: Kim Larsen (Aalborg University, DK) 10:00 - 10:30 Break 10:30 - 11:15 Ed Brinksma (University of Tweente, NL) 11:15 - 12:00 Andreas Podelski (Max Plank Inst. for Informatics, DE) 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Chris Hankin (Imperial College, UK) 14:30 - 15:00 Break 15:00 - 15:45 David Naumann (Stevens Institute of Technology, USA) 15:45 - 16:30 Wolfgang Weck (Oberon Microsystems, CH) 16:30 - 16:45 Break 16:45 - 17:30 Liu Zhiming (UNU-IIST, Macao) FRIDAY 5th, November 2004 9:00 - 10:00 Keynote: Samson Abramsky (Oxford University, UK) 10:00 - 10:30 Break 10:30 - 11:15 Luca de Alfaro (UC Santa Cruz, USA) 11:15 - 12:00 Luis Barbosa (Minho University, PT) 12:00 - 13:30 Lunch break 13:30 - 14:30 Keynote: Reinhard Wilhelm (Saarland University, DE) 14:30 - 15:00 Break 15:00 - 15:45 Olaf Owe (University of Oslo, NO) 15:45 - 16:30 Pierre Cointe (Ecole des Mines de Nantes, FR) ORGANIZING COMMITTEE F.S. de Boer (CWI and Utrecht University) M.M. Bonsangue (LIACS-Leiden University) S. Graf (Verimag) W.P. de Roever (CAU) From rrosebru@mta.ca Tue Mar 23 14:40:47 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Mar 2004 14:40:47 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5qmw-0001iW-00 for categories-list@mta.ca; Tue, 23 Mar 2004 14:38:10 -0400 From: Thomas Streicher Subject: categories: Domains VII (Call for Abstracts) To: categories@mta.ca Date: Tue, 23 Mar 2004 16:15:42 +0100 (CET) X-Mailer: ELM [version 2.4ME+ PL100 (25)] MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 36 Announcement and Call for Abstracts DOMAINS VII Darmstadt, August 29 - September 1 The WORKSHOP DOMAINS series is aimed at computer scientists and mathematicians alike who share an interest in the mathematical foundations of computation. It focusses on domains, their applications and related topics. The series was conceived and first realised by Klaus Keimel in Darmstadt in 1994. This seventh workshop returns to Darmstadt on the occasion of his 65th birthday. The following have declared their intention to participate: P.-L. Curien Univerity Paris 7 Yu.L. Ershov Sib. Branch of the Russian Academy of Science, Novosibirsk J.D. Lawson Louisiana State University M. Mislove Tulane University J.-E. Pin University Paris 7 D.S. Scott Carnegie Mellon University SCOPE Domain Theory has had applications to programming language semantics and logics (lambda calculus, PCF, LCF), recursion theory, general topology, topological algebra and analysis. As such Domain Theory is highly interdisciplinary. Part of the workshop will be devoted to the study of Continuous Phenomena in Computer Science, one of the themes of the APPLIED SEMANTICS II Network. Topics of interaction with Domain Theory for this workshop include, but are not limited to computation over the reals and other classical spaces probabilistic computation topology / locale theory constructive mathematics and its semantics computability theory program semantics program logics lambda calculus models of sequential computation LOCATION The Workshop will take place at Technische Universitaet Darmstadt located in the center of the city of Darmstadt. WORKSHOP SCHEDULE August 29 is the arrival day. There will be a barbecue in the afternoon or a dinner in the evening. The talks will take place from 9 a.m on Monday, August 30, until 5 p.m. on Wednesday, September 1. PARTICIPATION If you would like to participate in this workshop as we hope, please let us know: email: domains7@mathematik.tu-darmstadt.de Please indicate whether you would like to give a talk. SUBMISSION of ABSTRACTS One page abstracts should be submitted to domains7@mathematik.tu-darmstadt.de Shortly after an abstract is submitted (usually one or two weeks), the authors will be notified by the programme committee. Abstracts will be dealt with on a first come/first served basis. Submit as soon as possible, DEADLINE 30 June 2004 ACCOMODATION Participants will have to arrange accomodation by themselves in local hotels. There will be a certain number of hotel rooms made available through www.proregio-darmstadt.de. Details will be communicated later. FEES The Workshop is organized without institutional financial support. There will be a modest registration fee for covering expenses. PROGRAMME AND ORGANIZING COMMITTEE Achim Jung University of Birmingham Klaus Keimel Technische Universitaet Darmstadt Thomas Streicher Technische Universitaet Darmstadt From rrosebru@mta.ca Tue Mar 23 14:41:12 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Mar 2004 14:41:12 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B5qnn-0001lu-00 for categories-list@mta.ca; Tue, 23 Mar 2004 14:39:03 -0400 Date: Mon, 22 Mar 2004 18:37:07 -0800 (PST) From: Galchin Vasili Subject: categories: Cantor set/cantor dust and constructivism To: cat group MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 37 Hello, This question is a little bit afield; however, it is still tied in with intuitionistic logic. On my Linux machine, I have a screensaver that is a three-dimensional of a Cantor set. A Cantor set has prescription or algorithm as to how we "calculate" or build it. Question: From a constructivist viewpoint though, can we ever realize the actual object which is infinite? Regards, Bill Halchin From rrosebru@mta.ca Thu Mar 25 15:14:52 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 25 Mar 2004 15:14:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B6aIM-0004o0-00 for categories-list@mta.ca; Thu, 25 Mar 2004 15:13:38 -0400 X-Authentication-Warning: client122.comlab: lo owned process doing -bs Date: Thu, 25 Mar 2004 16:23:41 +0000 (GMT) From: Luke Ong To: categories@mta.ca Subject: categories: University Lectureship in Computer Science, Univ of Oxford Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=iso-8859-1 Content-Transfer-Encoding: QUOTED-PRINTABLE Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 39 =09=09=09UNIVERSITY OF OXFORD =09=09Mathematical and Physical Sciences Division =09=09=09Computing Laboratory =09=09in association with St John's College =09UNIVERSITY LECTURERSHIP IN COMPUTER SCIENCE Applications are invited for a University Lecturership in Computer Science from outstanding candidates working in any area of Computer Science. The successful candidate will be expected to engage in research of an internationally leading quality. He or she will be expected to lecture, supervise graduate students and perform other departmental duties under the direction of the Head of Department. The successful candidate will be appointed to a Tutorial Fellowship at St. John's College. As a Tutorial Fellow, the appointee will be expected to take a full part in the selection, academic progress and pastoral oversight of undergraduates reading Final Honours Schools involving Computer Science. The college requires a maximum of 6 hours teaching per week of full term (averaged over the three terms of the academic year) and performance of associated duties such as the pastoral care of Computer Science graduates. The combined University and College salary will be according to age on a scale up to =A342,900 (pay award pending). Further allowances may be payable from the college as detailed in its associated Further Particulars. Further Particulars of the post, including Selection Criteria and application method, together with more detailed descriptions of the Department and College duties are available at =09 www.comlab.ox.ac.uk/jobs by telephone request to +44 (0)1865 273831 or by email request to lecturership@comlab.ox.ac.uk. The closing date for receipt of applications is 21st April 2004. From rrosebru@mta.ca Thu Mar 25 15:14:52 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 25 Mar 2004 15:14:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B6aFh-0004V8-00 for categories-list@mta.ca; Thu, 25 Mar 2004 15:10:53 -0400 Date: Tue, 23 Mar 2004 14:50:00 -0500 (EST) From: Peter Freyd Message-Id: <200403231950.i2NJo0S8023836@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Re: Cantor set/cantor dust and constructivism Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 40 The question is: "From a constructivist viewpoint though, can we ever realize the actual object which is infinite?" Unless you specialize to decidable objects (excluded middle for equality) then there is no easy way to _avoid_ infinite objects. One may get a hint of this from the following example: in the category of N-sets (sets with a distinguished self-map) let A be the two-element set in which the distinguished self-map has a unique fixed point. In Cats and Allegories (1.925) we point out that A^A, the object of self-maps of A, falls apart as a coproduct (disjoint union) of continuously many N-sets. Hence if one takes the set of equivalence classes of A^A, where the equivalence relation is the double-negation of equality, one obtains a discrete set (it has trivial N-action) whose cardinality is the continuum. But that's just a hint. Consider the following theorem for topoi. Let T be the initial object in the category of topoi and logical morphisms. _No axiom of infinity._ Let 1 be the terminator in T. Let Omega be its power object. Let P be Omega^Omega, the object of self-maps on Omega. There's a subobject, M, of P on which we may define the structure of a rig (a ring without negation) that mimics the natural numbers. In particular the set of global sections, that is, maps from 1 to M with its inherited rig structure, is isomorphic to the standard natural numbers. Because the global sections can be misleading in an arbitrary topos it's worth pointing out that M mimics the natural numbers even from an internal point of view. For instance given any pair of polynomials, f(x_1,...,x_n), g(x_1,...,x_n) with natural-number coefficients the the first-order sentence \exists_{x_1,...,x_n\in M} f(x_1,...,x_n) = g(x_1,...,x_n) is satisfied in T iff it is satisfied in the standard natural numbers. An elaboration of this argument allows us to construct a topos-object in T whose topos of global sections is isomorphic to the free topos with natural-numbers-object. And, as for M, an equation in the theory of topoi is internally true for this internal topos iff it is true for the free topos with NNO. Lest one think I'm butting up against the consistency proof, the truth value in T of 0 = 1 in M is not totally false. (That is, M is "somewhat inconsistent" from the internal point of view.) From the external point of view, though, the two maps from the terminator to M that name 0 and 1 are not equal (from the internal point of view their equalizer is non-trivial). In particular we can find f,g so that the first-order sentence above has no standard solutions but is not totally false in T. The slogan: once you've given up excluded middle there's no gain in giving up the axiom of infinity. From rrosebru@mta.ca Fri Mar 26 12:54:38 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 26 Mar 2004 12:54:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B6uYp-0006NL-00 for categories-list@mta.ca; Fri, 26 Mar 2004 12:51:59 -0400 From: Thomas Streicher Message-Id: <200403261407.PAA29318@fb04209.mathematik.tu-darmstadt.de> Subject: categories: arithmetical and geometric reals in (models of) SDG To: categories@mta.ca Date: Fri, 26 Mar 2004 15:07:42 +0100 (CET) MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 41 Recently I was asking myself what is the relation between the arithmetical (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms (i.e. at least the Kock-Lawvere) axiom. >From the few things I know about models of SDG it seems to me as if in the usual sheaf models (over a site Loc of `loci') the Dedekind reals form a subring of R = y(\Re) (where \Re is the locus corresponding to the reals). As long as one considers just presheaves that's clear as the Dedekind reals are given by \Delta(\Re). Probably taking sheaves that situation isn't changed dramatically? So my impression is that in the usual sheaf models of SDG the real line R carries the structure of an algebra over the Dedekind reals. However, I don't see how to construct such an embedding of Dedekind reals into R based only on the usual axioms of SDG. Of course, when given an order on R and R is assumed as a Q-algebra then one has a good candidate for a function from R to R_D sending x \in R to {q \in Q | q.1 \leq x}. But how to define purely logically an embedding of R_D into R remains mysterious (to me). I am fully aware of the fact that my question is a bit `against the strain' of SDG but we know that both kinds of reals do coexist in topos models. But is this coexistence only peaceful or rather more collaborative? I am certain that people must have thought about this but I couldn't find anything at the usual places where to look. So I would be grateful for comments (on the correctness of my above speculations) or pointers to existing literature or folklore. Best, Thomas Streicher From rrosebru@mta.ca Fri Mar 26 12:54:38 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 26 Mar 2004 12:54:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B6uXA-0006Ee-00 for categories-list@mta.ca; Fri, 26 Mar 2004 12:50:16 -0400 Date: Fri, 26 Mar 2004 08:39:01 -0500 (EST) From: Peter Freyd Message-Id: <200403261339.i2QDd1h2017423@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Addendum Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 I wrote: [I]n the category of N-sets (sets with a distinguished self-map) let A be the two-element set in which the distinguished self-map has a unique fixed point....A^A, the object of self-maps of A, falls apart as a coproduct (disjoint union) of continuously many N-sets. Hence if one takes the set of equivalence classes of A^A, where the equivalence relation is the double-negation of equality, one obtains a discrete set (it has trivial N-action) whose cardinality is the continuum. It's more complicated. One obtains a disjoint union of "cycles", that is, objects of the form Z/nZ where the distinguished self-map is addition by 1. For each n > 0 there are only a finite number of copies of Z/nZ. The number of copies of Z = Z/0Z is uncountable. Postscript: The sequence of numbers of copies of each finite cycle is, of course, guaranteed to be in Neil Sloan's "On-Line Encyclopedia of Integer Sequences", but we didn't have that. Instead in 1.925 of Cats & Alligators we gave a ridiculous -- but both correct and unimprovable -- formula for that finite number. Preceding it was a sentence that begins "Bearing in mind that, as far as we know, this is computation for its own sake". Anyway, you can find the same formula at mathworld.wolfram.com/IrreduciblePolynomial.html under the name L_q(n) (with q = 2). With the exception of the zero'th value (in our case it's 2^{\Aleph_0}, for Lyndon words it's 1) here's what Sloan has to say: ID Number: A001037 (Formerly M0116 and N0046) URL: http://www.research.att.com/projects/OEIS?Anum=A001037 Sequence: 1,2,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710, 14532,27594,52377,99858,190557,364722,698870,1342176, 2580795,4971008,9586395,18512790,35790267,69273666, 134215680,260300986,505286415,981706806 Name: Degree n irreducible polynomials over GF(2); n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; binary Lyndon words of length n. Comments: Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence. References E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84. R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209. E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. M. A. Harrison, On the classification of Boolean functions by the general linear and affine groups, J. Soc. Indust. Appl. Math. 12 (1964) 285-299. M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79. G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol 4, no. 1, 1997, pp. 34-42, esp. p. 36. M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. G. Viennot, Algebres de Lie Libres et Monoides Libres, Lecture Notes in Mathematics 691, Springer verlag 1978. From rrosebru@mta.ca Mon Mar 29 09:37:39 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Mar 2004 09:37:39 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B7wsc-0003O9-00 for categories-list@mta.ca; Mon, 29 Mar 2004 09:32:42 -0400 Date: Sat, 27 Mar 2004 08:09:23 -0500 (EST) From: Peter Freyd Message-Id: <200403271309.i2RD9NkD027962@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Postscript^2 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 43 Cats & Alligators has just made it into Neil Sloane's "On-Line Encyclopedia of Integer Sequences"! Here's how the top of A001037 now reads: ID Number: A001037 (Formerly M0116 and N0046) URL: http://www.research.att.com/projects/OEIS?Anum=A001037 Sequence: 1,2,1,2,3,6,9,18,30,56,99,186,335,630,1161,2182,4080,7710, 14532,27594,52377,99858,190557,364722,698870,1342176, 2580795,4971008,9586395,18512790,35790267,69273666, 134215680,260300986,505286415,981706806 Name: Degree n irreducible polynomials over GF(2); n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; binary Lyndon words of length n. Comments: Also dimensions of free Lie algebras - see A059966, which is essentially the same sequence. References E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84. R. Church, Tables of irreducible polynomials for the first four prime moduli, Annals Math., 36 (1935), 198-209. P. J. Freyd and A. Scedrov, Categories, Allegories, North-Holland, Amsterdam, 1990. See 1.925. E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. M. A. Harrison, On the classification of Boolean functions by the general linear and affine groups, J. Soc. Indust. Appl. Math. 12 (1964) 285-299. M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79. G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol 4, no. 1, 1997, pp. 34-42, esp. p. 36. M. R. Nester, (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. G. Viennot, Algebres de Lie Libres et Monoides Libres, Lecture Notes in Mathematics 691, Springer verlag 1978. Postscript^3: For the record, this is the 8th item in which my name appears. The complete list: A000602 A000628 A000670 A001037 A067608 A067609 A067610 A067765 From rrosebru@mta.ca Wed Mar 31 11:53:57 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 31 Mar 2004 11:53:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B8hzk-0001TO-00 for categories-list@mta.ca; Wed, 31 Mar 2004 11:51:12 -0400 Message-ID: <40687CC2.5000606@doc.ic.ac.uk> Date: Mon, 29 Mar 2004 20:45:06 +0100 From: Concur 2004 User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.3) Gecko/20030312 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: CONCUR 2004: Final CFP - deadline approaching Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 44 Final Call for Papers for CONCUR 2004 [Apologies for multiple postings] The submission deadline for CONCUR 2004 is fast approaching: **** Friday 9 April 2004 **** New information since previous announcements: * David Harel (Weizmann Institute) will give an invited talk entitled "A Grand Challenge: Towards Full Reactive Modeling of a Multi-Cellular Animal" (see below for other invited speakers) * There will be a special issue of the journal Theoretical Computer Science associated with CONCUR 2004. * There will be a volume of Electronic Notes in Theoretical Computer Science associated with the workshops. ------------------------------------------------- CONCUR 2004, the 15th International Conference on Concurrency Theory, takes place at the Royal Society, London, Tuesday 31 August - Friday 3 September 2004. As well as the main event, we will also hold many workshops on Monday 30th August and Saturday 4th February, as detailed below. Further information is available at the conference's web site http://www.doc.ic.ac.uk/concur2004 Important dates Submission: Friday 9 April 2004 Notification: Monday 31 May 2004 Final version: Tuesday 15 June 2004 The purpose of the CONCUR conferences is to bring together researchers, developers and students in order to advance the theory of concurrency, and promote its applications. Interest in this topic is continuously growing, as a consequence of the importance and ubiquity of concurrent systems and their applications, and of the scientific relevance of their foundations. Submissions are solicited in all areas of semantics, logics and verification techniques for concurrent systems. Principal topics include (but are not limited to): - Basic models and logics of concurrent and distributed computation (such as process algebras, Petri nets, domain theoretic or game theoretic models, modal and temporal logics). - Specialized or enriched models (such as circuits, synchronous systems, real time and hybrid systems, stochastic systems, data bases, mobile and migrating systems, parametric protocols, security protocols). - Related verification techniques and tools (such as state-space exploration, model-checking, synthesis, abstraction, automated deduction, testing). - Related programming models (such as distributed, constraints or object oriented, graph rewriting, as well as associated type systems, static analyses, abstract machines, and environments). Authors are invited to submit an extended abstract not exceeding 15 pages electronically via the web submission form at the conference's web site. Submissions will be evaluated by the program committee for inclusion in the proceedings, which will be published by Springer-Verlag in the Lecture Notes in Computer Science series. Papers must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. Simultaneous submissions to other conferences are not allowed. There will be a special issue of the journal Theoretical Computer Science associated with Concur 2004. Invited speakers * David Harel (Weizmann Institute): Invited Talk entitled "A Grand Challenge: Towards Full Reactive Modeling of a Multi-Cellular Animal" * Sriram K. Rajamani (Microsoft Research): Invited Talk on Zing (a model-checking tool based on ideas from process algebra) * Peter O'Hearn (Queen Mary University of London) and Steve Brookes (Carnegie-Mellon): Tutorial on Local Reasoning about Concurrent Imperative Programs * Bengt Jonsson (Uppsala): Tutorial on Regular Model Checking Organisers * General chair: Philippa Gardner * Programme Committee co-chairs: Philippa Gardner, Nobuko Yoshida * Workshops organisers: Vladimiro Sassone, Julian Rathke * Local organisers: Iain Phillips, Sergio Maffeis, Alex Ahern Program committee Philippa Gardner, co-chair, UK Nobuko Yoshida, co-chair, UK Luca Aceto, Denmark Bruno Blanchet, Germany/France Steve Brookes, USA Luca De Alfaro, USA Paul Gastin, France Petr Jancar, Czech Republic Joost-Pieter Katoen, Netherlands Dietrich Kuske, Germany Cosimo Laneve, Italy Michael Mendler, Germany Ugo Montanari, Italy Catuscia Palamidessi, France Vladimiro Sassone, UK PS Thiagarajan, Singapore Antti Valmari, Finland Wang Yi, Sweden Concur 2004 gratefully acknowledges sponsorship from Microsoft Research. Monday workshops - Workshop on Structural Operational Semantics, SOS 2004 URL = http://www.cs.auc.dk/~luca/SOS-WORKSHOP Contact = Luca Aceto - 11th International Workshop in Expressiveness in Concurrency, EXPRESS 2004 URL = http://www.win.tue.nl/express04 Contact = Flavio Corradini - II Workshop on Object-Oriented Developments, WOOD 2004 URL = http://www.dsi.unive.it/wood2004 Contact = Viviana Bono - 3rd International Workshop on Foundations of Coordination Languages and Software Architectures, FOCLASA 2004 URL = http://www.info.fundp.ac.be/~jmj/Foclasa04 Contact = Jean-Marie JACQUET - 2nd International Workshop on Security Issues in Coordination Models, Languages and Systems, SECCO 2004 URL = http://www.cs.unibo.it/secco04/ Contact = Gianluigi Zavattaro - BIOCONCUR 2004 URL = http://www.imm.dtu.dk/bioconcur04 Contact = Anna Ingolfsdottir Friday afternoon + Saturday - Global Ubiquitous Computing URL = http://www.cogs.susx.ac.uk/fgc04 Contact = Julian Rathke Saturday workshops - 3rd International Workshop on Parallel and Distributed Methods in VerifiCation, PDMC 2004 URL = http://www.fi.muni.cz/~brim/PDMC04/ Contact = Martin Leucker - First International Workshop on Practical Applications of Stochastic Modelling, PASM 2004 URL = http://www.doc.ic.ac.uk/pasm2004 Contact = Jeremy Bradley - Infinity URL = http://www.lfcs.ed.ac.uk/Infinity04 Contact = Julian Bradfield - Fourth International Workshop on Automated Verification of Critical Systems, AVoCS 2004 URL = http://www.doc.ic.ac.uk/~mrh/AVoCS04.html Contact = Michael Huth * There will be a volume of Electronic Notes in Theoretical Computer Science associated with the workshops. From rrosebru@mta.ca Wed Mar 31 11:53:57 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 31 Mar 2004 11:53:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B8hz5-0001Jk-00 for categories-list@mta.ca; Wed, 31 Mar 2004 11:50:31 -0400 Date: Mon, 29 Mar 2004 11:03:30 -0500 (EST) From: Peter Freyd Message-Id: <200403291603.i2TG3UX6000400@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: on the axiom of infinity Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 45 Let *T* be the topos freely generated by its constants (a few postings ago I called it the initial object in the category of topoi and logical morphisms), let Omega be its subobject classifier and let P denote Omega^Omega. I wrote: There's a subobject, M, of P on which we may define the structure of a rig (a ring without negation) that mimics the natural numbers. In particular the set of global sections, that is, maps from 1 to M with its inherited rig structure, is isomorphic to the standard natural numbers... [Moreover we may] construct a topos-object in *T* whose topos of global sections is isomorphic to the free topos with NNO. And, as for M, an equation in the theory of topoi is internally true for this internal topos iff it is true for the free topos with NNO. It occurs to me that the proofs I have in mind rest on at least three considerations that may not be well known. (I have no idea about other proofs.) The first consideration stems from curiosity about the complete algebraic theory of Omega. In particular, what are its unary operators? Some would phrase that: What are the "modalities" implicit in higher-order intuitionistic logic? There's a particular operator that keeps popping up for me. In an arbitrary heyting algebra define x << y to mean that not only is x less than or equal to y, but the value of y -> x is as small as it can be, that is, y -> x = x. In a complete heyting algebra define an order-preserving, inflationary unary operation s by sx = inf{ y | x << y }. E.g.: on a linearly ordered set if { y | x < y } has a least element then that's what sx is. If there is no smallest element above x, then sx = x (even without the completeness hypothesis). In particular, note, there no assertion that x << sx. The subobject classifier in an elementary topos is complete in the relevant sense: s is definable. A quick description of the construction to follow is that we're going to turn s into the successor operation on an NNO. DIVERSION: The definition I just gave is the first I came across. The next incarnation for me was when I wanted a measure of the failure of booleaness. In any topos, *A*, there's a largest subterminator B with the property that the slice category *A*/B is boolean. But given any subterminator, U, we have its "closed sheaves", *A*_(U), the full subcat of objects A such that AxU --> U is an iso. (This is a subcategory of sheaves for a Lawvere-Tierney topology. Starting with a space X then Sheaves(X)_(U) may be identified with Sheaves(U'), where U' denotes the complement of U.) Note that the lattice of subterminators in *A*_(U) is isomorphic to the interval of subterminators in *A* from U up. We can define BU to be the largest subterminator in *A*_(U) such that *A*_(U)/BU is boolean. The interval of subterminators in *A* from U up to BU is boolean and in the relevant internal sense, BU is the largest such subterminator. We can, of course, translate this all to a unary operation on Omega. It's the same operator s. When one specializes this to a space X it becomes historically familiar if we dualize it it to a deflationary operator on closed subsets. It's the operation that removes isolated points. The very operation that got Cantor started. Hence the word "historically". END OF DIVERSION Back to *T*: in the monoid Omega^Omega define C to be the submonoid generated by s. We're going to use 0 : 1 --> C to denote the name of the identity element (you might think of it as s^0) and we're going to use suc : C -> C to denote the function that sends an element x to sx. It's easy to verify that the necessary and sufficient condition for C,0,suc to be an NNO is that -- in the internal sense -- it has no fixed points. The argument works for any one-generator partially- ordered monoid whose identity element is below the generator. Let E be the equalizer of 1, suc : C --> C. I'm saying that C,0,suc is an NNO iff E is empty. Let V be the negation of the image of E --> 1. It is the largest subterminator such that C,0,suc becomes an NNO in the slice category *T*/V. Now for the nice part. The sconing argument tells us that V, as is the case for any non-trivial negation, is an indecomposable projective -- that is, the covariant functor represented by V is an exact functor. And that means that 1 is an indecomposable projective in the slice category *T*/V. The "global-section" functor from *T*/V to *Sets* is exact. Hence (by my first theorem in this subject) it preserves the NNO and all the definable structure thereon. I advertised an object, not in a slice category but in *T*. So define M = C^V. Then it is easily verified that the global sections of M as defined in *T* are naturally isomorphic to the global sections of C/V as defined in *T*/V. Since it has an NNO we can construct all sorts of things in *T*/V. We'll settle here for the free-topos-object-with-NNO. That is, there is an object T in *T*/V with a binary partial operation that serves as the free topos with NNO in *T*/V. Since the global-section functor is exact we know that it preserves free structures in great generality (part of lore of exact functors between topoi). In particular it carries T to a standard free topos with NNO .And, finally, we know that T^V has that same last property as an object in *T*. (Better, actually, is to take T to be the free topos on one object. Every finitely presented topos is a slice thereof -- including the free-topos-with-NNO.) Surely this is all too easy. Yeah. I skipped over something. My point about negations being indecomposable projectives requires them to be non-trivial. (The covariant functor represented by 0 is not exact; it does not preserve the coterminator.) That is, we need to show that V is not 0. But for that it suffices (given the defining universal property for *T*) to show in at least one topos that the recipe for V yields something nontrivial. Equivalently, we need a topos in which V = 1. The most easily named example is the topos of N-sets, or as I described it a few postings ago, the topos of sets with distinguished self-maps. This topos is two-valued, that is, Sub(1) has just two elements and that can be an advantage. (We know that we must show that suc : C --> C has no fixed points.) In fact, I find it easier to work in the slice category (*Sets*^N)/N. (There is, anyway, a faithful representation of *Sets*^N into (*Sets*^N)/N.) The category (*Sets*^N)/N is equivalent to the category of covariant set-valued functors on the poset also denoted N. The advantage here is just the opposite: (*Sets*^N)/N is generated by its non-zero subterminators and they're all indecomposable projectives. DIVERSION: When I first finished writing this piece I noticed for the first time that there's a historically much better example. Take the category of sheaves on the reals (or for that matter, any space of geometric interest). Then follow in Cantor's footsteps. FOOTNOTE: One variation worth mentioning. Instead of taking C as the submonoid generated by s take it to be the smallest up-closed submonoid, that is, this new version of C will be closed with respect to sups. This C is trying to be a big ordinal number. Indeed, following just the path first taken by Cantor we can find Grothendieck topoi where C looks like an arbitrarily large initial section of ordinals. In the free topos with NNO this C seems to be none other than the ordinal (in some relevant sense) of all recursively enumerable ordinals. From rrosebru@mta.ca Wed Mar 31 11:53:57 2004 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 31 Mar 2004 11:53:57 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B8i0o-0001fE-00 for categories-list@mta.ca; Wed, 31 Mar 2004 11:52:18 -0400 Date: Wed, 31 Mar 2004 07:55:38 +0100 Message-Id: Subject: categories: New paper on Profunctors, Open Maps and Bisimulation MIME-Version: 1.0 X-Sensitivity: 3 Content-Type: text/plain; charset=iso-8859-1 From: "gianluca\.cattani" To: "categories" Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 46 This is to announce the availability of the following paper Profunctors, Open Maps and Bisimulation by Gian Luca Cattani and Glynn Winskel. The paper can be downloaded in postscript version from http://www.cl.cam.ac.uk/~gw104. Regards, Gian Luca Cattani