December 2015 - Categories - categories.org.au

type theory postdoc at Dalhousie University
by Peter Selinger 22 Jan '16

22 Jan '16

Dear colleagues, I invite applications for one postdoctoral position, starting between now and January 1, at Dalhousie University under my supervision. The successful applicant will work on a project entitled "Trusted Quantum Software via a Formally Verified Functional Quantum Programming Language". Specifically, the project will involve the design of a type system for a type-safe functional programming language for quantum computing, loosely modelled on the Quipper language (http://www.mathstat.dal.ca/~selinger/quipper/) It will also involve developing the meta-theory (including categorical semantics) of the language, and eventually the formalization of some of this meta-theory in a proof assistant. Familiarity with type theory (especially dependent type theory), programming language design, and/or semantics will be a prerequisite for this postdoc. Familiarity with quantum computing will be helpful, but is neither necessary nor sufficient for this position - the main emphasis is on programming languages and type systems. Although the position will be held at Dalhousie University in Canada, it will be an asset if the candidate is able to travel, because I will be spending one semester in Germany and one semester in the U.S. next year, and ideally I would like the postdoc to be able to accompany me. Funding for the project comes from the U.S. Air Force Office of Scientific Research, and the research project will be part of a team effort also involving collaborators from Tulane, Stanford, Oxford, the University of Iowa, and the University of Pennsylvania. The nominal start date for the project is September 30, 2015 (yes, this is today!), so the start date can be immediately, or some time in the near future. The position is initially for 1 year, and can be extended for an additional year. The salary is CAD $60,000 per year. Interested applicants should contact Peter Selinger at selinger(a)mathstat.dal.ca as soon as possible, and in any case before October 21. I can provide more details about the research project to interested applicants on request. Thanks, -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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a paper announcement
by Vladimir Voevodsky 31 Dec '15

31 Dec '15

Hello, I have posted on the arXiv a paper that I think might be of interest to many people on this list. It is about the connection between Lawvere theories and Cartmell’s contextual categories (called C-systems in the paper). Here is the link http://arxiv.org/abs/1512.08104 <http://arxiv.org/abs/1512.08104>. Vladimir. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Review of the book: Mathematics Without Apologies, NY Review
by Bob Rosebrugh 30 Dec '15

30 Dec '15

Mike Barr's post was transmitted badly, he offers to send a pdf to anyone interested. His address is: Michael Barr <barr(a)math.mcgill.ca> This is also an opportunity for me to remind you of the list policy from the list web page at: http://www.mta.ca/~cat-dist/ "The subject of this moderated mailing list is category theory. Items on another subject may be posted if judged by the moderator to be of interest to subscribers. Discussion of such posts is excluded. The list is to be used for concise discussion that is suitable for email distribution. Posts are not transmitted if, in the moderator's judgement, they are unsuitable for distribution. Postings to the categories list will be courteous and respectful. Expression of disagreement must be polite. A submission judged by the moderator to be offensive or defamatory will not be posted. Submissions judged to be related such material will not be posted. The categories mailing list distributes only text messages. Do not send attachments, messages containing html or excessively long postings (longer than about 10K). The default setting for some email clients is to compose messages containing html. This setting is easy to change. Postings with attachments or with html will be discarded." Best to all for 2016, Bob Rosebrugh [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Review of the book: Mathematics Without Apologies, NY Review=0A=
by Michael Barr 28 Dec '15

28 Dec '15

=0A= =0A= [Note from moderator: This is the full text of the book review discussed in= November.]=0A= =0A= This looks especially interesting to category theorists. The commented out= =3D=0A= stuff was graphical, including a photo of Grothendieck.=0A= =0A= =0A= In the Mountains of Mathematics=0A= Jim Holt=0A= =0A= December 3, 2015 Issue=0A= =0A= Mathematics Without Apologies: Portrait of a Problematic Vocation=0A= by Michael Harris=0A= Princeton University Press, 438 pp., $29.95=0A= =0A= =0A= =0A= =0A= The irony of pure mathematics begetting crass commercialism is not lost=0A= on Michael Harris, whose Mathematics Without Apologies irreverently=0A= echoes Hardy=3DE2=3D80=3D99s classic title. Harris is a distinguished midd= le-age=3D=0A= d=0A= American mathematician who works in the gloriously pure stratosphere=0A= where algebra, geometry, and number theory meet. =3DE2=3D80=3D9CThe guiding= probl=3D=0A= em=0A= for the first part of my career,=3DE2=3D80=3D9D he writes, was =3DE2=3D80= =3D9Cthe Conje=3D=0A= cture of=0A= Birch and Swinnerton-Dyer,=3DE2=3D80=3D9D which =3DE2=3D80=3D9Cconcerns the= simplest cl=3D=0A= ass of=0A= polynomial equations=3DE2=3D80=3D94elliptic curves=3DE2=3D80=3D94for which = there is no =3D=0A= simple way to=0A= decide whether the number of solutions is finite or infinite.=3DE2=3D80=3D9= D One =3D=0A= such=0A= elliptic curve is y2 =3D3D x3 - 25x, which can be shown to have infinitely= =0A= many rational solutions (that is, solutions that are whole numbers or=0A= fractions); another is y2 =3D3D x3 - x, which has only finitely many=0A= solutions. Though elementary in appearance, elliptic curves turn out to=0A= have a deep structure that makes them endlessly interesting.=0A= =0A= Harris has spent much of his research career in Paris, and it shows:=0A= his book is full of Gallic intellectual playfulness, plus references to=0A= figures like Pierre Bourdieu, Issey Miyake, and Catherine Millet (=3DE2=3D8= 0=3D9C=3D=0A= the=0A= sexual Stakhanovite=3DE2=3D80=3D9D), and mention of the endless round of Pa= risian=0A= champagne receptions where =3DE2=3D80=3D9Cmathematical notes are compared f= or the=0A= first glass or two, after which conversation reverts to university=0A= politics and gossip.=3DE2=3D80=3D9D It is rambling, sardonic (the term =3DE= 2=3D80=3D9Cf=3D=0A= uck-you=0A= money=3DE2=3D80=3D9D appears in the index), and witty. It contains fascina= ting=0A= literary digressions, such as an analysis of the occult mathematical=0A= structure of Thomas Pynchon=3DE2=3D80=3D99s novels, and lovely little inter= ludes =3D=0A= on=0A= elementary math, inspired by Harris=3DE2=3D80=3D99s gallant attempt to expl= ain nu=3D=0A= mber=0A= theory to a British actress at a Manhattan dinner party.=0A= =0A= Starting with the simple definition of a prime number, he builds, bit by=0A= bit, to an explanation of the aforementioned Birch-Swinnerton-Dyer=0A= conjecture=3DE2=3D80=3D94which, at a press conference given in Paris in the= year =3D=0A= 2000=0A= by an international group of leading mathematicians, was declared to be=0A= one of seven =3DE2=3D80=3D9CMillennium Prize Problems=3DE2=3D80=3D9D whose = resolution w=3D=0A= ould be=0A= rewarded by a million-dollar prize. The book takes an intimate look at=0A= the deepest developments in contemporary mathematics, especially the=0A= visionary work of the recently deceased Alexander Grothendieck. And it=0A= successfully conveys what Harris calls the =3DE2=3D80=3D9Cpathos=3DE2=3D80= =3D9D of the=0A= mathematician=3DE2=3D80=3D99s calling.=0A= =0A= Harris is rudely skeptical of the usual justifications for pure=0A= mathematics: that it is beautiful, true, or even much good, at least in=0A= the utilitarian =3DE2=3D80=3D9CGolden Goose=3DE2=3D80=3D9D sense. =3DE2=3D8= 0=3D9CIt is not on=3D=0A= ly dishonest but also=0A= self-defeating to pretend that research in pure mathematics is motivated=0A= by potential applications,=3DE2=3D80=3D9D he writes. He notes that public-= key=0A= cryptography, by making the world safe for Amazon, has destroyed the=0A= corner bookstore (in America, not France, where online retailers are=0A= prohibited by law from offering free shipping on discounted books). And=0A= he displays an olympian scorn for the sudden popularity of =3DE2=3D80=3D9Cf= inance=0A= mathematics,=3DE2=3D80=3D9D which offers a path to derivative-fueled wealth= on Wa=3D=0A= ll=0A= Street:=0A= =0A= A colleague boasted that Columbia=3DE2=3D80=3D99s mathematical finance = progra=3D=0A= m was=0A= underwriting the lavish daily spreads of fresh fruit, cheese, and=0A= chocolate brownies, when other departments, including mine in Paris,=0A= were lucky to offer a few teabags and a handful of cookies to=0A= calorie-starved graduate students.=0A= =0A= Even at France=3DE2=3D80=3D99s elite =3DC3=3D89cole Polytechnique, 70 perce= nt of the=0A= mathematics students today aspire to a career in finance.=0A= =0A= Nor is Harris impressed with the claim, voiced by Hardy and so many=0A= others, that pure mathematics is justified by its beauty. When=0A= mathematicians talk about beauty, he tells us, what they really mean is=0A= pleasure. =3DE2=3D80=3D9COutside this relaxed field, it=3DE2=3D80=3D99s con= sidered poor=3D=0A= form to=0A= admit that we are motivated by pleasure,=3DE2=3D80=3D9D Harris writes. =3DE= 2=3D80=3D9CA=3D=0A= esthetics is=0A= a way of reconciling this motivation with the =3DE2=3D80=3D98lofty habit of= mind.=3D=0A= =3DE2=3D80=3D99=3DE2=3D80=3D9D=0A= =0A= Why should society pay for a small group of people to exercise their=0A= creative powers on something they enjoy? =3DE2=3D80=3D9CIf a government min= ister =3D=0A= asked=0A= me that question,=3DE2=3D80=3D9D Harris writes,=0A= =0A= I could claim that mathematicians, like other academics, are needed=0A= in the universities to teach a specific population of students the=0A= skills needed for the development of a technological society and to keep=0A= a somewhat broader population of students occupied with courses that=0A= serve to crush the dreams of superfluous applicants to particularly=0A= desirable professions (as freshman calculus used to be a formal=0A= requirement to enter medical school in the United States).=0A= =0A= Although physicians don=3DE2=3D80=3D99t really need calculus, Harris at lea= st con=3D=0A= cedes=0A= that engineers, economists, and inventory managers couldn=3DE2=3D80=3D99t g= et by=0A= without a fair amount of math, even if it is trivial math by his lights.=0A= =0A= Finally, there is the presumed value of mathematical truth. Since the=0A= ancient Greeks, mathematics has been taken as a paradigm of knowledge:=0A= certain, timeless, necessary. But knowledge of what? Do the truths=0A= discovered by mathematics describe an eternal and otherworldly realm of=0A= objects=3DE2=3D80=3D94perfect circles and so forth=3DE2=3D80=3D94that exist= quite indep=3D=0A= endently of=0A= the mathematicians who contemplate them? Or are mathematical objects=0A= actually human constructions, existing only in our minds? Or, more=0A= radically still, could it be that pure mathematics doesn=3DE2=3D80=3D99t re= ally=0A= describe any objects at all, that it is just an elaborate game of formal=0A= symbols played with pencil and paper?=0A= =0A= The question of what mathematics is really about is one that continues=0A= to vex philosophers, but it does not much worry Harris. Philosophers=0A= who concern themselves with the problems of mathematical existence and=0A= truth, he claims, typically pay little attention to what mathematicians=0A= actually do. He invidiously contrasts what he calls =3DE2=3D80=3D9Cphiloso= phy of=0A= Mathematics=3DE2=3D80=3D9D (with a capital M)=3DE2=3D80=3D94=3DE2=3D80=3D9C= a purely hypotheti=3D=0A= cal subject invented=0A= by philosophers=3DE2=3D80=3D9D=3DE2=3D80=3D94with =3DE2=3D80=3D9Cphilosophy= of (small-m) math=3D=0A= ematics,=3DE2=3D80=3D9D which takes=0A= as its starting point not a priori questions about epistemology and=0A= ontology, but rather the activity of working mathematicians.=0A= =0A= Here, Harris is being a little unfair. He fails to remark that the=0A= standard competing positions in the philosophy of mathematics were=0A= originally staked out not by philosophers, but by mathematicians=3DE2=3D80= =3D94in=3D=0A= deed,=0A= some of the greatest of the last century. It was David Hilbert=3DE2=3D80= =3D94a=0A= =3DE2=3D80=3D9Csupergiant,=3DE2=3D80=3D9D in Harris=3DE2=3D80=3D99s estimat= ion=3DE2=3D80=3D94who or=3D=0A= iginated =3DE2=3D80=3D9Cformalism,=3DE2=3D80=3D9D which=0A= views higher mathematics as a game played with formal symbols. Henri=0A= Poincar=3DC3=3DA9 (another =3DE2=3D80=3D9Csupergiant=3DE2=3D80=3D9D), Herma= nn Weyl, and L.E=3D=0A= .J. Brouwer were=0A= behind =3DE2=3D80=3D9Cintuitionism,=3DE2=3D80=3D9D according to which numbe= rs and other=3D=0A= mathematical=0A= objects are mind-dependent constructions. Bertrand Russell and Alfred=0A= North Whitehead took the position known as =3DE2=3D80=3D9Clogicism,=3DE2=3D= 80=3D9D ende=3D=0A= avoring to=0A= show in their massive Principia Mathematica that mathematics was really=0A= logic in disguise. And =3DE2=3D80=3D9Cplatonism=3DE2=3D80=3D9D=3DE2=3D80= =3D94the idea that m=3D=0A= athematics describes=0A= a perfect and eternal realm of mind-independent objects, like Plato=3DE2=3D= 80=3D=0A= =3D99s=0A= world of Forms=3DE2=3D80=3D94was championed by Kurt G=3DC3=3DB6del.=0A= =0A= All of these mathematical figures were passionately engaged in what=0A= Harris slights as philosophy of Mathematics-with-a-capital-M. The=0A= debate among them and their partisans was fierce in the 1920s, often=0A= spilling over into personal animus. And no wonder: mathematics at the=0A= time was undergoing a =3DE2=3D80=3D9Ccrisis=3DE2=3D80=3D9D that had resulte= d from a ser=3D=0A= ies of=0A= confidence-shaking developments, like the emergence of non-Euclidean=0A= geometries and the discovery of paradoxes in set theory. If the old=0A= ideal of certainty was to be salvaged, it was felt, mathematics had to=0A= be put on a new and secure foundation. At issue was the very way=0A= mathematics would be practiced: what types of proof would be accepted=0A= as valid, what uses of infinity would be permitted.=0A= =0A= For reasons both technical and philosophical, none of the competing=0A= foundational programs of the early twentieth century proved=0A= satisfactory. (G=3DC3=3DB6del=3DE2=3D80=3D99s =3DE2=3D80=3D9Cincompletenes= s theorems,=3DE2=3D=0A= =3D80=3D9D in particular,=0A= created insuperable problems both for Hilbert=3DE2=3D80=3D99s formalism and= for=0A= Russell and Whitehead=3DE2=3D80=3D99s logicism: they showed=3DE2=3D80=3D94= roughly spea=3D=0A= king=3DE2=3D80=3D94that the=0A= rules of Hilbert=3DE2=3D80=3D99s mathematical =3DE2=3D80=3D9Cgame=3DE2=3D80= =3D9D could never =3D=0A= be proved consistent,=0A= and that a logical system like that of Russell and Whitehead could never=0A= capture all mathematical truths.) The issues of mathematical existence=0A= and truth remain unresolved, and philosophers have continued to grapple=0A= with them, if inconclusively=3DE2=3D80=3D94as witness the frank title that = Hilary=0A= Putnam gave to a 1979 paper: =3DE2=3D80=3D9CPhilosophy of Mathematics: Wh= y Noth=3D=0A= ing=0A= Works.=3DE2=3D80=3D9D=0A= =0A= To Harris, this looks a bit vieux jeu. The sense of crisis in the=0A= profession, so acute less than a century ago, has receded; the old=0A= difficulties have been patched up or papered over. If you ask a=0A= contemporary mathematician to declare a philosophical party affiliation,=0A= the joke goes, you=3DE2=3D80=3D99ll hear =3DE2=3D80=3D9Cplatonist=3DE2=3D80= =3D9D on weekdays =3D=0A= and =3DE2=3D80=3D9Cformalist=3DE2=3D80=3D9D on=0A= Sundays: that is, when they=3DE2=3D80=3D99re working at mathematics, mathe= matici=3D=0A= ans=0A= regard it as being about a mind-independent reality; but when they=3DE2=3D8= 0=3D99=3D=0A= re in=0A= a reflective mood, many claim to believe that it=3DE2=3D80=3D99s just a mea= ningle=3D=0A= ss=0A= game played with formal symbols.=0A= =0A= Today, as Harris observes, paradigm shifts in mathematics have less to=0A= do with =3DE2=3D80=3D9Ccrisis=3DE2=3D80=3D9D and more to do with finding su= perior metho=3D=0A= ds. It used=0A= to be thought, for example, that all mathematics could be constructed=0A= out of sets. Starting with the simple idea of one thing being a member=0A= of another, set theory shows how structures of seemingly limitless=0A= complexity=3DE2=3D80=3D94number systems, geometrical spaces, a never-ending= hiera=3D=0A= rchy=0A= of infinities=3DE2=3D80=3D94can be built up out of the most modest material= s. Th=3D=0A= e=0A= number zero, for example, can be defined as the =3DE2=3D80=3D9Cempty set=3D= E2=3D80=3D9D=3D=0A= : that is,=0A= the set that has no members at all. The number one can then be defined=0A= as the set that contains one element=3DE2=3D80=3D94zero and nothing else. = Two, i=3D=0A= n=0A= turn, can be defined as the set that contains zero and one=3DE2=3D80=3D94an= d so o=3D=0A= n,=0A= with the set for each subsequent number containing the sets for all the=0A= previous numbers. Numbers, instead of being taken as basic, can thus be=0A= viewed as pure sets of increasingly intricate structure.=0A= =0A= % holt_2-120315.jpg Cath=3DC3=3DA9rine Goldstein =3DC2=3DA9 2015 Artists R= ights So=3D=0A= ciety=0A= % (ARS), New York/ADAGP, Paris A mathematical formula for love by Isidore= =0A= % Isou, 1988; from Michael Harris=3DE2=3D80=3D99s Mathematics Without Apol= ogies=0A= =0A= In the 1930s, a cabal of brilliant young Paris mathematicians, including=0A= Andr=3DC3=3DA9 Weil, resolved to make the house of mathematics more secure = by=0A= rebuilding it on the logical foundation of set theory. The project,=0A= under the collective nom de guerre =3DE2=3D80=3D9CBourbaki,=3DE2=3D80=3D9D = went on for =3D=0A= decades,=0A= resulting in one fat treatise after another. Among its consequences,=0A= crazily enough, was the advent of the =3DE2=3D80=3D9Cnew math=3DE2=3D80=3D9= D education =3D=0A= reforms back=0A= in the 1960s, which so befuddled American schoolchildren and their=0A= parents by replacing intuitive talk of numbers with the alien jargon of=0A= sets.=0A= =0A= Physicists talk about finding the =3DE2=3D80=3D9Ctheory of everything=3DE2= =3D80=3D9D; w=3D=0A= ell, set=0A= theory is so sweeping in its generality that it might appear to be (as=0A= Harris quips) =3DE2=3D80=3D9Cthe theory of theories of everything.=3DE2=3D8= 0=3D9D It ce=3D=0A= rtainly=0A= appeared that way to the members of Bourbaki. Yet a few decades after=0A= their program got underway, the extraordinary Alexander Grothendieck=0A= came into their midst and transcended it. In doing so, he created a new=0A= style of pure mathematics that proved as fruitful as it was dizzyingly=0A= abstract. Long before his death last November at the age of eighty-six=0A= in a remote hamlet in the Pyrenees, Grothendieck had come to be regarded=0A= as the greatest mathematician of the last half-century. As Harris=0A= observes, he likely qualifies as the =3DE2=3D80=3D9Cmost romantic,=3DE2=3D8= 0=3D9D too: =3D=0A= =3DE2=3D80=3D9Chis life=0A= story begs for fictional treatment.=3DE2=3D80=3D9D=0A= =0A= The raw facts are astounding enough. Alexander Grothendieck was born in=0A= Berlin in 1928 to parents who were both active anarchists. His father,=0A= a Russian Jew, took part in the 1905 uprising against the tsarist regime=0A= and the 1917 revolution. He escaped imprisonment under the Bolsheviks;=0A= clashed with Nazi thugs on the streets of Berlin; fought on the=0A= Republican side in the Spanish civil war (as did Grothendieck=3DE2=3D80=3D9= 9s mot=3D=0A= her);=0A= and was deported, after the fall of France, from Paris to Auschwitz to=0A= be murdered.=0A= =0A= His mother, a gentile from Hamburg, raised Grothendieck in the south of=0A= France. There the boy showed talent both for numbers and for boxing.=0A= After the war, he made his way to Paris to study mathematics under the=0A= great Henri Cartan. Following early teaching stints in S=3DC3=3DA3o Paulo,= =0A= Kansas, and Harvard, Grothendieck was invited in 1958 to join the=0A= Institut des Hautes =3DC3=3D89tudes Scientifiques, which had just been foun= ded =3D=0A= by=0A= a private businessman outside Paris in the woods of Bois-Marie. There=0A= Grothendieck spent the next dozen years astonishing his elite colleagues=0A= and younger disciples as he recreated the landscape of higher=0A= mathematics.=0A= =0A= % Grothendieck was a physically imposing man, shaven-headed and handsome, = =3D=0A= as charismatic as he was austere. A staunch pacifist and antimilitarist, he= =3D=0A= refused to go to Moscow in 1966 (where the International Congress of Mathe= =3D=0A= matics was being held) to accept the Fields Medal, the highest honor in mat= =3D=0A= hematics. He did, however, make a trip the next year to North Vietnam, wher= =3D=0A= e he lectured on pure mathematics in the jungle to students who had been ev= =3D=0A= acuated from Hanoi to escape the American bombing. He remained (by choice) = =3D=0A= stateless most of his life, had three children by a woman he married and tw= =3D=0A= o more out of wedlock, founded the radical ecology group Survivre et Vivre,= =3D=0A= and once got arrested for knocking down a couple of gendarmes at a politic= =3D=0A= al demonstration in Avignon.=0A= % holt_3-120315.jpg=0A= % Erika Ifang=0A= % Alexander Grothendieck, 1988=0A= =0A= Owing to his unyielding and sometimes paranoiac sense of integrity,=0A= Grothendieck ended up alienating himself from the French mathematical=0A= establishment. In the early 1990s he vanished into the Pyrenees=3DE2=3D80= =3D94wh=3D=0A= ere,=0A= it was reported by the handful of admirers who managed to track him=0A= down, he spent his remaining years subsisting on dandelion soup and=0A= meditating on how a malign metaphysical force was destroying the divine=0A= harmony of the world, possibly by slightly altering the speed of light.=0A= The local villagers were said to look after him.=0A= =0A= Grothendieck=3DE2=3D80=3D99s vision of mathematics led him to develop a new= =0A= language=3DE2=3D80=3D94it might even be called an =3DE2=3D80=3D9Cideology= =3DE2=3D80=3D9D=3DE2=3D=0A= =3D80=3D94in which hitherto=0A= unimaginable ideas could be expressed. He was the first, Harris=0A= observes (breaking into emphatic boldface), =3DE2=3D80=3D9Cto be guided by = the=0A= principle that knowing a mathematical object is tantamount to knowing=0A= its relations to all other objects of the same kind.=3DE2=3D80=3D9D In othe= r word=3D=0A= s, if=0A= you want to know the real nature of a mathematical object, don=3DE2=3D80=3D= 99t lo=3D=0A= ok=0A= inside of it, but see how it plays with its peers.=0A= =0A= Such a peer group of mathematical objects is called, in a deliberate nod=0A= to Aristotle and Kant, a =3DE2=3D80=3D9Ccategory.=3DE2=3D80=3D9D One catego= ry might con=3D=0A= sist of=0A= abstract surfaces. These surfaces play together, in the sense that=0A= there are natural ways of going back and forth between them that respect=0A= their general form. For example, if two surfaces have the same number=0A= of holes=3DE2=3D80=3D94like a donut and a coffee mug=3DE2=3D80=3D94one surf= ace can, mat=3D=0A= hematically,=0A= be smoothly transformed into the other.=0A= =0A= Another category might consist of all the different algebraic systems=0A= that have an operation akin to multiplication; these algebras too play=0A= together, in the sense that there are natural ways of going back and=0A= forth between them that respect their common multiplicative structure.=0A= Such structure-preserving back-and-forth relations among objects in the=0A= same category are called =3DE2=3D80=3D9Cmorphisms,=3DE2=3D80=3D9D or someti= mes=3DE2=3D80=3D94=3D=0A= to stress their=0A= abstract nature=3DE2=3D80=3D94=3DE2=3D80=3D9Carrows.=3DE2=3D80=3D9D They de= termine the overal=3D=0A= l shape of the play=0A= within a category.=0A= =0A= And here=3DE2=3D80=3D99s where it gets interesting: the play in one catego= ry=3DE2=3D=0A= =3D80=3D94say, the=0A= category of surfaces=3DE2=3D80=3D94might be subtly mimicked by the play in= =0A= another=3DE2=3D80=3D94say, the category of algebras. The two categories th= emselv=3D=0A= es=0A= can be seen to play together: there is a natural way of going back and=0A= forth between them, called a =3DE2=3D80=3D9Cfunctor.=3DE2=3D80=3D9D Armed w= ith such a f=3D=0A= unctor, one=0A= can reason quite generally about both categories, without getting bogged=0A= down in the particular details of each. It might also be noticed that,=0A= since categories play with one another, they themselves form a category:=0A= the category of categories.=0A= =0A= Category theory was invented in the 1940s by Saunders Mac Lane of the=0A= University of Chicago and Samuel Eilenberg of Columbia. At first it was=0A= regarded dubiously by many mathematicians, earning the nickname=0A= =3DE2=3D80=3D9Cabstract nonsense.=3DE2=3D80=3D9D How could such a rarefied = approach to =3D=0A= mathematics,=0A= in which nearly all its classical content seemed to be drained away,=0A= result in anything but sterility? Yet Grothendieck made it sing.=0A= Between 1958 and 1970, he used category theory to create novel=0A= structures of unexampled richness. Since then, the heady abstractions=0A= of category theory have also become useful in theoretical physics,=0A= computer science, logic, and philosophy.*=0A= =0A= The project undertaken by Grothendieck was one that began with=0A= Descartes: the unification of geometry and algebra. These have been=0A= likened to the yin and yang of mathematics: geometry is space, algebra=0A= is time; geometry is like painting, algebra is like music; and so on.=0A= Less fancifully, geometry is about form, algebra is about structure=3DE2=3D= 80=3D=0A= =3D94in=0A= particular, the structure that lurks within equations. And as Descartes=0A= showed with his invention of =3DE2=3D80=3D9CCartesian coordinates,=3DE2=3D8= 0=3D9D equat=3D=0A= ions can=0A= describe forms: x2 + y2 =3D3D 1, for example, describes a circle of radius= =0A= 1. So algebra and geometry turn out to be intimately related, exchanging=0A= what Andr=3DC3=3DA9 Weil called =3DE2=3D80=3D9Csubtle caresses.=3DE2=3D80= =3D9D=0A= =0A= In the 1940s, thanks to Weil=3DE2=3D80=3D99s insight, it became apparent th= at the=0A= dialectic between geometry and algebra was the key to resolving some of=0A= the most stubbornly enduring mysteries in mathematics. And it was=0A= Grothendieck=3DE2=3D80=3D99s labor that raised this dialectic to such a pit= ch of=0A= abstraction=3DE2=3D80=3D94one that was said to leave even the great Weil da= unted=3D=0A= =3DE2=3D80=3D94that=0A= a new understanding of these mysteries emerged. Grothendieck laid the=0A= groundwork for many of the greatest mathematical advances in recent=0A= decades, including the 1994 proof of Fermat=3DE2=3D80=3D99s Last Theorem=3D= E2=3D80=3D94=3D=0A= a magnificent=0A= intellectual achievement of zero practical or commercial interest.=0A= =0A= Harris, whose own impressive work is much indebted to Grothendieck, is=0A= eloquent in praising his =3DE2=3D80=3D9Cruthless=3DE2=3D80=3DA6 minimalism,= =3DE2=3D80=3D9D wh=3D=0A= ich extended to his=0A= contempt for money and his monkish wardrobe. I only wish that Harris=0A= had more conspicuously credited another figure in the modern=0A= transformation of mathematics, Emmy Noether. It was Noether, born in=0A= Bavaria in 1882, who largely created the abstract approach that inspired=0A= category theory. Yet as a woman in a male academic world, she was=0A= barred from holding a professorship in G=3DC3=3DB6ttingen, and the classici= sts=0A= and historians on the faculty even tried to block her from giving unpaid=0A= lectures=3DE2=3D80=3D94leading David Hilbert, the dean of German mathematic= s, to=0A= comment, =3DE2=3D80=3D9CI see no reason why her sex should be an impediment= to he=3D=0A= r=0A= appointment. After all, we are a university, not a bathhouse.=3DE2=3D80=3D= 9D Noe=3D=0A= ther,=0A= who was Jewish, fled to the United States when the Nazis took power,=0A= teaching at Bryn Mawr until her death from a sudden infection in 1935.=0A= =0A= The intellectual habit of grappling with a problem by ascending to=0A= higher and higher levels of generality came naturally to Emmy Noether,=0A= and it was shared by Grothendieck, who said that he liked to solve a=0A= problem not by the =3DE2=3D80=3D9Chammer-and-chisel method,=3DE2=3D80=3D9D = but by letti=3D=0A= ng a sea of=0A= abstraction rise to =3DE2=3D80=3D9Csubmerge and dissolve=3DE2=3D80=3D9D it.= In his vis=3D=0A= ion, the=0A= familiar things dealt with by mathematicians, like equations, functions,=0A= and even geometrical points, were reborn as vastly more complex and=0A= versatile structures. The old things turned out to be mere shadows=3DE2=3D= 80=3D=0A= =3D94or,=0A= as Grothendieck preferred to call them, =3DE2=3D80=3D9Cavatars=3DE2=3D80=3D= 9D=3DE2=3D80=3D94o=3D=0A= f the new. (An=0A= avatar is originally an earthly manifestation of a Hindu god; as Harris=0A= notes, =3DE2=3D80=3D9Ca taste for Indian=3DE2=3D80=3DA6metaphysics inflecte= d [the] term=3D=0A= inology=3DE2=3D80=3D9D of=0A= many French mathematicians.)=0A= =0A= Nor is this a one-off process. Each new abstraction is eventually=0A= revealed to be but an avatar of a still-higher abstraction. As Harris=0A= puts it, =3DE2=3D80=3D9Cthe available concepts are interpreted as the avata= rs of =3D=0A= the=0A= inaccessible concepts we are trying to grasp.=3DE2=3D80=3D9D With the grasp= ing of=0A= these new concepts, mathematics ascends a kind of =3DE2=3D80=3D9Cladder=3DE= 2=3D80=3D9D =3D=0A= of increasing=0A= abstraction.=0A= =0A= This, Harris says, is what philosophers should be paying attention to.=0A= =3DE2=3D80=3D9CIf you were to ask for a single characteristic of contempora= ry=0A= mathematics that cries out for philosophical analysis,=3DE2=3D80=3D9D he wr= ites, =3D=0A= =3DE2=3D80=3D9CI=0A= would advise you to practice climbing the categorical and avatar ladders=0A= in search of meaning, rather than searching for solid Foundations.=3DE2=3D8= 0=3D9D=3D=0A= And=0A= what lies at the top of this ladder? Perhaps, Harris suggests with=0A= playful seriousness, there is One Big Theorem from which all of=0A= mathematics ultimately flows=3DE2=3D80=3D94=3DE2=3D80=3D9Csomething on the = order of sam=3D=0A= sara =3D3D=0A= nirvana.=3DE2=3D80=3D9D But since there are infinitely many rungs to climb,= it is=0A= unattainable.=0A= =0A= Here, then, is the pathos of mathematics. Unlike theoretical physics,=0A= which can aspire to a =3DE2=3D80=3D9Cfinal theory=3DE2=3D80=3D9D that would= account for=3D=0A= all the=0A= forces and particles in the universe, pure mathematics must concede the=0A= futility of its own quest for ultimate truth. As Harris observes,=0A= =3DE2=3D80=3D9Cevery veil lifted only reveals another veil.=3DE2=3D80=3D9D = The mathemat=3D=0A= ician is=0A= doomed to what Andr=3DC3=3DA9 Weil called an endless cycle of =3DE2=3D80=3D= 9Cknowledg=3D=0A= e and=0A= indifference.=3DE2=3D80=3D9D=0A= =0A= But it could be worse. Thanks to G=3DC3=3DB6del=3DE2=3D80=3D99s second inc= ompletenes=3D=0A= s=0A= theorem=3DE2=3D80=3D94the one that says, roughly, that mathematics can neve= r prov=3D=0A= e its=0A= own consistency=3DE2=3D80=3D94mathematicians can=3DE2=3D80=3D99t be fully c= onfident tha=3D=0A= t the axioms=0A= underlying their enterprise do not harbor an as-yet-undiscovered logical=0A= contradiction. This possibility is =3DE2=3D80=3D9Cextremely unsettling for= any=0A= rational mind,=3DE2=3D80=3D9D declared the Russian-born mathematician (and = Fields=0A= medalist) Vladimir Voevodsky, in a speech on the occasion of the=0A= eightieth anniversary of the Institute for Advanced Study. Indeed, the=0A= discovery of such an inconsistency would be fatal to pure mathematics,=0A= at least as we know it today. The distinction between truth and=0A= falsehood would be breached, the ladder of avatars would come crashing=0A= down, and the One Big Theorem would take a truly terrible form: 0 =3D3D 1.= =0A= Yet, oddly enough, e-commerce and financial derivatives would be left=0A= untouched.=0A= =0A= * A charming and improbably successful attempt to explain category=0A= theory in culinary terms is made by the mathematician Eugenia Cheng in=0A= her new book How to Bake Pi: An Edible Exploration of the Mathematics=0A= of Mathematics (Basic Books, 2015). ?=0A= =0A= Sender: categories(a)mta.ca Precedence: bulk Reply-To: Michael Barr <barr(a)math.mcgill.ca>=0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CMCS 2016 : Last Call for Papers
by fabio.zanasi＠ens-lyon.fr 21 Dec '15

21 Dec '15

Call for Papers 13th International Workshop on Coalgebraic Methods in Computer Science (CMCS'16) 2 - 3 April 2016, Eindhoven, The Netherlands http://www.coalg.org/cmcs16 Objectives and scope ------------------- Established in 1998, the CMCS workshops aim to bring together researchers with a common interest in the theory of coalgebras, their logics, and their applications. As the workshop series strives to maintain breadth in its scope, areas of interest include neighbouring fields as well. Topics of interest include, but are not limited to, the following: - The theory of coalgebras (including set theoretic and categorical approaches) - Coalgebras as computational and semantical models (for programming languages, dynamical systems, term rewriting, etc.) - Coalgebras in (functional, object-oriented, concurrent, and constraint) programming - Model checking, theorem proving and deductive verification using coalgebraic techniques - Coalgebraic data types, type systems and behavioural typing - Proof principles and (coinductive) definitions for coalgebras (e.g. with bisimulations or invariants) - Coalgebras and algebras - Coalgebraic specification and verification - Coalgebras and (modal) logic - Coalgebra and control theory (notably of discrete event and hybrid systems) - Coalgebra in quantum computing - Coalgebra and game theory - Tools exploiting colgebraic techniques Venue and event --------------- CMCS'16 will be held in Eindhoven, The Netherlands, co-located with ETAPS 2016 on 2 - 3 April 2016. Keynote Speaker --------------- Jiri Adamek, Braunschweig University of Technology, Germany Invited Speakers --------------- Andreas Abel, University of Gothenburg, Sweden Filippo Bonchi, CNRS/ENS Lyon, France Invited Tutorial Speakers ------------------------- There will be a special session on weighted automata, with invited tutorials by Borja Balle, Lancaster University, United Kingdom Alexandra Silva, University College London, United Kingdom Important dates --------------- Abstract regular papers 4 January 2016 Submission regular papers 13 January 2016 Notification regular papers 12 February 2016 Camera-ready copy 19 February 2016 Submission short contributions 22 February 2016 Notification short contributions 6 March 2016 Programme committee ------------------- Paolo Baldan, University of Padova, Italy Corina Cirstea, University of Southampton, United Kingdom Ugo Dal Lago, University of Bologna, Italy Ichiro Hasuo (chair), University of Tokyo, Japan Tom Hirschowitz, CNRS and University of Savoie, France Bart Jacobs, Radboud University Nijmegen, The Netherlands Shin-ya Katsumata, Kyoto University, Japan Bartek Klin, University of Warsaw, Poland Barbara Koenig, University of Duisburg-Essen, Germany Stefan Milius, FAU Erlangen-Nuernberg, Germany Matteo Mio, CNRS and ENS Lyon, France Rasmus Mogelberg, IT University of Copenhagen, Denmark Larry Moss, Indiana University, United States Fredrik Nordvall Forsberg, University of Strathclyde, United Kingdom Dirk Pattinson, Australian National University, Australia Daniela Petrisan, Paris Diderot University, France Jean-Eric Pin, CNRS and Paris Diderot University, France John Power, University of Bath, United Kingdom Jurriaan Rot, University of Leiden, the Netherlands Jan Rutten, CWI/Radboud University Nijmegen, The Netherlands Alexandra Silva, University College London, United Kingdom Joost Winter, University of Warsaw, Poland James Worrell, University of Oxford, United Kingdom Publicity chair --------------- Fabio Zanasi, Radbound University Nijmegen, The Netherlands PC chair -------- Ichiro Hasuo, University of Tokyo, Japan Submission guidelines --------------------- We solicit two types of contributions: regular papers and short contributions. Regular papers must be original, unpublished, and not submitted for publication elsewhere. They should not exceed 20 pages in length in Springer LNCS style. Short contributions may describe work in progress, or summarise work submitted to a conference or workshop elsewhere. They should be no more than two pages. Regular papers and short contributions should be submitted electronically as a PDF file via the Easychair system at http://www.easychair.org/conferences/?conf=cmcs2016. The proceedings of CMCS 2016 will include all accepted regular papers and will be published post-conference as a Springer volume in the IFIP-LNCS series. Accepted short contributions will be bundled in a technical report. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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PSSL 100
by Peter Johnstone 21 Dec '15

21 Dec '15

We are pleased to announce that the 100th meeting of the Peripatetic Seminar on Sheaves and Logic will be held in Cambridge on 21-22 May 2016, in the month which sees the fortieth anniversary of the first meeting of the seminar. Sender: categories(a)mta.ca Precedence: bulk Reply-To: Peter Johnstone <ptj(a)dpmms.cam.ac.uk> We are also delighted to announce that Dana Scott, who hosted the first PSSL in Oxford, will be present for this weekend, and he will be delivering the annual Mordell Lecture of the Cambridge Faculty of Mathematics on the afternoon of Friday 20 May. Further details and a registration form will be circulated nearer the time, but we thought people might wish to get the date into their diaries before the New Year. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Indiscrete objects in a functor category
by Aleks Kissinger 21 Dec '15

21 Dec '15

It's common to describe the category of (directed, multi-) graphs as a functor category Graph := [2, Set], where 2 here is the category with 2 objects and 2 parallel arrows (s & t). For a pair of sets (V,E), one can construct the indiscrete graph I(V,E) as a graph with vertices V and edges E x V x V, where the source and target maps are just the 2nd and 3rd projection respectively. This gives a right adjoint to the forgetful functor from Graph to pairs of sets. This enables one to construct a category of graphs with a fixed set of vetex/edge labels as a slice over Graph: Graph / I(Lv, Le) since a graph hm G --> I(Lv,Le) is the same as a map U(G) --> (Lv,Le), which is just a pair of functions assigning labels to the vertices and edges of G. This seems to me like a pretty standard trick, which extends to any functor category from a C which is in some sense "suitably acyclic". For instance, consider a category of "partitioned graphs" [3, Set], where 3 has objects (P,V,E) and arrows: E --s--> V, E --t--> V, and V --p--> P where, p assigns each of the vertices a partition. For a triple (P,V,E) we can form the indiscrete partitioned graph with: - partitions P - vertices V x P - edges E x (V x P) x (V x P) - p = pi2, s = pi2, t = pi3 which gives a right-adjoint to the forgetful functor from partitioned graphs to triples of sets. This is clearly an instance of a general recipe, whereby you start with the objects with no arrows out, and work your way backwards, always adding copies of the codomain of every out-arrow. Again one can attach labels to partitioned graphs by slicing: [3,Set] / I(Lp,Lv,Le) So, my question: Is the general case a known/studied construction? If so, could someone provide a reference? Best, Aleks [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CFP for Workshop on Semantic Spaces at the Intersection of NLP, Physics and Cognitive Science
by Mehrnoosh Sadrzadeh 18 Dec '15

18 Dec '15

==================================================================== Workshop on Semantic Spaces at the Intersection of NLP, Physics and Cognitive Science ----------------------------- 11th June 2016, University of Strathclyde, Glasgow, Scotland (co-located with QPL 2016) Website: https://sites.google.com/site/semspworkshop General enquiries: slpcs2016(a)easychair.org<mailto:slpcs2016@easychair.org> ==================================================================== CALL FOR PAPERS --------------- Since their introduction in the early 1970s, vector space models of meaning have evolved into a well-established area of research in Natural Language Processing (NLP). Their probabilistic nature and ability to exploit the abundance of large-scale resources such as the Web make them one of the most useful tools (arguably the most successful (Turney and Pantel, 2010)) for modeling what we broadly call meaning in language. The geometry provided by the angular distance between the vectors has been widely used as a representative of the degree of similarity of meaning in NLP. Another field in which vector space models play an important role is physics, and especially quantum theory. Though seemingly unrelated to language, intriguing connections have recently been uncovered. Some examples include models of compositionality in distributional semantics (Coecke et al. 2010), treatments of logical words in vector space models (Widdows, 2003), reasoning about the human mental lexicon in cognitive processes (Bruza et al., 2009), using vectors of queries and documents in information retrieval (Van Rijsbergen, 2004), and representing the meaning of words by density operators (Piedeleu et al., 2015). There is also a long-standing history of vector space models in cognitive science. Theories of categorization such as those developed by Nosofsky (1986); Smith, Osherson, Rips, & Keane, (1988), utilise notions of distance between concepts represented as feature vectors. More recently Gärdenfors (2004) has developed a model of concepts in which conceptual spaces provide geometric structures, and information is represented by points, vectors and regions in vector spaces. Exploiting the common ground provided by the concept of a vector space, the workshop aims to bring together researchers working at the intersection of NLP, cognitive science, and physics, offering to them an appropriate forum for presenting their uniquely motivated work and ideas. The interplay between these three disciplines will foster theoretically motivated approaches to understanding how meanings of words interact with each other in sentences and discourse, how diagrammatic reasoning depicts and simplifies this interaction, how language models are determined by input from the world, and how word and sentence meanings interact logically. Topics of interest include (but are not restricted to): * Reasoning in semantic spaces * Applications of quantum logic in natural language processing * Compositionality in semantic spaces and conceptual spaces * Links between conceptual spaces and natural language processing * Modeling functional words such as prepositions and relative pronouns in compositional distributional models of meaning * Diagrammatic reasoning for natural language processing SUBMISSION ---------- We solicit papers that describe original work. The submitted papers may consist of up to 6 pages of content (including references) in the EPTCS format (http://style.eptcs.org/) Authors are invited to submit their papers via EasyChair: https://easychair.org/conferences/?conf=slpcs2016 A selection of the accepted papers will be presented orally, and the rest of them as posters. Depending on participation and quality of submissions, the proceedings of the workshop will be published as an EPTCS (Electronic Proceedings in Theoretical Computer Science) volume. IMPORTANT DATES --------------- All deadlines are at 23:59 Howland Island time (UTC-12). * Paper submission deadline: 20 March 2016 * Reviewing period: 27 March 2016-17 April 2016 * Author notification: 24 April 2016 * Workshop: 11 June 2016 INVITED SPEAKERS ---------------- * Hans Briegel, University of Innsbruck * Peter Gärdenfors, University of Lund * Dominic Widdows, Microsoft (To be confirmed) REGISTRATION ------------ Restrigation will open closer to the date of the workshop, please check the website for more details. PROGRAMME COMMITTEE ------------------- * Peter Bruza, Queensland University of Technology * Stephen Clark, University of Cambridge * Bob Coecke, University of Oxford * Liane Gabora, University of British Columbia * Chris Heunen, University of Edinburgh * Peter Hines, University of York * Aleksandra Kislak-Malinowska, University of Warmia and Mazury in Olsztyn * Daniel Marsden, University of Oxford * Glyn Morrill, Universitat Politècnica de Catalunya * Valeria de Paiva, Nuance Communications, Inc * Stanley Peters, University of Stanford * Stephen Pulman, University of Oxford * Matthew Purver, Queen Mary University of London * Sebastian Riedel, University College London * Mehrnoosh Sadrzadeh, Queen Mary University of London * Frank Zenker, University of Konstanz ORGANIZATION ------------ * Dimitrios Kartsaklis, Queen Mary University of London * Martha Lewis, University of Oxford * Laura Rimell, University of Cambridge ADVISORY COMMITTEE ------------------ * Bob Coecke, University of Oxford * Mehrnoosh Sadrzadeh, Queen Mary University of London [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Rolland's MSc thesis
by David Roberts 18 Dec '15

18 Dec '15

Dear all, is Stanislas Rolland's MSc thesis "Essai sur les mathématiques algorithmiques" done at Université du Quebec á Montréal available in any electronic form? Many thanks, David -- David Roberts http://ncatlab.org/nlab/show/David+Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Reminder: 98th PSSL, January 8--10, Doorn, the Netherlands
by Oosten＠mlist.mta.ca 14 Dec '15

14 Dec '15

Dear all, please allow us to remind you that registration for the 98th PSSL, 8--10 Jan uary 2016, takes place preferably before December 21. For a web site for the meeting, see: https://staff.fnwi.uva.nl/b.vandenberg3/PSSL98/ Further details will appear on this page or be sent to registered participants. Best, Jaap, Ieke and Benno ________________________________________ From: Oosten, J. van [j.vanoosten(a)uu.nl] Sent: Wednesday, November 18, 2015 1:47 PM To: categories Subject: categories: 98th PSSL -- first announcement Dear all, We are happy to announce the 98th Peripatetic Seminar on Sheaves and Logic, which will take place during the weekend of 8-10 January 2016. As usual, we invite talks on every aspect of category theory or its applications in mathematics. We have booked a conference centre in a wooded area near the University of Utrecht, which offers full board and lodging and lecture room facilities. We can use some grant money so that the local expenses for the participants will be kept to a minimum. The web site of the centre is http://www.kaapdoorn.com/ (but precise travel instructions will be given in a further announcement). However, we could only make this arrangement for a maximum of 40 participants. We therefore ask participants to register NO LATER than December 17, by an e-mail sent to Benno van den Berg: bennovdberg(a)gmail.com. Registration will be dealt with on a first come first serve basis. We look forward to seeing you in the new year! Benno van den Berg Ieke Moerdijk Jaap van Oosten [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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