July 2010 - Categories - categories.org.au

Research positions
by Maria Manuel Clementino 26 Feb '14

26 Feb '14

5-YEAR RESEARCH POSITIONS The Centre for Mathematics of the University of Coimbra (CMUC) invites applications for five 5-year research positions in all areas of Mathematics. Applicants should have a PhD in Mathematics and at least three years of post-doctoral experience, with an international-level publication record and demonstrated ability to perform independent research. Preference will be given to candidates working in one of the areas in which the members of CMUC are currently active. A very good command of English is required. The appointments will be made for three years, renewable for two more. Successful applicants are expected to develop research within the areas of interest of CMUC. They should also contribute to graduate studies, supervising students from the Department of Mathematics. The basic salary is equivalent to the index 195 of the Portuguese research career (in the order of 42 500 euros a year), in accordance with the Ciência 2008 guidelines http://alfa.fct.mctes.pt/apoios/contratacaodoutorados/edital2008 Benefits include private health insurance and social security. Applicants should send • cover sheet (available from www.mat.uc.pt/cmuc/coversheet.pdf) including names and contact information of three persons who can provide a letter of recommendation • curriculum vitae (publication list included) • a 5-year research proposal (2 pages max), emphasizing their own research goals as pdf files to cmuc(a)mat.uc.pt. The subject of the e-mail message must include the job reference C2008 FCTUC CMUC v15-v19. The deadline for applications is *September 10, 2008.* ------------------------------ More information can be found at www.mat.uc.pt

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separable locale
by Thomas Streicher 17 May '13

17 May '13

Recently rereading Fourman's "Continuous Truth" I came across the term "separable locale" but could nowhere find an explanation. Does it mean a cHa A for which there exists a countable subset B such that ever a in A is the supremum of those b in B with b leq a. This would be the point free account of "second countable", i.e. having a countable basis. Of course, second countable T_) spaces are separable, i.e. have a countable dense set. Is this reading the "usual" one? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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What else do simplicial sets classify?
by Andrej Bauer 05 Oct '11

05 Oct '11

The presheaf category of simplicial sets is the classifying topos for the theory L of a bounded linear order. In general, there could be other theories which are "Morita equivalent" to L in the sense that their classifying toposes are equivalent to simplicial sets. Are any such known, preferably occurring in nature? With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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Re: Tensor of monads
by Prof. Peter Johnstone 10 Aug '10

10 Aug '10

On Fri, 30 Jul 2010, Sergey Goncharov wrote: > On 07/29/2010 03:24 PM, Prof. Peter Johnstone wrote: >> Sorry, the previous posting was nonsense -- a bisemilattice is the >> same thing as a semilattice, by the Eckmann-Hilton argument. >> However, if you leave out the zero, and consider the "set of >> nonempty subsets" monad, this time on the category of sets of >> cardinality 2^n - 1 for some n, you do get a counterexample. > This looks fine! But I guess, Eckmann-Hilton argument does not apply to your > previous example because it presupposes that the monoidal structures share > the unit, which was not the case there, was it? > Yes, it was: the fact that the unit for each semilattice structure is a homomorphism for the other forces them to be the same. Peter Johnstone P.S. -- You can enlarge the base category to contain all finite sets of odd cardinality (so that, for example, it's cartesian closed). The free bi-(semilattice-without-unit) on three generators has 20 elements. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Tensor of monads
by Sergey Goncharov 01 Aug '10

01 Aug '10

On 07/29/2010 03:24 PM, Prof. Peter Johnstone wrote: > Sorry, the previous posting was nonsense -- a bisemilattice is the > same thing as a semilattice, by the Eckmann-Hilton argument. > However, if you leave out the zero, and consider the "set of > nonempty subsets" monad, this time on the category of sets of > cardinality 2^n - 1 for some n, you do get a counterexample. This looks fine! But I guess, Eckmann-Hilton argument does not apply to your previous example because it presupposes that the monoidal structures share the unit, which was not the case there, was it? Thanks a lot, -- Sergey Goncharov, Junior Researcher DFKI Bremen Phone: +49-421-218-64276 Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov(a)dfki.de D-28359 Bremen Site: www.dfki.de/sks/staff/sergey ------------------------------------------------------------- Deutsches Forschungszentrum fuer Kuenstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschaeftsfuehrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Tensor of monads
by Michael Barr 31 Jul '10

31 Jul '10

I am afraid you are right. Here is what Manes says on the subject in LNM80: It is an open question whether or not $\t\otimes\widetilde{\t}$ always exists. A constructive proof can be given if both $\t$ and $\widetilde{\t}$ have a rank (in the sense of \cite[2.2.6]{man}) by generalizing Freyd's proof in \cite{fre}. What I had in mind involved two isomorphic, but distinct triples. The one coming from the underlying functor of complete sup semilattices (whose mu would be union) and the other coming from complete inf semilattices (whose mu is intersection). The problem comes when you make those operations commute with each other. It would seem to me that would force complete distributivity. But completely distributive boolean algebras do give a tripleable category. Perhaps the moderator could throw some light on this from his work on CCD lattices. Could this still be unknown? I guess it could. It is not a topic that has aroused a great deal of interest. Sorry, Michael On Sat, 31 Jul 2010, Richard Garner wrote: > Isn't P * P isomorphic to P, by the Eckmann-Hilton argument? > > On 29 July 2010 20:29, Michael Barr <barr(a)math.mcgill.ca> wrote: > >> There are examples in Ernie Manes's 1967 thesis. Perhaps the simplest >> (although it piggybacks on the non-existence of free complete boolean >> algebras that had been know for only a few years at the time) is that the >> tensor product of the complete sup semilattice triple with itself doesn't >> exist. The triple takes a set X to 2^X and can be interpreted also as the >> complete inf semilattice triple. On the other hand, I think Manes showed >> that the tensor product of the beta triple with itself exists, but is one >> of the two inconsistent triples, the one that fixes the empty set and >> takes all non-empty sets to one point. (The other inconsistent triple >> takes all sets to one point.) >> >> >> On Wed, 28 Jul 2010, Sergey Goncharov wrote: >> >> Dear categorists, >>> >>> in "Combining algebraic eects with continuations", by Hyland et al. the >>> >>> authors say carefully: "In general, the tensor product of two arbitrary >>> monads seems not to exist.." without providing a counterexample though, >>> presumably because they did not have any. Was there any progress reported >>> on >>> this issue since then? Or maybe someone can even make up a counterexample >>> right on the nail? >>> >>> Thanks, >>> >>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Tensor of monads
by Sergey Goncharov 31 Jul '10

31 Jul '10

Dear categorists, in "Combining algebraic eﬀects with continuations", by Hyland et al. the authors say carefully: "In general, the tensor product of two arbitrary monads seems not to exist.." without providing a counterexample though, presumably because they did not have any. Was there any progress reported on this issue since then? Or maybe someone can even make up a counterexample right on the nail? Thanks, -- Sergey Goncharov, Junior Researcher DFKI Bremen Phone: +49-421-218-64276 Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov(a)dfki.de D-28359 Bremen Site: www.dfki.de/sks/staff/sergey ------------------------------------------------------------- Deutsches Forschungszentrum fuer Kuenstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschaeftsfuehrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Tensor of monads
by Prof. Peter Johnstone 29 Jul '10

29 Jul '10

Sorry, the previous posting was nonsense -- a bisemilattice is the same thing as a semilattice, by the Eckmann-Hilton argument. However, if you leave out the zero, and consider the "set of nonempty subsets" monad, this time on the category of sets of cardinality 2^n - 1 for some n, you do get a counterexample. Peter Johnstone On Thu, 29 Jul 2010, Prof. Peter Johnstone wrote: > Here's a slightly artificial counterexample: let C be the category > of finite sets whose cardinality is a power of 2, and all functions > between them. The covariant power-set functor restricts to a > functor C --> C, and has a monad structure whose algebras are > semilattices. If the tensor product of this monad with itself > existed, its algebras would be bisemilattices, i.e. sets with two > semilattice structures which "commute with each other" in the > obvious sense. Free bisemilattices exist, but they don't > necessarily have cardinality a power of 2: by my calculation, the > free bisemilattice on two generators has seven elements. So the > free-bisemilattice functor doesn't exist as an endofunctor of C. > > Peter Johnstone > ----------------------- > On Wed, 28 Jul 2010, Sergey Goncharov wrote: > >> Dear categorists, >> >> in "Combining algebraic e?ects with continuations", by Hyland et al. the >> authors say carefully: "In general, the tensor product of two arbitrary >> monads seems not to exist.." without providing a counterexample though, >> presumably because they did not have any. Was there any progress reported >> on this issue since then? Or maybe someone can even make up a >> counterexample right on the nail? >> >> Thanks, >> >> -- >> Sergey Goncharov, Junior Researcher >> >> DFKI Bremen Phone: +49-421-218-64276 >> Safe and Secure Cognitive Systems Fax: +49-421-218-9864276 >> Cartesium, Enrique-Schmidt-Str. 5 Email: Sergey.Goncharov(a)dfki.de >> D-28359 Bremen Site: www.dfki.de/sks/staff/sergey >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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New email address
by Tom Leinster 27 Jul '10

27 Jul '10

Our university has been "restructured". This means among other things that we all have new email addresses. We're promised that the old ones will continue to work, but just in case that promise isn't kept, here's my new one: Tom.Leinster(a)glasgow.ac.uk effective from now. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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