October 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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Re: Proving enough injectives for modules over a Grothendieck topos
by Colin McLarty 26 Nov '10

26 Nov '10

This proof really fell through the cracks. The argument from injective Abelian groups to injective R-modules was known by 1960. The last piece was in place in 1974 with Barr's theorem on coverings with AC. The proof may be published somewhere but I can't find it and Peter suggests it is not. Its absence works mischief. Eisenbud COMMUTATIVE ALGEBRA (p. 621) proves modules (n Set) have enough injectives and then sends readers off to Hartshorne for the Godement construction to prove modules over topological spaces have enough injectives. But he has in effect already proved it for all Grothendieck toposes! He merely has to send readers off to Mike's paper or Peter's (1977) book for the AC result. While Eisenbud states results explicitly for module categories over Sets, he frames them to hold much more generally and he sends readers to sources for that generality. I am glad to hear it will be in the Elephant. best, Colin PS thanks to Carsten Butz for reminding me of Andreas Blass "Injectivity, projectivity, and the axiom of choice" (Trans. AMS Volume 255, November 1979) for both history and a proof that choice will be required. 2010/10/11 Prof. Peter Johnstone <P.T.Johnstone(a)dpmms.cam.ac.uk>: > Dear Colin, > > Yes, the argument is correct, and it'll be in volume 3 of the Elephant. > I don't know why it hasn't been published elsewhere. > > Peter > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CT2010 Proceedings: Deadline approaching/extension
by Bob Rosebrugh 03 Nov '10

03 Nov '10

We have received requests for an extension of the submission deadline for the CT2010 issue of JPAA. We will respond positively to a request to submit an article by November 30. We do ask that an intention to submit be made known to us by *November 8*. best to all, Jiri Adamek, George Janelidze, Pino Rosolini, Bob Rosebrugh [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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"schizophrenic" reference
by Fred E.J. Linton 31 Oct '10

31 Oct '10

At the Halifax Octoberfest just ended, I had mistakenly, in a private conversation, suggested looking in or near Chapter 4 section 5 of PTJ's Stone Spaces for occurrence of the word "schizophrenic." I regret that error. Better places to look might be page 268 of that book, or passim on pages 254-262, and thereabouts. Grosso modo: not Chapter 4 section 5, but Chapter 6 section 4. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CT and Physics at PSA in Montreal
by Steve Awodey 26 Oct '10

26 Oct '10

Some readers of this list may be interested in the following session of the upcoming Philosophy of Science Association meeting in Montreal: Symposium: Category Theoretical Reflections on the Foundations of Physics Saturday, November 6, 1:30-3:45 PM Speakers: Bob Coecke (Oxford University), “Quantum Picturalism” Klaas Landsmann (Radboud University), “Topos Theory and the Copenhagen Interpretation of Quantum Mechanics” Jeffrey Morton (University of Western Ontario), “Groupoidification in Physics” Gonzalo Reyes (University of Montreal), “From Laws of Motion to Vacuumfield Equations” The meeting takes place at the Hyatt Regency Montreal. Detailed information can be found on the meeting website: http://www.philsci.org/meetings/psa2010/index.html Feel free to contact the session organizers, Hans Halvorson and Steve Awodey, for further information. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: property_vs_structure
by Marta Bunge 26 Oct '10

26 Oct '10

Dear George, > Thanks for your interesting response. I contend that "unramified coverings" is not an appropriate expression to describe those "coverings" not associated with a Galois theory in your sense since there is a specific meaning attached to it. I would simply call them "generalized coverings". Such would be the unramified coverings or just the complete spreads. In fact also the branched coverings but there is a difference here. For branched coverings, the motivation of it all, a "branched Galois theory" already exists informally in the subject of knot groupoids. Moreover, this example gives you the reason concerning coverings and Galois theories. I devote one paragraph to each contention. > 1. Terminology "unramified". When R.H.Fox introduced spreads and their completions, he had in mind what the title of his paper says, that is, "coverings with singularities" so, not the traditional locally constant coverings. There could be "ramifications", or branchings over points of the base. But no folds. Specifically, he was thinking of branched coverings (branching over a knot in the base) as the spread completions of locally constant coverings, in which branching points were added to the domain space. This is what led him to define a notion of spread, and then perform a completion process. The branched coverings, and more generally the complete spreads of which they are the motivating example, are "ramified". Now, add the condition that the complete spread (e.g. a branched covering) be a local homeomorphism. This does not force it to be locally constant, as we know, but it cannot then have ramifications. Hence the expression "unramified coverings". That is why I suggested "generalized coverings" - for both the ramified and the unramified which are not locally constant. > 2. Do we need a generalized Galois theory to deal with the branched coverings? Perhaps but, if so, it would not be a big generalization. Indeed, the "branched coverings", which are very important in topology, although they do not correspond directly to any Galois theory in your sense, they do through a biequivalence. Indeed for a topos E (say), there is a biequivalence of the 2-categories of branched coverings of E branching over an object Y (the latter thought of as the complement of a knot K) on the one hand, and that of all locally constant coverings of the slice topos E/Y on the other (Bunge-Niefield 2000, Funk 2000, Bunge-Lack 2003). Now, C(E/Y) is in turn viewed as the knot groupoid G(K) of K. So, branched coverings branching over a knot is not an instance of coverings in your sense as they are not locally constant, yet there is a Galois theory associated with them, however, only through a non-trivial equivalence. A strange but not unmanageable situation. > With best regards, > Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge(a)mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: property_vs_structure
by Marta Bunge 26 Oct '10

26 Oct '10

Dear George, > Thanks for your interesting response. Let me just comment here on your "addition 1" below. It is my contention that "unramified coverings" is not an appropriate expression to describe those "coverings" not associated with a Galois theory in your sense, as first, there is a specific meaning attached to it, and secondly, a "branched Galois theory" already exists informally in the subject of knot groupoids. One may possibly generalize your Galois theories to include these phenomena. I devote one paragraph to each contention. > 1. When R.H.Fox ("Covering spaces with singularities", R.H. Fox et al, editors, Algebraic Geometry and Topology: A Symposium in honor of S. Lefschetz, Princeton University Press, 1957, 243-257) introduced spreads and their completions, he had in mind what the title of his paper says, that is, "coverings with singularities" so, not the traditional locally constant coverings. There could be "ramifications", or branchings over points of the base. But no folds. Specifically, he was thinking of branched coverings (branching over a knot in the base) as the spread completions of locally constant coverings, in which branching points were added to the domain space. This is what led him to define a notion of spread, and then perform a completion process leading to another spread singled out among all such corresponding to a given cosheaf on the base space. The branched coverings, and more generally the complete spreads of which they are the motivating example, are "ramified". Now, add the condition that the complete spread (e.g. a branched covering) be a local homeomorphism. This does not force it to be locally constant, as we know, but it cannot then have ramifications. Hence the expression "unramified coverings". > 2. As I said in my last posting, the "branched coverings", which are very important in topology, yet do not correspond to any Galois theory in your sense, should correspond to a "generalized Galois theory" or to a "branched Galois theory". To support my contention, note that, in (M. Bunge and S. Lack, van Kampen theorems for toposes, Advances in Mathematics 179/2 (2003) 291-317), we obtain, as an application of the van Kampen theorems we prove therein, a connection with the use of the automorphism group of a (universal) branched covering in the calculation of knot groups, as advocated by Fox. In particular, and the point I am making here, the expression "unramified coverings" does not describe them accurately, as there may be ramifications. For a topos E, there is a biequivalence of the 2-categories of branched coverings of E branching over an object Y (the latter thought of as the complement of a knot K) on the one hand, and that of all locally constant coverings of the slice topos E/Y on the other. The latter may in turn be viewed as the fundamental groupoid of E/Y, or as the knot groupoid G(K) of K. There is a "Galois theory" there not associated with coverings in your sense, that is, with locally constant coverings. > All of this requires further investigation, for which I will have no time possibly until December, due to my trip to Buenos Aires. > With best regards, > Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge(a)mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: property_vs_structure
by Marta Bunge 25 Oct '10

25 Oct '10

Dear George,> I haste to correct a possible misconception arising from my previous posting, and to propose an idea in connection with it. You wrote: > > I am not sure I fully understood what you say about unramified coverings > versus locally constant coverings. Are you even saying that you found a > Galois structure on TOP, or on any subcategory of TOP, whose coverings are > exactly the unramified coverings (and the situation is non-trivial in the > sense that unramified coverings are not the same as locally constant > coverings? That would be wonderful! > Let C = LoCo/E, defined as in your book Galois Theories (with F. Borceux). An object p of C with domain F is said to be a covering morphism if there exists a morphism e of effective descent in Top with codomain E such that (F,p) is split by e. A complete spread p of C with domain F - that is, an unramified morphism, need not be a covering morphism in C in your sense, as we know. Whether there is a Galois structure on C whose coverings are precisely the unramified coverings without it forcing them to be identified with the locally constant coverings does not seem likely. At least we know that the class of unramified coverings in C is stable under pullbacks and has other nice properties, so C is a natural choice of universe. The fact that we have called "coverings" the unramified morphisms may then be misleading if coverings are to be tied up with Galois theory. > Nevertheless, it is the case in topology that the notion of a covering in the traditional sense has been enlarged to include branchings but not folds. A branched covering of a locally connected space E (R.H. Fox 1957) is the spread completion of a locally constant covering on a pure open subspace U of E, thought off as "the complement of a knot". At least in this case it is meaningful to consider the "branched fundamental groupoid" (or "knot groupoid") of E with non-singular part U. It might be of interest to consider a notion of "generalized Galois theory" to encompass this notion of "generalized covering morphism". Let me know what you think. Relevant discussions in topos theory can be found in (Bunge-Niefield 2000), (Funk 2000), (Bunge-Lack 2003), as well as in (Bunge-Funk 2006, 2007). > With best regards, > Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Burnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge(a)mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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