August 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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String diagrams, adjunction and autonomous categories.
by David Leduc 07 Sep '10

07 Sep '10

As shown by Baez, in an autonomous category the isomorphism hom(A (X) B, C) = hom (B, A -o C), when drawn as a string diagram, is like the bending of the input wire A to make it an output. Now one can also draw string diagrams to represent the zigzag equations between the adjoint pair of functors _ (X) B and B -o _. How does the latter diagram relate to the former one? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Makkai's suggestion
by John Baez 05 Sep '10

05 Sep '10

David wrote: > http://ncatlab.org/johnbaez/show/Towards+Higher+Categories > > Thank you for the reference. But I don't know where to start. Start by reading the above book together with Cheng and Lauda's "Higher categories: an illustrated guidebook": http://www.cheng.staff.shef.ac.uk/guidebook/ and Leinster's "A Survey of Definitions of n-Category": http://arxiv.org/abs/math/0107188 Then try Lurie's "Higher Topos Theory": http://arxiv.org/abs/math/0608040 They're all free online! Expect to spend a decade on this stuff. Or, wait two decades for people to polish it up, and then spend half a decade learning the basics and half a decade learning what people have done in the next two decades. That may be more efficient. Is there a definitive definition of omega-categories somewhere in the > literature or is it still unknown? There are *several* definitions that are almost surely "right" and likely to be studied for many years hence. There is no particular reason to expect that one definition will be best for all applications - but there's a lot of reason to expect that all the "right" definitions will be shown to be equivalent (in a rather subtle sense). > Can it be stated in elementary terms (I mean in terms of object, arrows, > ... without references to simplicial sets or topology) ? > You should learn to love simplicial sets - they're way too important to avoid! If for some reason you're allergic to simplicial sets, you might like Batanin's definition of omega-categories. But then you need to like operads. You could state it without operads, but then it becomes quite long. The book by Cheng and Lauda takes various definitions and makes them less scary by illustrating how they work with lots of pictures. > In the definition of a bicategory, one could replace the coherence > axioms by the statement that all diagrams built from the canonical > ismorphisms commute. Can it be generalized to n=3, ... , omega. > You could say that's the basic idea behind Batanin's definition. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Another question on Grothendieck
by Michael Barr 04 Sep '10

04 Sep '10

Grothendieck introduces, on the top of p. 209 of the Tohoku paper, the notation U_{i_0..i_p} without explanation and uses it again over the next couple pages. Here {U_i} is an open cover of a space X and I have reason to believe that this stands for the intersection of U_{i_j}. Can anyone confirm this? Or give an alternate explanation? The context is that of a claim that (when A is a sheaf) and "every U_{i_0..i_p} is A-acyclic, then"... and that awfully like the definition of a simple cover. Michael [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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Makkai's suggestion
by David Leduc 01 Sep '10

01 Sep '10

In "Towards a categorical foundation of mathematics", Makkai wrote that: "This suggests that, possibly, the right approach to the definition of weak n-category is to aim at formulating all coherence conditions at once, regardless the fact that this might give a very "theoretical" definition. It would then be a separate, and still very important, project to find a (hopefully) finite and concise set of coherence conditions that would be enough to imply all coherence conditions." It was 15 years ago! Did this approach give something? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Symposium Philosophy of the Logic of Sheaves - Call for Participation
by Raul Leal Rodriguez 31 Aug '10

31 Aug '10

================================================================ Symposium Philosophy of the Logic of Sheaves October 19 - 21, 2010 Cali, Colombia http://staff.science.uva.nl/~rlealrod/ sheavessymposioum.html CALL FOR PARTICIPATION ================================================================ Classical Logic and Set Theory influenced deeply the emergence of Analytical Philosophy at the beginning of XXth Century. After a hundred years, it may be time that the Logic of Sheaves and Category Theory help to create alternative forms of practicing Philosophy. Closer to the concepts of movement, flow, passage, boundary and contamination, Sheaf Logic and Category Theory must reinvigorate some regions of Philosophy abandoned by the Analytical dogmae. “Naturalized” Phenomenology, “Mathematical” Metaphysics, “Semiotical” Hermeneutics, between others, may thus emerge with new instruments at hand. The Symposium Philosophy of the Logic of Sheaves hopes to establish the state of the question, profiting from a creator of the Logic of Sheaves (Caicedo) and some major specialists in the Philosophy and History of Category Theory (Krömer, Marquis, McLarty). Forty years after Grothendieck’s departure from the IHES, the Symposium will be devoted in part to his legacy. Speakers: - Colin McLarty (Case Western Reserve University) - Xavier Caicedo (U. de los Andes, Colombia) - Jean Pierre Marquis (Université de Montréal) - Andres Villaveces (U. Nacional, Colombia) - Ralf Kromer (Nancy Université / University of Siegen) -Fernando Zalamea (U. Nacional, Colombia) PROGRAMME: Colin McLarty Grothendieck's 'incorrigible naivety' in building worlds for Mathematics Xavier Caicedo The Model Theory of Sheaves or the Logic of Variable Structures Jean-Pierre Marquis Sheaves, Spaces and Logic Andrés Villaveces Bridging a gap between Physics and Logic: the role of Sheaves Ralf Krömer From Cantor to Sheaves. The development of the concepts of direct and inverse limits – a case study on shifts in mathematical methodology in the prehistory of Category Theory Fernando Zalamea Sheaf Logic: a mathematical conspectus and a philosophical prospectus REGISTRATION: Registration is free of charge. To register send an e-mail to Fernando Zalamea (fernandozalamea(a)gmail.com) LOCATION: UNIVERSIDAD DEL VALLE - CALI, COLOMBIA [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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21 September: Workshop on Categories, Logic and Physics, Birmingham UK
by Steve Vickers 31 Aug '10

31 Aug '10

============================================================================== 7th Workshop on Categories, Logic and Physics -- CLP7 Birmingham -- 21st September 2010 2nd Announcement http://categorieslogicphysics.wikidot.com/ ============================================================================== Categories, Logic and Physics is a workshop series bringing together researchers from the three fields to interact and collaborate, migrate results and cross fertilise. The 7th Workshop on "Categories, Logic and Physics" will be held on Tuesday September 21st, 2010 at the University of Birmingham. As always this is a one day event, there are no conference fees. A particular focus of this meeting will be topological aspects, but, as usual, other topics are covered either. DATE AND LOCATION: -- Tuesday 21st of September 2010 -- The University of Birmingham, School of Computer Science -- Room SPX-LT1 -- This Lecture Theater 1 is in the `Sports and Exercise Science' building Y14, about 5min walk from Computer Science (building Y9), see: http://www.about.bham.ac.uk/maps/pdfs/edgbaston-map-yellow-09.pdf REGISTRATION: If you plan to attend the workshop, please send an email as soon as possible to the local organizers: <S.J.Vickers(a)cs.bham.ac.uk> and/or <B.Fauser(a)cs.bham.ac.uk> so that we can make local arrangements. This is important for arranging for smooth lunch, so letting us know really helps you. TRAVEL INFORMATION: Birmingham is easily reached by train or plane (Birmingham International Airport). The train transfer from the airport to the University is approximately 45 minutes. For train travel, you should ask for the station "University" when purchasing tickets. Most routes include a change at Birmingham New Street to the line with destination Longridge or Redditch. ACCOMMODATION: The workshop is intended to be a one day event and most participants will not need to stay overnight. If you need help with an accommodation feel free to contact the local organizers. SCHEDULE: (Titles and abstracts see below) --------------------------------- 10:30-11:00 Welcome Coffee/Tea --------------------------------- 11:00-12:00 Martin Escado Maybe locales are made out of points after all 12:00-13:00 Christopher J. Mulvey Constructive Aspects of Gelfand Duality --------------------------------- 13:00-14:00 Lunch Break --------------------------------- 14:00-15:00 Ronnie Brown What is and what should be `higher dimensional group theory'? 15:00-16:00 Catherine Meusburger Higher categories and observables for generalised Turaev-Viro models --------------------------------- 16:00-16:30 Coffee/Tea Break --------------------------------- 16:30-17:30 Simon Willerton Two 2-traces (Tentative) 17:30-18:30 Cecilia Flori Topos Formulation of history Quantum Theory --------------------------------- 19:00- Pub Session --------------------------------- We are glad to see you in September, local organizers Steve Vickers + Bertfried Fauser workshop coordinators Bob Coecke + Andreas Doering ============================================================================== Titles and Abstracts, in alphabetical order ============================================================================== [Ronnie Brown, Bangor] Title: What is and what should be `higher dimensional group theory'? Abstract: The presentation will show, including some knot demos, some of the problems and intuitions which have led to this question, and how certain cubical algebraic structures with partial operations whose domains are given by geometric conditions have been found quite natural for expressing modes of higher dimensional subdivision and composition which are related to long term concerns in algebraic topology. [Martin Escado, Birmingham] Title. Maybe locales are made out of points after all. Abstract: Like topology in analysis, locale theory is about open sets, continuous functions, compact spaces, approximation and limit processes, and things like that. Both topology and locale theory start with opens. In topology, an open is made out of points, but in locale theory, a point is made out of opens. The localic view makes physical and computational sense: points are infinitely small (and carry an infinite amount of information), and hence are not directly observable, but each point is uniquely characterized by its (infinite) collection of observable properties. The opens are the observables, and locale theory takes the notion of observation as primitive, and all other notions, including that of point, as derived. (Moreover, some perfectly good spaces in locale theory have a rich supply of opens without allowing any point at all, but this is not what I will emphasize in my talk). Although the match of (physical or computational) reality with locale theory is arguably better than with topology, locale theory may be more mathematically demanding, or at least is certainly unfamiliar to most of us. In this talk I'll discuss how one can think of locales as if they were made out of points, like the spaces of classical analysis and geometry, trying to make them more familiar, manageable, and intuitive, without loss of rigour, so that we can reason and work with them efficiently. [Cecilia Flori, Perimeter] Title: Topos Formulation of history Quantum Theory Abstract: In this talk I will describe a topos formulation of consistent histories obtained using the topos reformulation of standard quantum mechanics put forward by Doering and Isham. Such a reformulation leads to a novel type of logic with which to represent propositions. In the first part of the talk I will introduce the topos reformulation of quantum mechanics. I will then explain how such a reformulation can be extended so as to include temporally-ordered collection of propositions as opposed to single time propositions. Finally I will show how such an extension will lead to the possibility of assigning truth values to temporal propositions. [Catherine Meusburger, Hamburg] Title: Higher categories and observables for generalised Turaev-Viro models Abstract: Generalised Turaev-Viro models that are formulated in terms of spherical categories play an important role in three-dimensional quantum gravity, where they are interpreted as discrete path integrals or state sum models of quantised three-manifolds. We discuss the role and interpretation of these models in quantum gravity and comment on the problem of defining observables for these models. We show how this problem can be addressed by using higher categories and discuss the mathematical properties and the physical interpretation of the resulting observables. The talk is based on joint work with John W. Barrett. [Christopher J. Mulvey, University of Sussex] Title: Constructive Aspects of Gelfand Duality Abstract: One of the important foundational aspects of recent approaches to developing quantum theories of space and time has been the existence of a constructive theory of Gelfand duality for commutative C*-algebras. In this talk, we shall outline the way in which this theory was developed, examine its application to the context of quantum physics, and consider its extension to the non-commutative case. [Simon Willerton, Sheffield] Title: Two 2-traces (Tentative) Abstract: Over recent years, in several areas of mathematics the notion of 'categorified trace' or '2-trace' has arisen. For instance, in higher representation theory where groups act on linear categories there is the notion of a '2-character'; in Khovanov knot homology the Hochschild homology is viewed as a categorical trace. It transpires that there are actually two orthogonal, and sometimes dual, notions of 2-trace in common usage and I will explain how they arise and give various examples from various areas of mathematics. ============================================================================== [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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