October 2016 - Categories - categories.org.au

Topological *-autonomous categories, revisited
by Michael Barr 02 Nov '16

02 Nov '16

In 2006 I published a paper called Topological *-autonomous categories (TAC, 16, 700-708). Looking it over early this year I realized it was badly flawed with gaps, incorrect proofs, and omissions (of things claimed in the intro). I have rewritten it, fixing all the flaws (I hope at any rate). Somewhat reluctantly, I submitted it to TAC, but the referee felt, doubtless correctly, that this revision was not really up to the quality of the leading journal in CT, but should be published elsewhere. Where for heaven's sake? Certainly not in one of the hundreds of vanity journals whose solicitations fill my mailbox daily. Anyway, it is the top paper that appears on the web site that contains virtually all the papers I have ever written as well as a few miscellaneous things (such as the Tohoku translation): http://www.math.mcgill.ca/barr/papers/. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Grothendieck toposes
by Steve Vickers 02 Nov '16

02 Nov '16

For some years now, I have been using the phrase "Grothendieck topos" - category of sheaves over a site - to allow the site to be in an arbitrary base elementary topos S (often assumed to have nno). Hence "Grothendieck topos" means "bounded S-topos". The whole study of Grothendieck toposes, as of geometric logic, is parametrized by choice of S. That's presumably not how Grothendieck understood it, and I know some of his results assumed S = Set, some classical category of sets. Moreover, the Elephant defines "Grothendieck topos" that way. On the other hand, if a topos is a generalized space, with a classifying topos being the space of models of a geometric theory, then that surely meant Grothendieck topos; and there are various reasons for wanting to vary S. For example, using Sh(X) as S gives us a generalized topology of bundles, fibrewise over X. I'm coming to suspect my usage may confuse. How do people actually understand the phase "Grothendieck topos"? Do they hear potential for varying an implicit base S, or do they hear a firm implication that S is classical? Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes
by Steve Vickers 01 Nov '16

01 Nov '16

Dear Marta, Thanks for your reply. My question was a survey of usage, so what follows is not meant disputatiously, but I thought your reasoning raised some interesting issues. As I see it you give two reasons here for taking Grothendieck toposes as being over Set = ZFC, both pertinent: (1) what Grothendieck meant, and (2) the (essential) uniqueness of geometric morphisms to Set. (1) is delicate, given Grothendieck's underlying implicit definition of topos as "that of which topology is the study". My understanding of this (you may know more - my knowledge of Grothendieck's work is almost all second-hand) is that there are two ways of viewing it, somewhat akin (respectively) to algebraic and general topology. The first is that he meant a topos to be a category with which to do sheaf cohomology, by forming an (exact) injective resolution of Abelian groups, taking global sections (becoming non-exact), and then extracting the ker/im cohomology groups. This is perhaps an algebraic topologist's idea of what topology means. Classicality of Set seems then to be needed in order to get injective hulls in the category of sheaves over a site. On the other hand, there is also the idea that global points can be recovered as sections of the geometric morphism to Set, and the topos simultaneously embodies both the points and the topology on them. Is this also part of what Grothendieck meant? It is closer to general topology, points and their continuous transformations. This idea then generalizes well to elementary toposes, replacing Set by S. Elementary toposes are not generalized spaces in themselves, but bounded geometric morphisms between them are, and many topological properties are redefined for geometric morphisms. (2) raises the question of why Set should have its privileged property. Technically, it is that every object of Set is a colimit of copies of 1, and that is preserved up to unique isomorphism by the inverse image part of any geometric morphism. But is that not because ZFC provides a 2-level structure of sets and classes, and we are implicitly using ZFC classes for our toposes? As we explore the foundational options then we should expect this uniqueness property for Set = ZFC to evaporate. All the best, Steve. > On 30 Oct 2016, at 20:17, Marta Bunge <martabunge(a)hotmail.com> wrote: > > Dear Steve, > > When an elementary (base) topos S is specified, I use "S-bounded topos" to mean the pair (E, e), with E an elementary topos and e: E--> S a (bounded) geometric morphism. When S = Set (a model of ZFC) and E an arbitrary elementary topos, then there is a most one geometric morphism e:E---> Set, so in that case the latter need not be specified. I therefore use "E is a Grothendieck topos" to mean "E is an elementary topos bounded over Sets". The latter has been shown to be equivalent to what Grothendieck meant by it. > > Best regards, > Marta > ************************************************ > Marta Bunge > Professor Emerita > Dept of Mathematics and Statistics > McGill University > Montreal, QC, Canada H3A 2K6 > Home: (514) 935-3618 > marta.bunge(a)mcgill.ca > http://www.math.mcgill.ca/people/bunge > ************************************************ > From: Steve Vickers <s.j.vickers(a)cs.bham.ac.uk> > Sent: October 27, 2016 7:07:52 AM > To: Categories > Subject: categories: Grothendieck toposes > > For some years now, I have been using the phrase "Grothendieck topos" - > category of sheaves over a site - to allow the site to be in an > arbitrary base elementary topos S (often assumed to have nno). Hence > "Grothendieck topos" means "bounded S-topos". The whole study of > Grothendieck toposes, as of geometric logic, is parametrized by choice of S. > > That's presumably not how Grothendieck understood it, and I know some of > his results assumed S = Set, some classical category of sets. Moreover, > the Elephant defines "Grothendieck topos" that way. > > On the other hand, if a topos is a generalized space, with a classifying > topos being the space of models of a geometric theory, then that surely > meant Grothendieck topos; and there are various reasons for wanting to > vary S. For example, using Sh(X) as S gives us a generalized topology of > bundles, fibrewise over X. > > I'm coming to suspect my usage may confuse. > > How do people actually understand the phase "Grothendieck topos"? Do > they hear potential for varying an implicit base S, or do they hear a > firm implication that S is classical? > > Steve Vickers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes
by Marta Bunge 01 Nov '16

01 Nov '16

Dear Steve, Thank you for your interesting comments. Grothendieck did not come up with the notion of an elementary topos - Lawvere and Tierney did, so much so that he informally referred to the subobjects classifier as "the Lawvere object". Nevertheless, the basic idea of Grothendieck of a category of sheaves on a site is indeed captured by the more general (and certainly less controversial) notion of an S-bounded elementary topos, where S is an arbitrary elementary topos with an NNO. I think that this is what you had in mind. As for the word "topos", I believe that it ought to be specified in any context where one uses it. I find this way of proceeding preferable to identifying it with "Grothendieck topos" as Joyal suggests. Best regards, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University Montreal, QC, Canada H3A 2K6 Home: (514) 935-3618 marta.bunge(a)mcgill.ca http://www.math.mcgill.ca/people/bunge ************************************************ ________________________________ From: Steve Vickers <s.j.vickers(a)cs.bham.ac.uk> Sent: October 31, 2016 7:27:57 AM To: Marta Bunge Cc: categories(a)mta.ca Subject: categories: Re: Grothendieck toposes Dear Marta, Thanks for your reply. My question was a survey of usage, so what follows is not meant disputatiously, but I thought your reasoning raised some interesting issues. As I see it you give two reasons here for taking Grothendieck toposes as being over Set = ZFC, both pertinent: (1) what Grothendieck meant, and (2) the (essential) uniqueness of geometric morphisms to Set. (1) is delicate, given Grothendieck's underlying implicit definition of topos as "that of which topology is the study". My understanding of this (you may know more - my knowledge of Grothendieck's work is almost all second-hand) is that there are two ways of viewing it, somewhat akin (respectively) to algebraic and general topology. The first is that he meant a topos to be a category with which to do sheaf cohomology, by forming an (exact) injective resolution of Abelian groups, taking global sections (becoming non-exact), and then extracting the ker/im cohomology groups. This is perhaps an algebraic topologist's idea of what topology means. Classicality of Set seems then to be needed in order to get injective hulls in the category of sheaves over a site. On the other hand, there is also the idea that global points can be recovered as sections of the geometric morphism to Set, and the topos simultaneously embodies both the points and the topology on them. Is this also part of what Grothendieck meant? It is closer to general topology, points and their continuous transformations. This idea then generalizes well to elementary toposes, replacing Set by S. Elementary toposes are not generalized spaces in themselves, but bounded geometric morphisms between them are, and many topological properties are redefined for geometric morphisms. (2) raises the question of why Set should have its privileged property. Technically, it is that every object of Set is a colimit of copies of 1, and that is preserved up to unique isomorphism by the inverse image part of any geometric morphism. But is that not because ZFC provides a 2-level structure of sets and classes, and we are implicitly using ZFC classes for our toposes? As we explore the foundational options then we should expect this uniqueness property for Set = ZFC to evaporate. All the best, Steve. > On 30 Oct 2016, at 20:17, Marta Bunge <martabunge(a)hotmail.com> wrote: > > Dear Steve, > > When an elementary (base) topos S is specified, I use "S-bounded topos" to mean the pair (E, e), with E an elementary topos and e: E--> S a (bounded) geometric morphism. When S = Set (a model of ZFC) and E an arbitrary elementary topos, then there is a most one geometric morphism e:E---> Set, so in that case the latter need not be specified. I therefore use "E is a Grothendieck topos" to mean "E is an elementary topos bounded over Sets". The latter has been shown to be equivalent to what Grothendieck meant by it. > > Best regards, > Marta > ************************************************ > Marta Bunge > Professor Emerita > Dept of Mathematics and Statistics > McGill University > Montreal, QC, Canada H3A 2K6 > Home: (514) 935-3618 > marta.bunge(a)mcgill.ca > http://www.math.mcgill.ca/people/bunge > ************************************************ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Internal categories in monoidal categories
by Mike Stay 30 Oct '16

30 Oct '16

I'm having trouble understanding cotensors and the role they play in categories internal to a monoidal category. Say we take the data from a category internal to Set and apply the "free complex-linear sums" functor to it. The result is not quite a category in Hilb, since the functor doesn't preserve limits: the product in Set gets mapped to the tensor product in Hilb. In particular, the composition function gets mapped to the composition linear transformation o: C^{Mor sxt Mor} -> C^Mor However, since we have a chosen basis, we can take a superposition of composable morphisms and produce a superposition of compositions. Is this an example of a category internal to a monoidal category, and if so, is the Hilbert space C^{Mor sxt Mor} an example of a cotensor? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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[CFP] RAMiCS 2017
by Damien Pous 27 Oct '16

27 Oct '16

Second Call for Papers ----------------------------------------------------- 16th International Conference on RELATIONAL AND ALGEBRAIC METHODS IN COMPUTER SCIENCE (RAMiCS 2017) May 15-19, 2017 Lyon, France http://ramics-conference.org ----------------------------------------------------- GENERAL INFORMATION: For more than two decades, the RAMiCS conferences series has been the main venue for research on relation algebras, Kleene algebras and similar algebraic formalisms, and their applications as conceptual and methodological tools in computer science and beyond. TOPICS: We invite submissions in the general field of algebraic structures relevant to computer science and on applications of such algebras. Topics of the conference include, but are not limited to the following -theory * algebraic structures from semigroups, residuated lattices and semirings to Kleene algebras, relation algebras and quantales * other algebras relevant to the theory of automata, concurrency, formal languages, games, networks, programming languages and social choice * algorithmic, category-theoretic, coalgebraic or proof-theoretic methods for such algebras * their formalisation with automated and interactive theorem provers -applications * tools and techniques for the verification and correctness of sequential and concurrent programs * quantitative and qualitative models for computing systems * logics of programs, e.g., modal, dynamic, interval, temporal or resource logics, logics for games, social choice and distributed systems * design of algorithms, network protocol analysis, optimisation and control INVITED SPEAKERS: * Annabelle McIver (Macquarie University, Sydney) * Jean-Eric Pin (CNRS, IRIF, Paris) * Alexandra Silva (University College, London) IMPORTANT DATES: Abstract Submission: 2016, Nov 25 Paper Submission: 2016, Dec 2 Author Notification: 2017, Feb 3 Camera-ready papers: 2017, Feb 24 RAMiCS 2017: 2017, May 15-19 SUBMISSION INSTRUCTIONS: Submission is via EasyChair at https://www.easychair.org/conferences/?conf=ramics16 All papers will be peer reviewed by at least three referees. The proceedings will be published in an LNCS volume by Springer, ready at the conference. Submissions must be in English, in PDF format and should not exceed 16 pages in LNCS style. Submissions must be unpublished, not under review for publication elsewhere and provide sufficient information to judge their merits. Additional material may be provided in a clearly marked appendix or by a reference to a manuscript on a web site. Experimental data, software or mathematical components for theorem provers must be available in sufficient detail for reviewers. Deviation from these requirements may lead to rejection. Accepted papers must be produced with LaTeX. One author of each accepted paper is expected to present the paper at the conference. Formatting instructions and LNCS style files can be obtained at: http://www.springer.de/comp/lncs/authors.html. COMMITTEES: Conference Chair: Damien Pous, CNRS, France Programme Chairs: Peter Höfner, Data61, Australia Georg Struth, U Sheffield, UK, Programme Committee: Luca Aceto, Reykjavik U, Iceland Rudolf Berghammer, U Kiel, Germany Filippo Bonchi, CNRS, France Jules Desharnais, U Laval, Canada Hitoshi Furusawa, Kagoshima U, Japan Tim Griffin, U Cambridge, UK Walter Guttmann, U Canterbury, New Zealand Robin Hirsch, UCL, UK Peter Höfner, Data61, CSIRO, Australia Marcel Jackson, LaTrobe U, Australia Jean-Baptiste Jeannin, Samsung, USA Peter Jipsen , Chapman U, USA Christian Johansen, U Oslo, Norway Wolfram Kahl, McMaster U, Canada Dexter Kozen, Cornell U, USA Szabolcs Mikulas, Birkbeck U, UK Bernhard Möller, U Augsburg, Germany José N. Oliveira, U Minho, Portugal Damien Pous, CNRS, France Georg Struth, U Sheffield, UK, Pascal Weil, CNRS, France Michael Winter, Brock U, Canada [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Categories Logic and Physics - 30 November
by Ross Duncan 26 Oct '16

26 Oct '16

CATEGORIES LOGIC AND PHYSICS SCOTLAND Wednesday 30 November 2016 Department of Computer and Information Sciences, University of Strathclyde http://homepages.inf.ed.ac.uk/cheunen/clapscotland/ Following our successful meeting in April, we are pleased to announce the second meeting of CLAP Scotland. The final programme is still under construction, but will be available on the website soon. Currently speakers include: * Kevin Dunne (University of Strathclyde) * Martti Karvonen (University of Edinburgh) * Ulrich Kraehmer (University of Glasgow) We invite contributions to the meeting! If you would like to give a talk, please send us an email. Registration is free. For catering purposes, please email the local organiser <ross.duncan(a)strath.ac.uk> as soon as possible if you plan to attend. Best wishes, Chris Heunen Ross Duncan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Chaire de la Vallee Poussin 2017
by Tim Van der Linden 26 Oct '16

26 Oct '16

Dear Colleagues, We are very pleased to announce that professor George Janelidze will be the recipient of the Chaire de la Vallee Poussin 2017 (see https://www.uclouvain.be/453620.html for more information about this chair). Professor Janelidze has kindly accepted to give a series of invited lectures at the Universite catholique de Louvain during the week starting on May 29, 2017. We plan to organize an international workshop at the end of that week (June 1-3). More details concerning the subject of the talks and the scientific programme of that week will be communicated later on. With best regards, Marino Gran, Tim Van der Linden, and Enrico Vitale [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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12-month post-doctoral grant in Padova
by Maria Emilia Maietti 26 Oct '16

26 Oct '16

Dear all a 12 months postdoctoral grant is offered at the University of Padova to work within the project "Algebraic set theory of the Minimalist Foundation" under my supervision and in collaboration with the other members of the Padova logic group at the Department of Mathematics. The announcement and the application form can be found at the bottom page in http://www.math.unipd.it/en/news/?id=1920 The deadline is November 10th 2016. Preference will be given to candidates with expertise in categorical logic. Many thanks for your attention Maria Emilia Maietti [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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John C. Reynolds Doctoral Dissertation Award
by Jeremy.Gibbons＠cs.ox.ac.uk 26 Oct '16

26 Oct '16

Dear all, If this year one of your PhD students has graduated and the thesis was outstanding, please consider proposing it for the John C. Reynolds Doctoral Dissertation Award. See http://www.sigplan.org/Awards/Dissertation/ for details. Best wishes, Susan -- Professor Susan Eisenbach Department of Computing Imperial College London London SW7 2AZ, U.K. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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