November 2016 - Categories - categories.org.au

Predicativity in Categories
by Sergey Goncharov 02 Dec '16

02 Dec '16

Dear Community, I am recently racking my brains in trying to understand what (im-)predicativity means from a categorical perspective, but failing short, hence calling for you help. Certain constructive principle are easy to accommodate (and understand!) within category theory, e.g. the axiom of choice (all epis are split), excluded middle (every mono is a coproduct summand), however this does not seem to be the case with the notion of predicativity. The rough intuition says that impredicative definitions attempt to define things topdown: first by introducing a class, and then defining elements of this class by their characteristic properties -- arguably, a controversial idea, if you strive for foundations. That intuition is sort of helpful to deal with extreme cases, e.g. to accept that toposes are impredicative, and Martin-L??f type theory is predicative, but it does not seem to give a complete answer. The discussion and the papers I found on the web rather seem to reflect the set-theoretic tradition in categorical terms, than to explain the issue intrinsically. Let me clarify. We agree that toposes are impredicative (or, at least not compatible with predicative approach) because they include subobject classifiers, and those arguably classify to much, for according to Bishop, we have no idea a priory, what an arbitrary subset of given set is like. On the other hand, classifying *something* can be OK, e.g. in extensive categories we can classify summands: 2 = 1 + 1 is a summand classifier, and extensive categories are certainly not considered to be impredicative, for every pretopos is extensive. Now, there is a spectrum of classes of monomorphisms that could be classified, depending on a category. I could e.g. try to classify r.e-subsets in an effective topos. How can I know, if it is still OK or not OK from a predicative perspective? (I realize that if we reject impredicativity in foundations, we probably just can not construct any example of a topos, including the effective one, but suppose, I still can make sense of this idea of classifying r.e.-subsets somehow without constructing a topos). Sorry, if that all sounded jumbled, but maybe you can understand my concerns. I presume that a complete answer just does not exist, and would be a subject to an open research, but I would be happy to understand at least informally what sort of justification is used to claim that one category is predicative (or predicative-friendly) and another one is not. Looking forward to your answers. Sergey [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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A paper on double categories
by Marco Grandis 23 Nov '16

23 Nov '16

The following paper, in press in a volume dedicated to B. Banaschewski, is downloadable online: M. Grandis - R. Paré, Span and cospan representations of weak double categories, Categ. Gen. Algebr. Struct. Appl., Vol. 6 n. 5, in press Published online (18 Nov 2016) at: http://www.cgasa.ir/? _action=press&issue=-1&_is=Articles%20in%20Press Abstract. We prove that many important weak double categories can be `represented' by spans, using the basic higher limit of the theory: the tabulator. Dually, representations by cospans via cotabulators are also frequent. Marco Grandis and Bob Paré [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Plural form of topos
by Johannes Huebschmann 20 Nov '16

20 Nov '16

Dear All A minor comment: The correct plural form of "topos" is "topoi", as the plural form of "logos" is "logoi". I vaguely remember that the plural form of "topos" was already discussed on this board. In ancient Greek, "(ho) topos" is a noun. Here is a citation: "Ton de hyperouranion topon ..." (Plato, Phaidros, Bibliotheca Oxoniensis, Tomus II, 247 c). This citation shows that "topos" is a noun. Likewise, in algebraic geometry, it is common to use "zero loci" as the plural form of "zero locus". Johannes HUEBSCHMANN Johannes Professeur ?m?rite USTL, UFR de Math?matiques UMR 8524 Laboratoire Paul Painlev? 59 655 VILLENEUVE d'ASCQ Cedex/France http://math.univ-lille1.fr/~huebschm TEL. (33) 3 20 43 41 97 (33) 3 20 43 42 33 (s?cr?tariat) (33) 3 20 43 48 50 (s?cr?tariat) Fax (33) 3 20 43 43 02 Johannes.Huebschmann(a)math.univ-lille1.fr Date: Wed, 09 Nov 2016 17:53:24 +0200 From: Patrik Eklund <peklund(a)cs.umu.se> To: Marta Bunge <martabunge(a)hotmail.com> Cc: categories(a)mta.ca, wlawvere <wlawvere(a)buffalo.edu> Subject: categories: Re: Grothendieck toposes Heraclitus of Ephesus used 'logos' for what is pretty well-known. Even more known is that 'logos' is in the first sentences of the Gospel of John. I'm not claiming anybody should care, but using 'logos' for a math concept must be justified, not for what I mentioned, but for what it is. What is this discussion really about? Just wondering. Patrik On 2016-11-09 04:35, Marta Bunge wrote: > Dear all, > > > Indeed, as pointed out by Bill Lawvere, the term "logos" was > introduced and is central to the book by Freyd and Scedrov. In > addition to that of Walter Tholen there is a review of it by myself > > > Categories, Allegories, by Peter J. Freyd; Andrej Scedrov > > Review by Marta C. Bunge, > > The Journal of Symbolic Logic 56-1 (March 1993) 352-354 > > > Best wishes, > > Marta [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Third Nordic Logic Summer School (NLS) 2017, First Announcement
by Erik Palmgren 18 Nov '16

18 Nov '16

[Includes course on categorical logic -- see below] Third Nordic Logic Summer School (NLS) 2017, First Announcement --------------------------------------------------------------------------------------------- Stockholm, August 7 - 11, 2017 The third Nordic Logic Summer School is arranged under the auspices of the Scandinavian Logic Society (http://scandinavianlogic.org/) The two previous schools were organized in Nordfjordeid, Norway (2013) and Helsinki (2015). The intended audience is advanced master students, PhD-students, postdocs and experienced researchers wishing to learn the state of the art in a particular subject. The school is co-located with Logic Colloquium 2017 (14-20 August) and Computer Science Logic 2017 (21-24 August). The school will consist of 10 five-hour courses, running in two parallel streams. In addition, there will be short student presentations and poster sessions. Lecturers and courses ------------------------------ The following lecturers and course topics are confirmed. Mirna Dzamonja (Univeristy of East Anglia) -- Set Theory Martin Escardo (Birmingham) -- Topological and Constructive Aspects of Higher-Order Computation Henrik Forssell (Oslo) -- Categorical Logic Volker Halbach (Oxford) -- Formal Theories of Truth Larry Moss (Indiana University, Bloomington) -- Natural Logic Anca Muscholl (LaBRI, Universit? Bordeaux) -- Logic in Computer Science - Control and Synthesis, from a Distributed Perspective Eric Pacuit (University of Maryland) -- Logic and Rationality Peter Pagin and Dag Westerst?hl (Stockholm University) -- Compositionality Sara L. Uckelman (Durham) -- Medieval Logic Andreas Weiermann (Ghent) -- Proof Theory Certificates for participation will be provided. There will be possibilities to take official credits for some of the courses. Venue: Kr?ftriket Campus, Stockholm University. Important dates --------------------- Registration Registration opens: March 6, 2017 Early registration: June 2, 2017 Late registration: August 4, 2017. Submission of abstracts for presentations and posters Opening: March 6, 2017 Closing: May 2, 2017 Notification of acceptance: May 16, 2017 The registration fee will be at most 2000 SEK (approx 200 Euros) per participant, and includes lunches, coffee breaks and conference materials. It does not cover accommodation, but there will special offers at hostels and hotels (in the range 700 -1200 SEK/night for single rooms, and much lower for shared hostel rooms) available when the registration opens. Some participation/fee waiver grants may be available. Further information ------------------------- Further information about submissions, registration and accommodation possibilities will (in due time) be available on the NLS webpage: https://www.sls17.conf.kth.se Enquiries: nls2017(a)philosophy.su.se Committees ---------------- Program Committee of NLS 2017: Thierry Coquand (G?teborg), Ali Enayat (G?teborg), Mai Gehrke (IRIF, Paris), Nina Gierasimczuk (Copenhagen), Valentin Goranko (Stockholm U), Lauri Hella (Tampere), Lars Kristiansen (Oslo), Juha Kontinen (Helsinki), ?ystein Linnebo (Oslo), Sara Negri (Helsinki), Erik Palmgren (chair, Stockholm U). Local Organizing Committee of NLS 2017: Valentin Goranko (co-chair), Dilian Gurov, Roussanka Loukanova, Peter LeFanu Lumsdaine, Anders Lundstedt, Erik Palmgren (co-chair). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Giraud_Elementary_?
by Eduardo Julio Dubuc 17 Nov '16

17 Nov '16

Hi, in this posting I will use the terminology used by most people in the list. There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos. Grothendieck are Giraud and Elementary, my question is: Are Elementary Giraud topos which are not Grothendieck ? Examples ? greetings e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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partial adjoints
by Peter Bubenik 16 Nov '16

16 Nov '16

Hello all, Sender: categories(a)mta.ca Precedence: bulk Reply-To: Peter Bubenik <peter.bubenik(a)gmail.com> Has anyone studied pairs of functors for which there exists a unit, but not a counit? Is there a name for such things? In the case of interest, the functors are between Cat and Met, the category of (extended pseudo-) metric spaces and 1-Lipschitz maps. The unit provides a characterization of certain coherent maps of metric spaces. Further details may be found in https://arxiv.org/abs/1603.07406. Thanks, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes
by Thomas Streicher 11 Nov '16

11 Nov '16

Dear Andr'e, of course, logos would be a natural name when understood as elementary toposes and logical morphisms. In the Elephant one finds Top, BTop and LogTop as names for elementary toposes and various different notions of morphisms. There is a problem with just geometric morphisms since they dont correspond to Grothendieck toposes - only bounded ones do. I personally find logical morphisms the basic notion and geometric morphisms as a derived notion. As I learnt from Benabou quite some time ago one should consider geometric morphisms as fibred toposes. If SS is a (base) topos then P : EE -> SS is a fibration of toposes iff P is a Grothendieck fibration whose fibers are toposes and whose reindexing functors are logical. One may characterize them as fibrations P such that P is a logical morphism of toposes. Fibrations of cocomplete toposes over SS correspond to finite limit preserving functors from SS to some topos (namely Delta(I) = \coprod_I 1_I). Fibrations of locally small cocomplete toposes correspond to finite limit preserving functors to some topos which, moreover, have a right adjoint. The bounded gm's to SS correspond to fibrations of locally small cocomplete toposes which moreover admot a small generating family. This, for me justifies the notion of bounded gm's. The fibrational point of view doesn't suggest how to compose them (see discussion pp.66-67 of my notes on fibrations). But the corresponding Delta's are obviously closed under composition. I then would add the "observation" that these functors maybe understood as "glorified" frame morphisms. This, of course, doesn't reflect history but that's not the issue here. It's about "rational reconstruction" of a notion. Thomas PS That's what I think Benabou had in mind with his remark after one of your lectures at IHES last November. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes
by Marta Bunge 10 Nov '16

10 Nov '16

Dear all, Indeed, as pointed out by Bill Lawvere, the term "logos" was introduced and is central to the book by Freyd and Scedrov. In addition to that of Walter Tholen there is a review of it by myself Categories, Allegories, by Peter J. Freyd; Andrej Scedrov Review by Marta C. Bunge, The Journal of Symbolic Logic 56-1 (March 1993) 352-354 Best wishes, Marta ________________________________ From: wlawvere <wlawvere(a)buffalo.edu> Sent: November 8, 2016 8:32:16 AM To: categories(a)mta.ca Subject: categories: Re: Grothendieck toposes The term 'logos' already has a well-established meaning. See Tholen's review of the 1990 book by Freyd and Scedrov: Categories, allegories ...(a logos is a regular category in which the subobjects of an object form a lattice, and in which each inverse-image map has a right adjoint) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Grothendieck toposes
by Thomas Streicher 10 Nov '16

10 Nov '16

On Wed, Nov 09, 2016 at 02:44:45PM +0000, Paul B Levy wrote: > > > On 09/11/16 10:48, Thomas Streicher wrote: >> What Eduardo means by "Giraud topos" is a category validating all >> conditions of the Giraud theorem with the exception of having a small >> generating family. These guys can be elementary toposes or not. >> There is Freyd's example where objects are set X with an ordinal >> indexed family of bijections from X to X > > So they don't form a class. > > Is there an example where the objects form a class? > > (I like categories to have this property!) That's not the problem. Objects do form a class! Eduardo rather thought that Freyd's example were not locally small. But it is! Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Giraud_Elementary_?
by Eduardo Julio Dubuc 09 Nov '16

09 Nov '16

By Giraud topos I mean all the assumptions in Giraud's theorem, exept a small set of generators. What Grothendieck call "faux topos". See SGA4 Exposse IV Theoreme 1.2 (Giraud's theorem) and Example 2.8 (faux topos). best e.d. I guess I was wrong when I thought that "Giraud Topos" was established terminology in the cat-list. On 11/8/16 09:59, Daniil Frumin wrote: > What is actually a Giraud topos? I cannot find a reference for this on > the internet. > > On Mon, Nov 7, 2016 at 10:03 PM, Eduardo Julio Dubuc <edubuc(a)dm.uba.ar > <mailto:edubuc@dm.uba.ar>> wrote: > > Hi, in this posting I will use the terminology used by most people in > the list. > > There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos. > > Grothendieck are Giraud and Elementary, my question is: > > Are Elementary Giraud topos which are not Grothendieck ? > > Examples ? > > greetings e.d. > > > [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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