August 2017 - Categories - categories.org.au

modules over a group
by Michael Barr 02 Sep '17

02 Sep '17

If pi is a group, are the categories of right pi-modules and of 2-sided pi-modules equivalent? The reason I raise this question is that I was looking at Example 5, p. 43 in the TAC reprint of Jon Beck's thesis. He identifies the abelian group objects of Gp/pi as right pi-modules via the following construction. Let M be a right pi-module and let Y = pi x M with multiplication (x,m)(x',m') = (xx',m+m'x). This is an abelian group object in Gp/pi and for any Z --> pi, Hom(Z,Y) = Der(Z,M) the group of derivations d: Z --> M meaning that d(zz') = (dz)z' + d(z'). Here M becomes a right Z-module using the map Z --> Y. For the converse, if Y --> pi is an abelian group object in Gp/pi, then the kernel of the map is an abelian normal subgroup M of Y and the conjugation action of Y on M descends to pi since M is abelian and this conjugation action is a right pi-module structure (see Beck's thesis for all details). But suppose M is a 2-sided pi-module. Now let Y = pi x M and define (x,m)(x',m') = (xx',mx'+xm'). This is just as above an abelian group structure in Gp/pi. For any Z --> pi, Hom(Z,Y) = Der(Z,M) as above but now Der means 2-sided derivations d(zz') = (dz)z' + z(dz'), still an abelian group. But the kernel of Y --> pi is M but now with a new right module structure m*x = x^{-1}mx. So there is a functor from 2-sided to right modules by replacing the 2-sided operation by conjugation which is now a right operation, but could this possibly be an equivalence? I don't see how. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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UPDATE: Dana Scott at Octoberfest 2017
by Steve Awodey 01 Sep '17

01 Sep '17

UPDATE: There will be a special, pre-Octoberfest lecture by Prof. Dana Scott on Friday 27 October, 4:30 - 6pm. *** Second Announcement *** Category Theory Octoberfest 2017 28-29 October 2017 Carnegie Mellon University The 2017 Category Theory Octoberfest will be held on the weekend of Saturday the 28th and Sunday the 29th at Carnegie Mellon University in Pittsburgh. Following the tradition of past Octoberfests, this is intended to be an informal meeting, covering all areas of category theory and its applications. Talks by PhD students and young researchers are particularly encouraged! Details and travel information can be found here: https://www.andrew.cmu.edu/user/awodey/CToctoberfest/Octoberfest.html Registration: There is no registration fee. Registration is optional, but participants are requested to contact the organizers in advance, especially if they would like to give a talk. To register and/or submit a talk, please send email to the organizers with the following information: your name, will you give a talk (yes or no), the title of your talk (if yes). Organizers: Steve Awodey (awodey(a)cmu.edu) Jonas Frey (jonasf(a)andrew.cmu.edu) Hope to see you in Pittsburgh! Steve and Jonas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Categories with specified pullbacks
by David Roberts 31 Aug '17

31 Aug '17

Dear all, I'm trying to track down a paper I once saw that I recall constructed, from a category with pullbacks, an equivalent category with *specified* pullbacks. I don't believe there was any strictness, in the sense that pulling back along fg was the same as the result of pulling back along one then the other. The feature I'm interested in is that I think that this construction required no global choice, except for one direction of the equivalence, which for me is no big deal. I may have misremembered this last bit, so I need to check the paper. Thanks, David -- David Roberts http://ncatlab.org/nlab/show/David+Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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CT Octoberfest 2017 at CMU
by Steve Awodey 30 Aug '17

30 Aug '17

** Second Announcement ** Category Theory Octoberfest 2017 28-29 October 2017 Carnegie Mellon University The 2017 Category Theory Octoberfest will be held on the weekend of Saturday the 28th and Sunday the 29th at Carnegie Mellon University in Pittsburgh. Following the tradition of past Octoberfests, this is intended to be an informal meeting, covering all areas of category theory and its applications. Talks by PhD students and young researchers are particularly encouraged! Details and travel information can be found here: https://www.andrew.cmu.edu/user/awodey/CToctoberfest/Octoberfest.html <https://www.andrew.cmu.edu/user/awodey/CToctoberfest/Octoberfest.html> Registration: There is no registration fee. Registration is optional, but participants are requested to contact the organizers in advance, especially if they would like to give a talk. To register and/or submit a talk, please send email to the organizers with the following information: your name, will you give a talk (yes or no), the title of your talk (if yes). Organizers: Steve Awodey (awodey(a)cmu.edu) Jonas Frey (jonasf(a)andrew.cmu.edu) Hope to see you in Pittsburgh! Steve and Jonas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: modules over a group
by George Janelidze 27 Aug '17

27 Aug '17

Could you please make more precise what you are saying? Meanwhile, let me go back to your: > ....But the kernel of Y --> pi is M but now with a new right > module structure m*x = x^{-1}mx. So there is a functor from 2-sided to > right modules by replacing the 2-sided operation by conjugation which is > now a right operation, but could this possibly be an equivalence? I > don't see how. and put my "of course not" also here: Let pi = be the 2-element cyclic group, written as pi={1,x}, and let A and B be 2-sided pi-modules defined as follows: - A and B have the same underlying group, which is the additive group of integers; - pi acts trivially on A, that is, xa=a=ax; - pi acts on B by xb=-b=bx. Then your functor will send A and B to the same pi-module, while A and B are obviously not isomorphic. Therefore your functor is not an equivalence. George From: Michael Barr Sent: Saturday, August 26, 2017 7:00 PM To: George Janelidze Cc: Categories list" <Categories list> Subject: Re: categories: modules over a group Okay, your argument seems sound. But consider the following. If Z is a free group, say Z = F(X), then you can show that for any Z-module M (if M is a pi-module a homomorphism Z --> pi makes M into a Z-module of course), then for either a right or a 2-sided module, it is trivial to see that Der(Z,M) = M^X. That is, any function X --> M extends in a unique way to a derivation Z --> M. Now resolve an arbitrary Z with G^2Z ===> GZ ---> Z and then we get an equalizer Der(Z,M) ---> Der(GZ,M) ===> Der(G^2Z,M) or Der(Z,M) ---> Der(GC,M) ===> Der(G^2Z,M) which is Der(GZ,M) ---> M^Z ===> M^{GZ} for whichever kind of module you have. But, the ismorphisms are not natural (except with respect to maps of the basis, which G\epsilon is, but \epsilon G isn't) and so the two versions of Der(Z,M) may well be different and hence the two versions of Y. I guess Jon didn't correctly describe the category of pi-modules. Michael ----- Original Message ----- From: "George Janelidze" <George.Janelidze(a)uct.ac.za> To: "Michael Barr" <barr(a)math.mcgill.ca>, "Categories list\" <Categories list>" <categories(a)mta.ca> Sent: Saturday, August 26, 2017 6:47:42 AM Subject: Re: categories: modules over a group Dear Michael, Although the answer to your question is "Of course not!", it is a very interesting question because it inspires very interesting questions about internal actions in semi-abelian categories. In this message I shall, however, only explain why is it "Of course not!" (with "!"). 1. The category of pi-modules can be identified with the category of Z[pi]-modules, where Z[pi] is the group ring over pi. Now, suppose, for simplicity, that pi is abelian. Then you question can be stated as: Are the rings Z[pi] and Z[pi x pi] Morita equivalent? Of course not, because two commutative rings are Morita equivalent if and only if they are isomorphic. Well, for some funny groups they are, but I know you are not asking asking about funny groups. 2. Your functor from 2-sided modules to split epis is fine, but it is not an equivalence, e.g. because your structure on Y will not allow you to recover the pi-module structure on M from the Beck module structure on Y-->pi. In fact one can prove the following: Two 2-sided pi-module structures * and # on M give the same Beck modules structures on a chosen pi-->Y-->pi if and only if x*m*(inverse of x) = x#m#(inverse of x) for every x in pi and every m in M. For example if we assume that both left actions are trivial, then the right actions will have to coincide, as in Beck's theory. Remark: but you never mention what happened after Beck - I mean Bourn protomodularity and internal actions... Best regards, George -------------------------------------------------- From: "Michael Barr" <barr(a)math.mcgill.ca> Sent: Saturday, August 26, 2017 12:18 AM To: "Categories list" <Categories list>" <categories(a)mta.ca> Subject: categories: modules over a group > If pi is a group, are the categories of right pi-modules and of 2-sided > pi-modules equivalent? > > The reason I raise this question is that I was looking at Example 5, p. > 43 in the TAC reprint of Jon Beck's thesis. He identifies the abelian > group objects of Gp/pi as right pi-modules via the following > construction. Let M be a right pi-module and let Y = pi x M with > multiplication (x,m)(x',m') = (xx',m+m'x). This is an abelian group > object in Gp/pi and for any Z --> pi, Hom(Z,Y) = Der(Z,M) the group of > derivations d: Z --> M meaning that d(zz') = (dz)z' + d(z'). Here M > becomes a right Z-module using the map Z --> Y. For the converse, if Y > --> pi is an abelian group object in Gp/pi, then the kernel of the map > is an abelian normal subgroup M of Y and the conjugation action of Y on > M descends to pi since M is abelian and this conjugation action is a > right pi-module structure (see Beck's thesis for all details). > > But suppose M is a 2-sided pi-module. Now let Y = pi x M and define > (x,m)(x',m') = (xx',mx'+xm'). This is just as above an abelian group > structure in Gp/pi. For any Z --> pi, Hom(Z,Y) = Der(Z,M) as above but > now Der means 2-sided derivations d(zz') = (dz)z' + z(dz'), still an > abelian group. But the kernel of Y --> pi is M but now with a new right > module structure m*x = x^{-1}mx. So there is a functor from 2-sided to > right modules by replacing the 2-sided operation by conjugation which is > now a right operation, but could this possibly be an equivalence? I > don't see how. > > Michael > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Disclaimer - University of Cape Town This e-mail is subject to UCT policies and e-mail disclaimer published on our website at http://www.uct.ac.za/about/policies/emaildisclaimer/ or obtainable from +27 21 650 9111. If this e-mail is not related to the business of UCT, it is sent by the sender in an individual capacity. Please report security incidents or abuse via csirt(a)uct.ac.za [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: modules over a group
by Michael Barr 27 Aug '17

27 Aug '17

Okay, your argument seems sound. But consider the following. If Z is a free group, say Z = F(X), then you can show that for any Z-module M (if M is a pi-module a homomorphism Z --> pi makes M into a Z-module of course), then for either a right or a 2-sided module, it is trivial to see that Der(Z,M) = M^X. That is, any function X --> M extends in a unique way to a derivation Z --> M. Now resolve an arbitrary Z with G^2Z ===> GZ ---> Z and then we get an equalizer Der(Z,M) ---> Der(GZ,M) ===> Der(G^2Z,M) or Der(Z,M) ---> Der(GC,M) ===> Der(G^2Z,M) which is Der(GZ,M) ---> M^Z ===> M^{GZ} for whichever kind of module you have. But, the ismorphisms are not natural (except with respect to maps of the basis, which G\epsilon is, but \epsilon G isn't) and so the two versions of Der(Z,M) may well be different and hence the two versions of Y. I guess Jon didn't correctly describe the category of pi-modules. Michael ----- Original Message ----- From: "George Janelidze" <George.Janelidze(a)uct.ac.za> To: "Michael Barr" <barr(a)math.mcgill.ca>, "Categories list\" <Categories list>" <categories(a)mta.ca> Sent: Saturday, August 26, 2017 6:47:42 AM Subject: Re: categories: modules over a group Dear Michael, Although the answer to your question is "Of course not!", it is a very interesting question because it inspires very interesting questions about internal actions in semi-abelian categories. In this message I shall, however, only explain why is it "Of course not!" (with "!"). 1. The category of pi-modules can be identified with the category of Z[pi]-modules, where Z[pi] is the group ring over pi. Now, suppose, for simplicity, that pi is abelian. Then you question can be stated as: Are the rings Z[pi] and Z[pi x pi] Morita equivalent? Of course not, because two commutative rings are Morita equivalent if and only if they are isomorphic. Well, for some funny groups they are, but I know you are not asking asking about funny groups. 2. Your functor from 2-sided modules to split epis is fine, but it is not an equivalence, e.g. because your structure on Y will not allow you to recover the pi-module structure on M from the Beck module structure on Y-->pi. In fact one can prove the following: Two 2-sided pi-module structures * and # on M give the same Beck modules structures on a chosen pi-->Y-->pi if and only if x*m*(inverse of x) = x#m#(inverse of x) for every x in pi and every m in M. For example if we assume that both left actions are trivial, then the right actions will have to coincide, as in Beck's theory. Remark: but you never mention what happened after Beck - I mean Bourn protomodularity and internal actions... Best regards, George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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ETAPS 2018 joint call for papers
by Tarmo Uustalu 24 Aug '17

24 Aug '17

NEW! The proceedings of ETAPS 2018 will appear in *gold open access*. As an exception for this year only, the important dates of POST differ from the other member conferences! ****************************************************************** JOINT CALL FOR PAPERS 21st European Joint Conferences on Theory And Practice of Software ETAPS 2018 Thessaloniki, Greece, 14-21 April 2018 http://www.etaps.org/2018 ****************************************************************** -- ABOUT ETAPS -- ETAPS is the primary European forum for academic and industrial researchers working on topics relating to software science. ETAPS, established in 1998, is a confederation of five main annual conferences, accompanied by satellite workshops. ETAPS 2018 is the twenty-first event in the series. -- MAIN CONFERENCES (16-20 April) -- * ESOP: European Symposium on Programming (PC chair Amal Ahmed, Northeastern University, USA) * FASE: Fundamental Approaches to Software Engineering (PC chairs Alessandra Russo, Imperial College London, UK, and Andy Schürr, Technische Universität Darmstadt, Germany) * FoSSaCS: Foundations of Software Science and Computation Structures (PC chairs Christel Baier, Technische Univ. Dresden, Germany, and Ugo Dal Lago, Università di Bologna, Italy) * POST: Principles of Security and Trust (PC chairs Lujo Bauer, Carnegie Mellon University, USA, Ralf Küsters, University of Stuttgart, Germany) * TACAS: Tools and Algorithms for the Construction and Analysis of Systems (PC chairs Dirk Beyer, Ludwig-Maximilian-Univ. München, Germany, and Marieke Huisman, Universiteit Twente, The Netherlands) TACAS '18 hosts the 7th Competition on Software Verification (SV-COMP). -- INVITED SPEAKERS -- * Unifying speaker: Martin Abadi (Google Research & University of California at Santa Cruz, USA) * FASE invited speaker: Pamela Zave (AT&T Labs, USA) * ESOP invited speaker: Derek Dreyer (MPI-SWS, Germany) * POST invited speaker: Benjamin C. Pierce (University of Pennsylvania, USA) -- IMPORTANT DATES (all member conferences except POST) * Abstracts due: 13 October 2017 * Papers due: 20 October 2017 * Rebuttal (ESOP, FoSSaCS only): 6-8 December 2017 * Notification: 22 December 2017 * Camera-ready versions due: 23 February 2018 IMPORTANT DATES for POST * Abstracts due: 22 November 2017 * Papers due: 24 November 2017 * Rebuttal: 12-16 January 2018 * Notification: 25 January 2018 * Camera-ready versions due: 23 February 2018 -- SUBMISSION INSTRUCTIONS -- ETAPS conferences accept two types of contributions: research papers and tool demonstration papers. Both types will appear in the proceedings and have presentations during the conference. ESOP and FoSSaCS accept only research papers. A condition of submission is that, if the submission is accepted, one of the authors attends the conference to give the presentation. Submitted papers must be in English presenting original research. They must be unpublished and not submitted for publication elsewhere (this does not apply to abstracts). In particular, simultaneous submission of the same contribution to multiple ETAPS conferences is forbidden. Submissions must follow the formatting guidelines of Springer's Lecture Notes in Computer Science and be submitted electronically in pdf through the EasyChair author interface of the respective conference. Submissions not adhering to the specified format and length may be rejected immediately. FASE and POST will use a double-blind review process. The proceedings of ETAPS 2018 will be published in *gold open access*. The copyright of the papers will remain with the authors. It is most likely that the proceedings will be published in the Advanced Research in Computing and Software Science (ARCoSS) subline of Springer's Lecture Notes in Computer Science series. - Research papers FASE, FoSSaCS and TACAS have a page limit of 15 pp (excluding bibliography of max 2 pp) for research papers, whereas POST allows at most 20 pp (excluding bibliography of max 2 pp) and ESOP 25 pp (excluding bibliography of max 2 pp). Additional material intended for the referees but not for publication in the final version - for example, details of proofs - may be placed in a clearly marked appendix that is not included in the page limit. ETAPS referees are at liberty to ignore appendices and papers must be understandable without them. In addition to regular research papers, TACAS solicits also *case study papers* (at most 15 pp, excluding bibliography of max 2 pp). Both TACAS and FASE solicit also *regular tool papers* (at most 15 pp, excluding bibliography of max 2 pp). - Tool demonstration papers Submissions should consist of two parts: * The first part, at most 6 pages, should describe the tool presented. Please include the URL of the tool (if available) and provide information that illustrates the maturity and robustness of the tool. (This part will be included in the proceedings.) * The second part, at most 6 pages, should explain how the demonstration will be carried out and what it will show, including screen dumps and examples. (This part will be not be included in the proceedings, but will be evaluated. ESOP and FoSSaCS do not accept tool demonstration papers. -- SATELLITE EVENTS (14-15 April, 21 April) -- A number of satellite workshops will take place before and after the main conferences: CMCS, CREST, DICE, FAEPAS, GALOP, HotSpot, LiVe, MARS, MeTRiD, SNR, SynCoP, VerifyThis, VPT, VSSE, WRLA. -- HOST INSTITUTION -- ETAPS 2018 is hosted by the School of Informatics of the Aristotle University of Thessaloniki, the largest university in Greece. -- ORGANIZERS Panagiotis Katsaros (general chair), Ioannis Stamelos, Lefteris Angelis, Nick Bassiliades, Alexander Chatzigeorgiou, George Rahonis, Ezio Bartocci, Simon Bliudze, Petros Stratis, Emmanouela Stachtiari, Kyriakos Georgiadis -- FURTHER INFORMATION -- Please do not hesitate to contact the organizers at katsaros(a)csd.auth.gr. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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PSSL101: final announcement
by Nicola Gambino 17 Aug '17

17 Aug '17

Dear colleagues and friends, I am writing with a reminder that registration for the 101st Peripatetic Seminar on Sheaves and Logic (PSSL 101) will close on Friday 1st September. If you wish to attend or give a talk, please follow the instructrions at the web page http://www1.maths.leeds.ac.uk/~pmtng/pssl101.html The meeting will be held on the weekend of September 16th-17th at the University of Leeds, UK. In the afternoon of Friday 15th September there will be special lectures by Jiri Rosicky and Andrew Brooke-Taylor on accessible categories, set theory and model theory. Please do not hesitate to contact me if you have any queries. With best wishes, Nicola [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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An elementary question
by Dana Scott 17 Aug '17

17 Aug '17

The category of posets (= partially ordered sets) and monotone maps is often used as an easy example -- different from the category of sets -- that has products, coproducts, and is cartesian closed but not a topos. Let P and Q be two posets. Define (P (<) Q) as the modified coproduct where all the elements of P are made less than all the elements of Q. QUESTION. Does (P (<) Q) have a nice categorical definition as a functor in the category of posets? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: An elementary question
by Mike Stay 16 Aug '17

16 Aug '17

On Mon, Aug 14, 2017 at 10:50 PM, Patrik Eklund <peklund(a)cs.umu.se> wrote: > So in that text, readable on the monitor, there are indeed partial orders, > and if you, Mike, want to focus on the carriage return, fine. The rendering of text or graphics on a screen made of scan lines is just one easy example of serialization, a nontrivial practical task on which millions of man hours have been spent. The lexicographic ordering is well-defined and well-motivated, and it's reasonable to ask for a nice characterization. -- 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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