September 2014 - Categories - categories.org.au

Partial functor
by David Leduc 16 Mar '15

16 Mar '15

Hi, A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

7 6

Re: Reference search: new categories by replacing morphisms with diagrams
by Jason Erbele 26 Sep '14

26 Sep '14

Ah, it figures I would leave something out in my first post here. My apologies to all of you that were scratching your heads over the missing rules for composition. At least five of you have responded with essentially the same question, so this will be sort of a blanket response. Composition of (f,g,h): A --> B through X and (f',g',h'): B --> C through Y will go through the biproduct X (+) Y with (f'f, [f'g, g'], [h, h'(f+gh)]^T): A --> C. That is, the first component composes in the usual way, the second component is a row matrix, and the third component is a column matrix. In the original category, the various morphisms in the composed triple correspond to A --> B --> C, directly; X --> B --> C and Y --> C; and A --> X and A --> B --> Y, where this A --> B is f+gh rather than f. Identity morphisms in the new category are those with f=id and the zero object for X, which uniquely determines g and h. The embedding I mentioned of the original category into the new category is a functor, after all. Best, Jason On Wednesday, September 24, 2014, Tom Hirschowitz <tom.hirschowitz(a)univ-savoie.fr> wrote: > > > Dear Jason, > > How do your triples compose? > > Best, > Tom > > Jason Erbele <erbele(a)math.ucr.edu> writes: > >> Dear all, >> >> I built a category from another category by keeping "the same" objects >> and taking the morphisms to be diagrams from the old category that >> satisfy certain properties. The closest thing to what I'm doing that >> I have been able to find is factorization systems, but there are some >> major differences. >> >> To be more specific, I am starting with an Abelian category. If there >> are morphisms f: A --> B, g: X --> B, and h: A --> X, it makes sense >> to talk about the morphism f+gh: A --> B, which can be represented >> with a non-commutative triangle. I don't know how to draw that in >> plain text, so I will depict it as the ordered triple (f,g,h). The >> category I built takes this type of non-commutative triangle as a >> morphism (f,g,h): A --> B. >> >> That is, the new category is storing extra information in the >> morphisms by distinguishing between the part that goes directly from A >> to B and the part that takes a detour through an intermediate object, >> X. So while it may be possible for f+gh = f'+g'h' in the original >> category, (f,g,h) and (f',g',h') would be different morphisms in the >> new category unless f=f', g=g', and h=h'. One nice feature of this >> construction is the original category can be embedded in the new >> category by taking X to be the zero object. >> >> The people I have shown this to have told me they have never seen >> anything like my construction. I am at a loss for search terms -- >> everything I have tried either turns up nothing or thousands of >> unrelated articles. The closest I've found is factorization systems, >> which involve a commutative triangle, f=gh, for some g and h with >> certain properties. >> >> If any of you know a reference or keyword associated with expanding a >> category by replacing the morphisms with diagrams (with a specified >> property/shape), I would greatly appreciate the assistance. >> >> Sincerely, >> Jason Erbele >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

4 3

Soundness of commutative diagram proofs
by Aleks Kissinger 26 Sep '14

26 Sep '14

A common style of proof in CT papers is to draw a huge commutative diagram, number some subset of the faces, and justify why each of these faces commute. However, such an argument alone doesn't imply that the overall diagram commutes. Consider for example a triangle of arrows with three additional arrows connecting each of the corners to a fourth object in the centre. It is very easy to find examples where the three little triangles commute, but not the big outside triangle. E.g. take the three inward-pointing arrows to be 0 morphisms, then we can take f,g,h to be arbitrary on the outside. So, my question is: Is there a simple way of judging soundness for a commutative diagram proof? One answer is to determine what constitutes a legal pasting of diagrams, then only admit those which were obtained inductively by legal pastings of commuting faces. However, its not immediately obvious that, given a diagram without such a decomposition into legal pastings, we can obtain the decomposition efficiently. Has this problem been studied formally somewhere? Best, Aleks [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

3 2

Dietmar Schumacher
by Robert Pare 25 Sep '14

25 Sep '14

It is with great sadness that I announce that my longtime friend and collaborator, Dietmar Schumacher, died on Wednesday September 17, 2014, of a massive stroke. It was in 1971, when he took up his position at Acadia University, that he left a note on my office door introducing himself and suggesting we meet to discuss category theory. We did and that was the start of our long collaboration. We also started, in 1972, the category seminar which still runs today, 42 years later. For many years he would drive the 100 kms from Wolfville every week to attend the seminar, and drive back afterwards. He would sometimes speak in the seminar or, if not, stimulated discussion with his penetrating questions. The last few years, he came less often as he found the drive tiring, but invariably he would read up on the topic of the week and always emailed me a question or two about it. He kept up his interest in category theory until the end. He came to the first seminar this year, just two weeks ago, looking strong and fit as ever. We'll miss his wry sense of humour. His way of apologizing for some little thing, and then apologizing for apologizing. Our thoughts are with his family at this difficult time. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

7 7

5Yr Fellowships
by Neil Ghani 25 Sep '14

25 Sep '14

In the heart of Free Glasgow, in the soon to be People's Republic of Scotland, there is a chance to join the University of Strathclyde with a 5 year research fellowship leading to a permanent full time position. All you need is a 5-page research plan … a lot less than many Fellowship avenues. The area is data science and/or security but we intend to interpret this broadly. Very broadly. More details can be found here. strathvacancies.co.uk All the best Neil [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

tenure(track) position in Nijmegen, The Netherlands
by Ieke Moerdijk 24 Sep '14

24 Sep '14

Dear colleagues, Our institute is advertising for a new position in the section "Algebra and Topology". Priority will be given to candidates working in an area related to algebraic topology. In principle, the position is tenure track, but for more senior candidates other arrangements can presumably be made. All the details can be found at http://www.ru.nl/newstaff/working_at_radboud/details/details-vacature?taal=… Ieke Moerdijk. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

Reference search: new categories by replacing morphisms with diagrams
by Jason Erbele 23 Sep '14

23 Sep '14

Dear all, I built a category from another category by keeping "the same" objects and taking the morphisms to be diagrams from the old category that satisfy certain properties. The closest thing to what I'm doing that I have been able to find is factorization systems, but there are some major differences. To be more specific, I am starting with an Abelian category. If there are morphisms f: A --> B, g: X --> B, and h: A --> X, it makes sense to talk about the morphism f+gh: A --> B, which can be represented with a non-commutative triangle. I don't know how to draw that in plain text, so I will depict it as the ordered triple (f,g,h). The category I built takes this type of non-commutative triangle as a morphism (f,g,h): A --> B. That is, the new category is storing extra information in the morphisms by distinguishing between the part that goes directly from A to B and the part that takes a detour through an intermediate object, X. So while it may be possible for f+gh = f'+g'h' in the original category, (f,g,h) and (f',g',h') would be different morphisms in the new category unless f=f', g=g', and h=h'. One nice feature of this construction is the original category can be embedded in the new category by taking X to be the zero object. The people I have shown this to have told me they have never seen anything like my construction. I am at a loss for search terms -- everything I have tried either turns up nothing or thousands of unrelated articles. The closest I've found is factorization systems, which involve a commutative triangle, f=gh, for some g and h with certain properties. If any of you know a reference or keyword associated with expanding a category by replacing the morphisms with diagrams (with a specified property/shape), I would greatly appreciate the assistance. Sincerely, Jason Erbele [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

K-Theory Foundation
by Anthony Bak 23 Sep '14

23 Sep '14

Readers of this list may be interested in the following slightly abridged version of remarks on the background and founding of the K-Theory Foundation. The remarks were delivered at the opening ceremony of the first conference to be sponsored by the Foundation. The conference was titled 'K-theory and related topics' and took place in Beijing in August of this year as an ICM satellite conference. At the conference, the Foundation awarded for the first time its prizes to the most deserving young mathematicians working in K-theory and related fields to Joseph Ayoub of the University of Zurich and Moritz Kerz of the University of Regensburg. ________________________________________________ Remarks on the background of the K-theory Foundation Dear Colleagues, Thank you for participating and contributing to this conference. Special thanks go to the organizers of the conference Professor Guoping Tang and Professor Jianzhong Pan and the sponsors of the conference, the Chinese Academy of Sciences, the University of the Chinese Academy of Sciences, the Chinese National Science Foundation and the K-Theory Foundation, Inc. For the K-Theory Foundation, it is a special occasion. It is the first conference the Foundation is sponsoring and it is the first conference at which the Foundation will award prizes for outstanding work related to K-theory. The ceremony awarding the prizes will follow immediately after the first talk of this morning. The funds which the Foundation is using to support the conference and to award the prizes were derived from sales of the Journal of K-Theory. There was a time, however, when there was no journal for the subject K-theory and not even a major listing for the subject in the American Mathematical Society's Mathematical Subject Classification scheme. Let us go back in time, specifically to the year 1982, when this was the case and recall events which led to the creation 5 years later of a journal for K-theory, namely the journal 'K-Theory' and then 20 years after that the founding of the 'Journal of K-Theory ' and finally 3 years after that the founding of the K-Theory Foundation. 1982 was a time when K-theory had established itself as a very fashionable topic and had played a major role in solving problems of great interest to mathematicians in various areas of mathematics. Many Fields Medalists of the preceding 20 years had contributed to or used K-theory in their work: Atiyah, Connes, Grothendieck, Milnor, Quillen and Serre. Still the vast majority of mathematicians remained skeptical that K-theory would be more than a passing vogue, a fashionable topic whose interest and application would dwindle, certainly by now. In 1982, Grothendieck was 52 years old and the average age of a participant at a K-theory conference was between 30 and 40 years old. K-theory was a subject for young people. Against this background, I approached an academic publisher, Reidel, to ask if it might be interested in publishing a journal in a new, but rapidly growing subject of mathematics. I explained that K-theory was a new discipline of mathematics, embracing concepts, methodologies and problems central to many major disciplines of mathematics and that the aim of the proposed journal would be to further the cause of crosspollinization in mathematics by bringing together under one roof work having close conceptual and methodological relationships, work which had hitherto been scattered in the literature. (Further details regarding this are contained in the Editorial to the first issue of 'K-Theory'.) Reidel's response was cautious, but there was some optimism, which was supported by my close contact with one of its advisors. To make a long story short, let me simply say that it took 5 years before the first issue of 'K-Theory' appeared in 1987, the key development during this period of time being a genuine swing in attitude of the mathematical community towards K-theory, from a fashionable topic to a subject in its own right. Despite misgivings of some colleagues even in K-theory, the journal was an immediate success right from the beginning. In 1990 the journal 'K-Theory' played a role in the American Mathematical Society's admitting K-theory as a major subject index in its Mathematical Subject Classification scheme and K-theory was the first subject the AMS spoke of as interdisciplinary within mathematics. Many of you might have never heard of the publishing house Reidel, the first publisher of 'K-Theory'. It was small, but fine. It does not exist anymore. It was taken over in the late 1980's by Kluwer Academic Publishers, also seated in the same city as Reidel, namely Dordrecht, Holland. Kluwer was considerably bigger than Reidel. The commercial aspects of publishing a scientific journal were felt more strongly at Kluwer than at Reidel. This may be attributed to the size of the company. The size aspect was enhanced in 2003-04 when Springer Science & Business Media took over Kluwer. This was exacerbated by the fact that both Kluwer and Springer were migrating at that time from print to print-and-online. It was noticed by editors, authors and readers that the quality of the publication process and of the final product was declining considerably during this time, for example the quality of the typesetting and the timeliness of the appearance of issues of the journal were suffering. Many complaints were received from authors and this prompted in 2004 the idea among editors of the journal to begin looking for a new publisher. At the same time, prices for the journal were increasing systematically. Some of you will recall that Rob Kirby had been voicing concern regarding high prices charged by commercial academic publishers and had gathered broad agreement among mathematicians on this issue. Furthermore, the publisher began violating the copyright embedded in the contract with Reidel, which had not been changed for 20 years. Attempts to mutually resolve these problems with the publisher failed. It was decided to at least resolve the copyright violation issue by filing a law suit against Springer, which was done in 2006. This made moving to a new publisher/distributor of the journal at a later date less problematic. The move to a new publisher happened a year later. To make once again a long story short, let me say that in 2007 a contract was negotiated with Cambridge University Press to bring out a successor to 'K-Theory' called the 'Journal of K-Theory'. Almost all the editors of the old journal were present on the Editorial Board of the new journal and a couple younger ones were added. However, a new era began with the Journal of K-Theory. The continual complaints the old journal had been receiving regarding shoddy typesetting and delayed issues disappeared overnight, being replaced although not as frequently as the former complaints, by emails of thanks and praise. In the course of starting the new journal, it was conceived that the new journal should not only publish papers, but support in general the scientific community producing the papers it publishes and in particular use financial gains from publishing the journal for supporting K-theory activities. Accordingly, the K-Theory Foundation was set up in 2010 to carry out the philanthropic mission above. Its first Board of Trustees consisted in alphabetical order of Tony Bak (President), Max Karoubi, Tim Porter, Jonathan Rosenberg (Secretary) and Chuck Weibel (Treasurer). Through the Boards pioneering work, the Foundation received the basis of its current organization as a corporation and the formulation of operational rules for implementing the mission of the Foundation. The latter was accomplished last year by setting up 2 sets of rules, one for supporting conferences in K-theory and related subjects and the other for awarding prizes for outstanding work related to K-theory. This brings us to this moment in this room, to the opening ceremony of the first conference the Foundation is supporting and to the first awarding of the Foundation prizes. Let us extend our thanks and gratitude to the first Board of Trustees for bringing this about. Some organizational parts of the K-Theory Foundation still have to be completed, but are on the way. Finally, words of thanks go to the University of Bielefeld and the Faculty of Mathematics for their support for over 30 years of the projects above and to Cambridge University Press for its contribution as distributor of the Journal of K-Theory. Anthony Bak President of the K-Theory Foundation, Inc. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

2 1

CSR 2015: First Call for Papers
by Edward A. Hirsch 18 Sep '14

18 Sep '14

(apologies if you receive multiple copies) CSR-2015: First Call for Papers The 10th International Computer Science Symposium in Russia July 13-17, 2015, Listvyanka (Lake Baikal), Russia http://logic.pdmi.ras.ru/csr2015 In partnership with European Association for Theoretical Computer Science Program Committee Chair: Lev Beklemishev (Steklov Inst./Moscow) Program Committee: Eric Allender (Rutgers) Sergei Artemov (U. of New York) Andrei Bulatov (Simon Fraser U.) Harry Buhrman (U. of Amsterdam) Nachum Dershowitz (Tel Aviv U.) Edward A. Hirsch (Steklov Inst./St.Petersburg) Bahkadyr Khoussainov (U. of Auckland) Gregory Kucherov (CNRS and U. Marne-la-Vallee) Sergei O. Kuznetsov (Higher School of Economics/Moscow) Daniel Leivant (Indiana U.) Georg Moser (U. of Innsbruck) Damian Niwinski (U. of Warsaw) Prakash Panangaden (McGill U.) Jean-??ric Pin (CNRS and U. Paris-Diderot) Alexander Razborov (U. of Chicago and Steklov Inst./Moscow) Andre Scedrov (U. of Pennsylvania) Alexander Shen (LIRMM/Montpellier and IITP/Moscow) Wolfgang Thomas (Aachen U.) Helmut Veith (TU Vienna) Nikolai K. Vereshchagin (MCCME and Moscow State U.) Mikhail Volkov (Ural Federal U.) Michael Zakharyaschev (U. of London) Distinguished opening lecture: Moshe Y. Vardi (Rice U.) Invited Speakers TBA. Important Dates: Submission: December 14, 2014 Notification: before ICALP deadline Topics: include, but are not limited to: algorithms and data structures algorithms for concurrent and distributed systems, networks automata theory and formal languages automated reasoning automatic structures and applications computational complexity proof complexity Kolmogorov complexity combinatorial optimization constraint solving cryptography combinatorics in computer science computational models and concepts database theory description logics formal concept analysis games and program synthesis modal logic model checking proof theory quantum computation reasoning about security and privacy term rewriting verification Submission: Authors are invited to submit original (and not previously published) research. Submissions consist of two parts: the main paper and an appendix (which might be empty). The main paper must be at most 14 pages in length, including references. All proofs omitted from the main paper due to space constraints should be given in the appendix or made accessible through a reliable link to a freely available electronic preprint. The papers must be submitted in English, in the LNCS format (LaTeX, as pdf; final version with source) with page numbering turned on using the \pagestyle{plain} command; instructions are here: http://www.springer.com/computer/lncs?SGWID=0-164-7-72376-0 Simultaneous submission to journals or to other conferences with published proceedings is not allowed. The proceedings of the symposium will be published in Springer's LNCS series. Accepted papers MUST be presented at the symposium. Submissions should be uploaded to the EasyChair Conference system: http://www.easychair.org/conferences/?conf=csr2015 Yandex Awards for the best paper and for the best student paper will be given by the PC. Conference Chair: Daniil Musatov (Moscow Institute of Physics and Technology) Organized by Irkutsk State University Moscow Center for Continuous Mathematical Education Further information and contacts: Web: http://logic.pdmi.ras.ru/csr2015 Email: csr2015 "at" googlegroups.com UNSUBSCRIPTION: If you do not wish to receive any news regarding CSR conferences, please reply to this mail and we will remove you from the mailing list. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0

papers on colimits of monads available
by Jiri Adamek 17 Sep '14

17 Sep '14

I would like to announce two papers available on arxiv: ---------------------- J. Adamek, N. Bowler, P. Levy and S. Milius: Coproducts of Monads on Set (http://arxiv.org/abs/1409.3804) (This is an abstract, extended by proofs in the appendix, of a talk presented at LICS 2012.) A monad M on Set is proved to have a coproduct with every monad in the category Monad(Set) iff M is a submonad of either the terminal monad (constant to 1) or an exception monad (sending X to X+E). Calling such monads trivial, we prove that a coproduct of nontrivial monads exists iff the monads have arbitrarily large joint pre-fixpoints. (A pre-fixpoint of an endofunctor M is an object X such that MX is a subobject of X.) A surprisingly simple formula for coproducts of monads in Set is presented -------------------------- J. Adamek: Colimits of Monads (http://arxiv.org/abs/1409.3805) For "set-like" categories A the category Monad(A) is proved to have coequalizers. It also has a colimit of every diagram such that arbitrarily large joint pre-fixpoints of all the monads exist. This is stronger than the well-known fact that accessible monads admit all colimits. Somewhat surprisingly, the category of monads on Gra does not have coequalizers. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

1 0