May 2012 - Categories - categories.org.au

Partial functor
by David Leduc 17 Mar '15

17 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/ ]

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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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Bourbaki & category theory
by Staffan Angere 15 Jun '12

15 Jun '12

Dear categorists, and also, hello everyone, since this is my first post here! I'm wondering about the connection of Bourbaki to category theory. The copy of "Theory of Sets" that I have says it's written in 1970. Yet, Dieudonné famously saiid that the theory of functors subsumed Bourbaki's theory of structures... and, also, Bourbaki's theory of structures is very clearly a theory of a type of concrete categories. On the other hand, I've seen claims that the categorists' use of "morphism" comes from Bourbaki. So who was first? Does anyone here know when Bourbaki's theory of structures was really conceived? I guess this might be self-evident to anyone born during the 1st half of the 20th century, but it has turned out to be really hard to find out for me. Thanks in advance, staffan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Participation, registration, and programme: 4WFTop and HDACT, June 15-20 2012, Ljubljana
by Andrej Bauer 31 May '12

31 May '12

***** REGISTRATION AND PROGRAMME ***** FOURTH WORKSHOP ON FORMAL TOPOLOGY (4WFTop) June 15-19 2012 and HIGHER DIMENSIONAL ALGEBRA, CATEGORIES AND TYPES (HDACT) June 20 2012 Ljubljana (Slovenia) http://4wft.fmf.uni-lj.si/ PARTICIPATION AND REGISTRATION ============================== The 4WFTop and HDACT registration is now open at http://4wft.fmf.uni-lj.si/registration/ Please help us plan the event and register today if you are planning to come. You can change your registration details later. PROGRAMME ========= A preliminary programme is available at http://4wft.fmf.uni-lj.si/programme/ ABOUT THE WORKSHOP ON FORMAL TOPOLOGY ===================================== The workshop on formal topology is an international meeting dedicated to formal topology and related topics, including constructive and computable topology, point-free topology, and other generalizations of topology. This is the fourth of a series of successful meetings on the development of Formal Topology and its connections with related approaches. The first three have been held in Padua (1997), Venice (2002), and Padua (2007). KEYNOTE SPEAKERS * Per Martin-Löf (Stockholm University) * Ieke Moerdijk (Radboud University Nijmegen) * Vladimir Voevodsky (Institute of Advanced Studies) INVITED SPEAKERS * Olivia Caramello (University of Cambridge) * Francesco Ciraulo (University of Padua) * Hajime Ishihara (Japan Advanced Institute of Science and Technology) * Davorin Lešnik (TU Darmstadt) * Claudio Sacerdoti Coen (University of Bologna) * Alex Simpson (University of Edinburgh) * Bas Spitters (Radboud University Nijmegen) * Paul Taylor (United Kingdom) TUTORIAL DAY June 15th 2012 will be dedicated to tutorials on topics in and around formal topology, suitable for researchers and graduate students without in-depth knowledge of the area. They will be given by: * Thierry Coquand (Chalmers University) * Erik Palmgren (Stockholm University) * Giovanni Sambin (University of Padua) * Steven Vickers (University of Birmingham) PROGRAM COMMITTEE * Andrej Bauer (chair, University of Ljubljana) * Thierry Coquand (Chalmers University) * Maria Emilia Maietti (University of Padua) * Giovanni Sambin (University of Padua) * Peter Schuster (co-chair, University of Leeds) The workshop is supported by: * Faculty of Mathematics and Physics, University of Ljubljana * Institute of mathematics, physics and mechanics, Ljubljana * Department of Mathematics, University of Padua, Italy ABOUT THE WORKSHOP ON HIGHER DIMENSIONAL ALGEBRA, CATEGORIES AND TYPES ====================================================================== The importance of higher dimensional structure for mathematics and computer science has been recognized for a long time. However, recent connections between homotopy theory and type theory have spurred new activity that is connecting previously unrelated topics. The workshop will be a place for meeting and dissemination of ideas between researchers from different areas who all use higher dimensional structures. INVITED SPEAKERS * Thorsten Altenkirch * Steve Awodey * Emily Riehl PROGRAM COMMITTEE * Nicola Gambino * Neil Ghani * Zhaohui Luo * Erik Palmgren * Randy Pollack * Urs Schreiber * Bas Spitters * Thomas Streicher [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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The Idea of Structure as Data and Conditions
by Ellis D. Cooper 30 May '12

30 May '12

In the 1952 document at http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only mathematician "pr\'{e}sent" referenced by first name only is Sammy. I was permitted to audit a graduate course on category theory guided by Sammy at Columbia University in the early 1960s. I recall his insistence that mathematical structure is given by data and conditions. Is that idea implicit or explicit in Bourbaki? Has that idea been superceded? How does it relate to the development of algebraic theories as understood by Lawvere, Linton, Barr-Wells, the Elephant, and so on? Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Mac Lane's Idea of Structure versus Weil's
by Colin McLarty 28 May '12

28 May '12

On Sat, May 26, 2012 at 7:48 PM, Michael Barr <barr(a)math.mcgill.ca> wrote: > Let me point out that not every structure comes with an obvious notion of > morphism. This is entirely true on Bourbaki's theory of structures. Mike gives topological spaces as a good example. Should open sets be preserved by morphisms? or reflected? or both? Each Bourbaki structure comes with a unique obvious notion of isomorphism: a bijection which preserves and reflects all the data. Each one comes with as many possible definitions of morphism as there are ways to choose which data to preserve and which to reflect. Weil correctly understood this, as shown in the quote: \begin{quotation} As you know, my honorable colleague Mac~Lane maintains every notion of structure necessarily brings with it a notion of homomorphism, which consists of indicating, for each of the data that make up the structure, which ones behave covariantly and which contravariantly [\dots] what do you think we can gain from this kind of consideration? (Weil letter to Chevalley 1951).\end{quotation} But Saunders was not thinking of any formal definition of "structure". He meant that in fact wherever you see mathematicians using some notion of space or algebra, or whatever, you will see a notion of homomorphism used with it. And he was largely right, though he also helped to make this more strictly true by convincing people it was a useful perspective. There was by then a well-established notion of topological space with continuous functions as morphisms (reflecting open sets). An important subclass of morphisms was open functions, defined as continuous and preserving open sets. Nobody studied topological spaces plus functions which merely preserve open sets (without also reflecting them). And Saunders knew that homomorphisms in his sense need not be functions in the strict sense. He knew algebraic geometry used functions which are not everywhere defined. Functional analysis used "functions" which have no well defined value at any single point. Weil correctly understood that a structure in his sense admits many different notions of morphism. But he was committed to a general set theoretic theory of "structure," so he struggled with that, and got Bourbaki to struggle with it (as seen in Le Tribu in 1952), and Bourbaki failed to make anything useful of it. The more categorical members of Bourbaki never achieved a general theory of "structure" either. But they created many extremely useful and now widely used categorical tools. I will mention that recent philosophers of mathematics have been satisfied with simpler set theoretic theories of structure where data are always preserved. These philosophers are little aware of topological spaces, let alone say uniform spaces or local rings, and so unaware of the complexities Bourbaki faced. colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: The Idea of Structure as Data and Conditions
by Charles Wells 27 May '12

27 May '12

The sentence "contains all the info about any model" is ambiguous. I mean the theory contains everything you can say that is correct about every model. --C On Sun, May 27, 2012 at 9:02 AM, Charles Wells <charles(a)abstractmath.org>wrote: > Data and conditions constitute a presentation. The graph, diagrams, cones > & cocones of a sketch are a presentation. This idea has not been > superseded, not at all, but it has been completed (in two senses) by the > concept of theory, which is the object generated by the presentation: The > theory of a sketch, the classifiying topos, the algebraic theory in the > sense of Lawvere, and so on. This object contains all the information > about any model. > > That idea is in some way the other face of, or the complementary point of > view about, data and conditions. > > Charles > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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"Generic figures and their glueings" book ..... please help
by Vasili I. Galchin 26 May '12

26 May '12

Hello, I think the above book by Reyes, Reyes and Zolfaghari is quite intriguing. At the same time I can't get my mind around the notions of "generic figures" and "change of figures"! The first seems to be objects in the base category and the latter morphisms in the base category. Whoever has read this book and understands it please give me some pointers. Sorry the vague call for help. Kind regards, Vasili [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Bourbaki & category theory
by George Janelidze 24 May '12

24 May '12

I am ready to take back my criticism and apologize, if the "longer sentence" is correct. But is it? I am certainly not an expert in Bourbaki history, and, as far as I remember, they say no word about morphisms in the historical part of "Theory of Sets" and give no references on categories. But I think they "always" believed that structures determine isomorphisms but not morphisms, and I don't think they changed their mind between 1951 and 1957. When I say "they" I mean "those of them who made main decisions about the Bourbaki tractate". Because I hope (!) that not all of them were happy that categories are not even defined in the tractate. In my previous message I wrote "Removing Bourbaki's formalism..." but in fact that "formalism" is (not nice but) serious, in the sense that it takes us further away from abstract categories. Anyway, we need to know, if it is still possible, how exactly did Bourbaki definition of morphism(s) came up. George -------------------------------------------------- From: "Colin McLarty" <colin.mclarty(a)case.edu> Sent: Wednesday, May 23, 2012 1:06 AM To: "George Janelidze" <janelg(a)telkomsa.net> Cc: <categories(a)mta.ca> Subject: Re: categories: Re: Bourbaki & category theory > On Tue, May 22, 2012 at 5:51 PM, George Janelidze <janelg(a)telkomsa.net> > wrote: >> Dear Colleagues, >> >> I don't think it is good to say that "Bourbaki had a notion of >> isomorphism >> but no general notion of morphism", even in a brief message! > > It would not be good -- unless it was part of a longer sentence. > > I wrote "Prior to encountering category theory, Bourbaki had a notion > of isomorphism but no general notion of morphism." The Bourbaki > passage you quote was first published in 1957, at least 6 years after > Bourbaki encountered category theory as shown by the letter from Weil > that i quoted. > > best, Colin > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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