December 2009 - Categories - categories.org.au

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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Foundations?
by F W Lawvere 03 Sep '10

03 Sep '10

Down with "Foundations"! Up with algebra! Dear Friends, Presumably I am among those who are being "vilified" as "lovers of categories as foundation". By avoiding any precise definition, such a formulation might appeal to the widespread justified boredom induced by the past hundred years of "foundations as justification". Whenever I used the word "foundation" in my writings over the past forty years, I have explicitly rejected that reactionary use of the term and instead used the definition implicit in the work of my teachers Truesdell and Eilenberg. Namely, an important component of mathematical practice is the careful study of historical and contemporary analysis, geometry, etc. to extract the essential recurring concepts and constructions; making those concepts and constructions (such as homomorphism, functional, adjoint functor, etc.) explicit provides powerful guidance for further unified development of all mathematical subjects, old and new. What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra! Nodal points in the progress of this kind of research occur when, as is the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra. I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory Had it not been for the century of excessive attention given to the alleged possibility that mathematics is inconsistent, with the accompanying degradation of the F-word, we would still be using it in the sense known to the general public: the search for what is "basic". We, who supposedly know the explicit algebra of homomorphisms, functionals, etc. are long remiss in our duty to find ways to utilize those concepts also in guiding high school calculus. Best wishes, Bill Bibliography: - The Category of Categories as a Foundation for Mathematics, La Jolla conference 1965, Springer (1966) - Adjointness in Foundations, Dialectica, (1969), to be reprinted in TAC - Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic, (2003) vol 9, pp 213-224 - Sets for Mathematics, w/ Bob Rosebrugh, Cambridge Univ. Press, (2003) ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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Re: evil
by Toby Bartels 07 Jan '10

07 Jan '10

[Note from moderator: Several messages to categories apparently hung in a mail system for several days. With apologies to posters, I am about to post four from late Decemeber in what should have been their posting order. Sorry about the delay, Bob] A dagger structure on a category should not really be considered evil at all. If you have a functor F: C^op -> C and ask whether it is a dagger structure, then this is (taken literally) an evil question; the answer is yes iff F^2 = 1 and F is the identity on objects, both evil conditions. More precisely, two isomorphic functors may have different answers. (A non-evil version is to ask whether F is isomorphic to a dagger structure.) However, it's not necessary to define a dagger-category as a category C equipped with a functor F: C^op -> C such that F satisfies these conditions. In lower-level language, we ask instead that C be equipped with an operation that takes each morphism f: x -> y to a morphism f^\dag: y -> x such that id^\dag = id, (f g)^\dag = g^\dag f^\dag, and (f^\dag)^\dag = f. Nothing here refers to equality of objects; it can be formulated in a language that (like FOLDS) does not have this concept. Given a dagger structure on C, defined in this elementary way, we can construct a functor \dag: C \to C^op that satisfies the evil property. (Of course, it also satisfies the non-evil version of that property.) But that is neither here nor there as to whether dagger structures are evil. There is some new discussion on the nLab: http://ncatlab.org/nlab/show/evil#daggers In particular, Mike Shulman shows how to translate dagger structures along equivalences of categories, proving that they are not evil. My previous post on this subject should probably be ignored. While any concept ~can~ be de-evilled in the way shown there, this does not necessarily give you the concept that you want, and indeed it need not even preserve already non-evil concepts. (And in this case specifically, it does not seem to be correct, as others have already argued here.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Quantum computation and categories
by John Baez 04 Jan '10

04 Jan '10

Dear Dusko - You wrote: bob coecke proposed to add quantum computing to andre joyal's list of > important directions of categorical research, but andre rejected it. most results in quantum computing are theorems about hilbert spaces. quantum > computing is a *tensor calculus*. but it is a tensor calculus of a special > kind: it attempts to describe a wildly unintuitive world. even the greatest > contributors, like von neumann and feynman, deplored the gap between the > quantum world, imposed on us in the lab, and the intuitions imposed on us in > everyday life. now category theory often helps where the common intuitions > fail. many of its applications demonstrate this. so > quantum computation might be an opportunity for an effective application of > *geometry of tensor calculus*. > Exactly! Samson Abramsky, Bob Coecke, Peter Selinger and others have been doing great work along these lines. I think this line of research will eventually be the key to understanding quantum gravity, because string diagrams reveal the common features of the tensor category of Hilbert spaces (Hilb, fundamental to quantum theory) and the tensor category of cobordisms (nCob, fundamental to our traditional notion of spacetime). I argued this case here, in a nontechnical way: http://math.ucr.edu/home/baez/quantum/ And I think that regardless of whether quantum computers or quantum gravity ever work, this line of research is very interesting. > is it really wise to reject an attempt to develop this, as objectionable as > it might be in any details? Andre didn't precisely "reject an attempt to develop" these ideas. He said "I am not convinced that quantum computing can contribute significantly to category theory". And that's fine. The bold researchers listed above will now redouble their efforts to convince Andre by proving lots of wonderful theorems. Here's one point where work on quantum computing, quantum gravity, and TQFT could have a radical effect on category theory. Researchers in these subjects have been forced by the nature of the material to embrace "dagger-categories". I explain why in my article above, but I called them "*-categories" instead of dagger-categories. A dagger-category is a category C with a functor F: C -> C^{op} which is the identity on objects and has F^2 = 1. Category theorists will note that the above definition is "evil", in the technical sense of that term: http://ncatlab.org/nlab/show/evil Namely, it imposes equations between objects, so we cannot transport a dagger-category structure along an equivalence of categories. Often evil concepts (like the concept of "strict monoidal category") have non-evil counterparts (like the concept of "monoidal category"). But in this particular case I know no way to express the idea without equations between objects. Both Hilb and nCob are dagger-categories. This fact is important. Try saying it in a non-evil way! Once Andre told me some ideas about this, relating to the case of Hilb, but unfortunately I don't see that how they could apply to nCob. I was very interested at Mark Weber's reaction to this problem. He said, roughly, "So dagger-categories aren't really categories with extra structure. Okay: they're something else! And that's fine." (I'd be happy for him to correct my rough summary and make his point more precisely.) I like this bold attitude, especially coming from someone like Mark, who knows enough category theory to carry it off. This could lead to really new developments. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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evil (fwd) Re: Quantum computation and categories
by Dusko Pavlovic 03 Jan '10

03 Jan '10

[yesterday john baez sent his message only to me, and i replied only to him. he actually meant to send it to the list, and encouraged me to resend the reply. i apologize for posting so much these days. -- dusko (in bed with a flu and a computer)] hi john, thanks for your note. the notion of evil is an interesting challenge in any context. > A dagger-category is a category C with a functor > > F: C -> C^{op} > > which is the identity on objects and has F^2 = 1. > > Category theorists will note that the above definition is "evil", in the > technical sense of that term: > > http://ncatlab.org/nlab/show/evil > > Namely, it imposes equations between objects, so we cannot transport a > dagger-category structure along an equivalence of categories. > > Often evil concepts (like the concept of "strict monoidal category") have > non-evil counterparts (like the concept of "monoidal category"). But in > this particular case I know no way to express the idea without equations > between objects. Both Hilb and nCob are dagger-categories. This fact is > important. Try saying it in a non-evil way! let me try. DEF. Let CC be a monoidal category. a *dagger* on CC is a functor P: CC^op ---> CC which is * self-adjoint * equivalence * given together with the dinaturals ** e_X : X (x) PX ---> I ** h_X : I--->PX (x) X which make PX -| X. LEMMA. Suppose that every object in CC comes with a Frobenius algebra structure. Then there are coherent natural isomorphisms PX-->X. PROOF. The Frobenius algebra structure induces ** ee_X : X (x) X ---> I ** hh_X : I---> X (x) X which make X-|X. the natural isomorphisms PX-->X are composed from the adjunction equipment (along the proof that an adjoint is unique up to a coherent iso). QED DEF. A strict dagger is a functor D:CC^op ---> CC obtained by transferring a dagger along the canonical isomorphisms from the lemma. COROLLARY. strict daggers are not evil: they are preserved under the equivalences. PROOF. daggers are obviously preserved. frobenius algebras are preserved. hence the canonical isomorphisms are preserved. FACT 1. a frobenius algebra structure on a hilbert space is just a choice of a basis. (hence we can a non-evil adjoint by first defining a preadjoint to be the conjugate of the dual operator, and then transferring along the isomorphism X^* ---> X induced by the chosen basis.) FACT 2. a frobenius algebra structure in nCob is the underwear structure. -- dusko PS the hope is that this provides a nonevil view of the daggers in FHilb and nCob. i guess the general suggestion might be to define dagger compact structure by a self-adjoint equivalence, plus a requirement that every object admits a frobenius algebra structure. that structure is not evil, and it is carried by all examples considered so far. i don't think that there is a general solution for the problem of evil in categories: we can only pin down a particular object, as an element of an isomorphism class, in the lucky cases when there is some additional structure that characterizes it. but in general, evil exists. every functor can be factored as an identity-on-the-objects-functor (ioof), followed by an embedding. the embedding is good, but ioofs are evil, and i think that they deserve their name. lord knows how much we use them. in a sense, category theory can be distinguished from set theory by the presence of evil. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: replacement message
by Bob Rosebrugh 01 Jan '10

01 Jan '10

Hi Peter - On Thu, 31 Dec 2009, Peter Selinger wrote: > I just bounced the missing message to categories, using the original > date (so it may drop down in your inbox depending on how you sort your > mail). now I'm baffled - I haven't received it in the folder where my cats mail gets filtered to, and it's not in the spam bucket. Both of the other messages have arrived fine for the second time - no idea what is wrong. If you try sending directly to me that might work out, sorry about this; will fwd the others now. best, Bob > Yes, "in defense of evil" is still the subject (I am using the > American spelling). And yes, I meant hyperbole, thanks for pointing > this out. > > You know, for your convenience, I'll just bounce the other two > (corrected) messages to you @ categories as well. > > Thanks, -- Peter > > Bob Rosebrugh wrote: > > > > Hi Peter - > > > > Unless I made a mistaken deletion, I didn't see this: > > > > > Subject: Re: categories: Re: Quantum computation and categories > > > To: toby+categories(a)ugcs.caltech.edu > > > Date: Wed, 30 Dec 2009 10:52:47 -0400 (AST) > > > Cc: categories(a)mta.ca (categories) > > > > Will forward when you resend, as well as the second and modified third - > > do you want to keep "in defence of evil" as subject. I also suspect you > > mean "hyperbole". > > > > best, Bob > > On Thu, 31 Dec 2009, Peter Selinger wrote: > > > > > Bob Rosebrugh wrote: > > > > > > > > Hi Peter, > > > > > > > > No problem - I went to bed early last night :-) > > > > > > > > I assume you do want me to post the first message? > > > > > > > > ps - you referred to a third message, but I see only two? did one get > > > > lost? > > > > > > Hi Bob, > > > > > > I sent three messages to categories(a)mta.ca yesterday. They were: > > > > > > ---------------------------------------------------------------------- > > > Subject: Re: categories: Re: Quantum computation and categories > > > To: toby+categories(a)ugcs.caltech.edu > > > Date: Wed, 30 Dec 2009 10:52:47 -0400 (AST) > > > Cc: categories(a)mta.ca (categories) > > > > > > Several people attempted to give a "non-evil" definition of a dagger > > > category. Not much of this makes sense. > > > > > > [snip] > > > ---------------------------------------------------------------------- > > > Subject: in defense of strictness > > > To: categories(a)mta.ca (Categories List) > > > Date: Wed, 30 Dec 2009 21:54:44 -0400 (AST) > > > > > > Here is a bit of hyperbola. Hopefully entertaining. > > > > > > [snip] > > > ---------------------------------------------------------------------- > > > Subject: in defense of evil > > > To: Dusko.Pavlovic(a)comlab.ox.ac.uk > > > Date: Wed, 30 Dec 2009 22:01:59 -0400 (AST) > > > Cc: categories(a)mta.ca > > > > > > Dear all, > > > > > > I apologize for sending out a third (long) email about dagger > > > [snip] > > > ---------------------------------------------------------------------- > > > > > > It is this last one (to Dusko) that I wanted to suppress, or rather > > > replace with the friendlier version. > > > > > > It seems that you didn't receive one of the two others. Please let me > > > know which one, and I'll resend it! -- Peter > > > > > > >

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Re: replacement message
by Bob Rosebrugh 01 Jan '10

01 Jan '10

Hi Peter - Unless I made a mistaken deletion, I didn't see this: > Subject: Re: categories: Re: Quantum computation and categories > To: toby+categories(a)ugcs.caltech.edu > Date: Wed, 30 Dec 2009 10:52:47 -0400 (AST) > Cc: categories(a)mta.ca (categories) Will forward when you resend, as well as the second and modified third - do you want to keep "in defence of evil" as subject. I also suspect you mean "hyperbole". best, Bob On Thu, 31 Dec 2009, Peter Selinger wrote: > Bob Rosebrugh wrote: > > > > Hi Peter, > > > > No problem - I went to bed early last night :-) > > > > I assume you do want me to post the first message? > > > > ps - you referred to a third message, but I see only two? did one get > > lost? > > Hi Bob, > > I sent three messages to categories(a)mta.ca yesterday. They were: > > ---------------------------------------------------------------------- > Subject: Re: categories: Re: Quantum computation and categories > To: toby+categories(a)ugcs.caltech.edu > Date: Wed, 30 Dec 2009 10:52:47 -0400 (AST) > Cc: categories(a)mta.ca (categories) > > Several people attempted to give a "non-evil" definition of a dagger > category. Not much of this makes sense. > > [snip] > ---------------------------------------------------------------------- > Subject: in defense of strictness > To: categories(a)mta.ca (Categories List) > Date: Wed, 30 Dec 2009 21:54:44 -0400 (AST) > > Here is a bit of hyperbola. Hopefully entertaining. > > [snip] > ---------------------------------------------------------------------- > Subject: in defense of evil > To: Dusko.Pavlovic(a)comlab.ox.ac.uk > Date: Wed, 30 Dec 2009 22:01:59 -0400 (AST) > Cc: categories(a)mta.ca > > Dear all, > > I apologize for sending out a third (long) email about dagger > [snip] > ---------------------------------------------------------------------- > > It is this last one (to Dusko) that I wanted to suppress, or rather > replace with the friendlier version. > > It seems that you didn't receive one of the two others. Please let me > know which one, and I'll resend it! -- Peter >

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Between Cape Town and Genova
by George Janelidze 01 Jan '10

01 Jan '10

Dear Colleagues, The Cape Town Year is about to end and the Genova Year is about to begin... So, please look at http://ct2009.info/ and at http://ct2010.disi.unige.it/ The website http://ct2009.info/ is now updated and has abstracts of the talks in it. In particular I am asking the participants of CT2009 to look at their abstracts and send me any corrections by email please. Please also look at photos, and send me a replacement if you do not like your photo (or just tell me that you do not like it, and we shall decide something together). In some cases this is obvious: sorry to say, we could not take all photos during talks and used the excursion photos not all of which are sufficiently good. We do not have many big CT Conferences Proceedings Volumes in recent years - only a few, although there were also several good Special Volumes dedicated to our colleagues. And I was not sure about CT2009. I would like to report to you that I received "NO" from "Lecture Notes in Mathematics" and from "Journal of Algebra". It was nicely formulated NO's but it feels sad: the reason from LNM was "...About 15 years ago, we stopped publishing conference proceedings volumes in the Lecture Notes in Mathematics series. We found that, even when they were very well edited, the response to such books in terms of sales, etc. was rather low, and they were rated as second priority for acquisition in libraries, for instance. Thus they were hardly a viable proposition for us anymore....", while the reason from JA was "...not only is it quite unusual for us to publish Proceedings (unless agreed in advance, on a subject in the heart of the editorial board, and with a special editorial board built a priori), but the Journal of Algebra has already much too many submissions to consider extending its field to Category theory...". But, in spite of all problems, soon or late - and better soon that late - the Golden Age of CT1990 should come back (should not it?) and, also having in mind beautiful mathematical results presented at CT2009, we are considering a possibility of the joint CT2009+CT2010 Proceedings Volume. The organizers of CT2010 will surely tell us more about that soon. George Janelidze [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: replacement message
by Bob Rosebrugh 01 Jan '10

01 Jan '10

Hi Peter, No problem - I went to bed early last night :-) I assume you do want me to post the first message? cheers, Bob

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