January 2011 - 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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What else do simplicial sets classify?
by Andrej Bauer 04 Oct '11

04 Oct '11

The presheaf category of simplicial sets is the classifying topos for the theory L of a bounded linear order. In general, there could be other theories which are "Morita equivalent" to L in the sense that their classifying toposes are equivalent to simplicial sets. Are any such known, preferably occurring in nature? With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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source, sinks, and ?
by Michael Shulman 04 Feb '11

04 Feb '11

A family of morphisms { x_i --> y }_{i \in I} in some category, all with the same codomain, is called a "sink" or a "cocone". A family { x --> y_j }_{j \in J} all with the same domain is called a "source" or a "cone". Is there a name for a family of the form { x_i --> y_j }_{i \in I, j \in J} ? A cylinder? Or a frustrum (since I \neq J)? Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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colimits of polynomial functors
by Ondrej Rypacek 03 Feb '11

03 Feb '11

Dear all Does the category of (dependent) polynomial functors and strong natural transformation have all/some colimits ? In general, what is known about them ? Many thanks! Ondrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Fibrations in a 2-Category
by Marta Bunge 29 Jan '11

29 Jan '11

Dear Mike, I have no objection to anything you actually say. Yes, of course any functor is an anafunctor and, under AC, every anafunctor is isomorphic to a functor. But that is what I meant. It was just that, emphasizing the fact that anafunctors generalize functors (without AC), I wanted to note that distributors (not necesarily representable) also do, even with AC, and that in this alone lies their importance, plus the fact that they can be composed etc. I do not intend to modify your excellent exposition of anafunctors, a subject that I "learnt" just by reading it. I did not mean it as a criticism. Many thanks, Marta > Date: Sat, 29 Jan 2011 11:02:19 -0800 > Subject: Re: categories: Re: Fibrations in a 2-Category > From: mshulman(a)ucsd.edu > To: marta.bunge(a)mcgill.ca > CC: categories(a)mta.ca > > Dear Marta, > > The discussion of the equivalence in the nLab article you mention was > added 5 days ago by me, by extracting and condensing a bit from > Jean's, my, and David's emails a week ago. I thought this discussion > interesting enough that it ought to be preserved. > > On Sat, Jan 29, 2011 at 9:45 AM, Marta Bunge <marta.bunge(a)mcgill.ca> wrote: >> In that article, it is furthermore pointed out that each version has its advantages over the other, and that therefore both are of interest for category theory in a topos S without AC, where they generalize ordinary functors. But, even in the presence of AC, distributors (profunctors) generalize ordinary functors, a fact that I have known for 45 years, whereas anafunctors do not. > > This confused me for a minute until I realized you probably meant > "generalize" to mean "strictly generalize." (It is of course still > true under AC that every functor is an anafunctor; the difference > being rather that under AC every anafunctor is isomorphic to a > functor.) > > As David pointed out, since the nLab is a wiki, anyone who feels the > discussion currently existing there has defects should feel free to > take it upon themselves to remedy those defects. > > Best, > Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Fibrations in a 2-Category
by Michael Shulman 29 Jan '11

29 Jan '11

Dear Marta, The discussion of the equivalence in the nLab article you mention was added 5 days ago by me, by extracting and condensing a bit from Jean's, my, and David's emails a week ago. I thought this discussion interesting enough that it ought to be preserved. On Sat, Jan 29, 2011 at 9:45 AM, Marta Bunge <marta.bunge(a)mcgill.ca> wrote: > In that article, it is furthermore pointed out that each version has its advantages over the other, and that therefore both are of interest for category theory in a topos S without AC, where they generalize ordinary functors. But, even in the presence of AC, distributors (profunctors) generalize ordinary functors, a fact that I have known for 45 years, whereas anafunctors do not. This confused me for a minute until I realized you probably meant "generalize" to mean "strictly generalize." (It is of course still true under AC that every functor is an anafunctor; the difference being rather that under AC every anafunctor is isomorphic to a functor.) As David pointed out, since the nLab is a wiki, anyone who feels the discussion currently existing there has defects should feel free to take it upon themselves to remedy those defects. Best, Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Fibrations in a 2-Category
by Marta Bunge 29 Jan '11

29 Jan '11

Dear Jean (and Mike), Your guess/proof that anafunctors and representable distributors are equivalent notions, and by an equivalence which extends to an equivalence of categories between Rep(A,B) and Cat_ana(A,B), is not only correct but also fully discussed in: http://ncatlab.org/nlab/show/anafunctor In that article, it is furthermore pointed out that each version has its advantages over the other, and that therefore both are of interest for category theory in a topos S without AC, where they generalize ordinary functors. But, even in the presence of AC, distributors (profunctors) generalize ordinary functors, a fact that I have known for 45 years, whereas anafunctors do not. This ought to be pointed out by the authors of the article mentioned above. In my thesis ("Categories of Set-Valued Functors", University of Pennsylvania, 1966), inspired by a monograph of Michel Andre ("Categories of Functors and Adjoint Functors", Batelle Report, Geneve, 1964), I discuss what you later called "distributors", including their composition as a generalized matrix product. It was Bill Lawvere who pointed out to me the importance of "distributors", well before you introduced them. I gave an expose of a portion of my thesis at the Oberwolfach meeting in 1966. My main interest therein was the equivalence between the category of profunctors between two small categories and that of those functors between the corresponding set-valued functor categories that have an adjoint or a coadjoint. In turn, this led to Morita equivalence theorems. The results are valid for an arbitrary (co)complete topos S but, even in the case of Set, which satisfies AC, this shows that profunctors generalize ordinary functors. With best regards to all, Marta > To: mshulman(a)ucsd.edu; categories(a)mta.ca > From: jean.benabou(a)wanadoo.fr > Subject: categories: Re: Fibrations in a 2-Category > Date: Sat, 22 Jan 2011 11:25:55 +0100 > > ANAFUNCTORS VERSUS DISTRIBUTORS > > Dear Mike, > > (I apologize for using in a few places capital letters, where > normally I would have used italics, but html is not accepted in the > Category List) > > In your mail about fibrations in a 2-category, dated Jan.14, you say: > > "One way to deal with the difficulty you mention is by using > "anafunctors," which were introduced by Makkai precisely in order to > avoid the use of AC in category theory". > > There is "another way", which I prefer. It is using distributors, > which do much more than merely "avoid the use of AC", and apply to > more general situations than the ones you consider. Let me first > give a very simple definition: > > Let M: A -/-> B be a distributor, identified with a functor A --> (B > °, Set) = B^. > I say that M is "representable" iff for every object a of A the > presheaf M(a) is. With AC, such an M is isomorphic to a functor F: A > --> B, which is unique up to a unique isomorphism. But my definition > doesn't need any reference to AC. > I shall denote by Rep(A,B)) the full subcategory of Dist(A,B) having > as objects the representable distributors. > "Corepresentable" distributors are defined by the canonical duality > of Dist, and I denote by Corep(A,B) the corresponding category. > > 1- In your example you say: > > "let P --> 2 be a fibration, with fibers B and A. Then there is > (without AC) an anafunctor A --> B, where the objects of F are the > cartesian arrows of P over the nonidentity arrow of 2, and the > projections assign to such an arrow its domain and codomain" > > What I can say with distibutors is: > > 1' - Let P --> 2 be an ARBITRARY functor with fibers B and A. Then > there is, without AC, a canonical distributor A -/-> B associated to > this functor. Moreover the the functor is a fibration iff the > associated distributor is representable, and a cofibration (I think > you'd say "op-fibation") iff this distributor is corepresentable. . > (again no AC). > Which statement do you prefer? > > 2- A little bit further on you say: > > "More generally, if Cat_ana denotes the bicategory of categories and > anafunctors, then from any fibration P --> C we can construct (without > AC) a pseudofunctor C^{op} --> Cat_ana." > > With distibutors I can say: > > 2' - Let F: P --> C be an ARBITRARY functor. From F, I can construct, > without AC, a normalized lax functor D(F) : C^(op) --> Dist . Then we > have, without AC: > (i) F is a Giraud functor (GIF) iff D(F) is a pseudo functor. > (ii) F is a prefibration iff for every map c of C the distributor D(F) > (c) is representable > (iii) F is a fibration iff it satisfies (i) and (ii) > (Iv) F is a cofibration if it is a GIF and the D(F!(c)'s are > corepresentable. > > Which statement do you prefer ? > In (iv) I insist on the fact that it is the same D(F). Is there a > notion of "ana-cofunctor"? > Note moreover that many other important properties of F can be > characterized by very simple properties of D(F), again without AC! > > 3- You also say: > > "An anafunctor is really a simple thing: a morphism in the bicategory > of fractions obtained from Cat by inverting the functors which are > fully faithful and essentially surjective". > > Woaoo, you call this a simple thing! Ordinary categories of fractions > are very complicated, unless you have a calculus of right (or left) > fractions. Is there, precisely defined, and without neglecting the > coherence of canonical isomorphisms, such a "calculus" defined. Does > it apply to the "simple thing" of anafunctors. > > 4- In guise of conclusion you say: > > In general, it seems to me that there are two overall approaches to > doing category theory without AC (including with internal categories > in a topos): > > 1) Embrace anafunctors as "the right kind of morphism between > categories" in the absence of AC > 2) Insist on using only ordinary functors, so that we can work with > the strict 2-category Cat, which is simpler and stricter than Cat_ana. > "Personally, while there is nothing intrinsically wrong with (2), I > think (1) gives a more satisfactory theory." > > Sorry,but your approaches 1) and 2) are not the only ones. I opt for > the following one: > 3) Work with distributors. > > I still have to see precise mathematical applications anafunctors.. > Do I have to mention applications of distributors? Do I have to point > out that distributors can, not only be internalized, but also be > "enriched"? > > 5 - You are a very persuasive person Mike, but I'm not "buying" > anafunctors, unless you give me convincing examples of what > anafunctors can do, which distributors cannot do much better. > And if you want to generalize functors, without going all the way to > arbitrary distributors, good candidates, for me, instead of > anafunctors, are representable distibutors, which are very simple to > define rigorously and easy to work with. And of course don't use AC., > I have a very strong guess that anafunctors are "the same thing" as > representable distributors. I can even sketch a proof of my guess. > (i) You say that an anafunctor can be represented by a span A <-- F -- > > B where the left leg, say p, is full and faithful and surjective > on objects and the right leg, say q, is arbitrary functor. > In Dist you can take the composite: q p*: A -/-> F --> B, where p* is > the distributor right adjoint to the functor p. It is easy to see > that his composite is representable. > Thus we get a map on objects, u: Cat_ana(A,B) --> Rep(A,B) > (ii) Conversely, suppose M: A -/-> B is representable. By 1' we get a > fibration P --> 2 thus by 1 an anafunctor A --> B . > Thus we get a map on objects, v: Rep(A,B) --> Cat_ana(A,B) . > It should be routine that u and v extend to functors U and V and give > an equivalence of categories between Rep(A,B) and Cat_ana(A,B) > I didn't write a complete proof because, in order to do so, I'd have > to know a little more than what you wrote about the category Cat_ana > (A,B) and I'm not ready to spend much time on the study of anafunctors. > Is my guess correct? If it isn't, where does my "sketch of proof" > break down? > In particular what is the category of anafunctors with domain the > terminal category 1 and codomain a category C? > I'd be very grateful if you could answer these questions, and some of > the ones I asked in 1) and 2). > > I'm sure I didn't convince you. All I hope for, is that a few > persons, after reading this mail, and your future answer of course, > will think twice before they abandon "old fashioned" Category Theory > with its functors, AND DISTRIBUTORS, and rush to anafunctors, with > the belief that they are unavoidable foundations for the future AC- > free "New Category Theory". > > Regards, > Jean, > > [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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Stone duality for generalized Boolean algebras
by Andrej Bauer 27 Jan '11

27 Jan '11

The well known Stone duality says that there is an equivalence between Boolean algebras (BA) and the opposite of Stone spaces and continuous maps. Here a Stone space is a Hausdorff zero-dimensional compact space. Furthermore, Boolean algebras correspond to Boolean rings with unit. How exactly does this extend to generalized Boolean algebras? A generalized Boolean algebra (GBA) is an algebra with 0, binary meet, binary join, and relative complement in which meets distribute over joins. Equivalently it is a Boolean ring (possibly without a unit). I have seen it stated that the dual to these are (the opposite of) locally compact zero-dimensional Hausdorff spaces and proper maps, e.g., it is stated in Benjamin Steinberg: "A groupoid approach to discrete inverse semigroup algebras", Advances in Mathematics 223 (2010) 689-727. Another source to look at is Givant & Halmos "Introduction to Boolean algebras", but there this material is covered in exercises and the duality is stated separately for objects and for morphisms, and I can't find an exercise that treats the morphisms, so I wouldn't count that as a reliable reference. Stone's original work does not seem to speak about morphisms very clearly (to me). Unless I am missing something very obvious, it cannot be the case that GBA's correspond to locally compact 0-dimensional Hausdorff spaces and proper maps, for the following reason. The space which corresponds to the GBA 2 = {0,1} is the singleton. The space which corresponds to the four-element GBA 2 x 2 is the two-point discrete space 2. There are _four_ GBA homomorphisms from 2 to 2 x 2 (because a GBA homomorphism preserves 0 but it need not preserve 1), but there is only one continuous map from 2 to 1. Or to put it another way, there are _four_ ring homomorphisms from Z_2 to Z_2 x Z_2 (because they need not preserve 1), but there is only one continuous map from 2 to 1. So, either the spectrum of a GBA is not what I think it is (namely maximal ideals), or we should be taking a more liberal notion of maps on the topological side. For example, there are _four_ partial maps from 2 to 1. If someone knows of a reliable reference, one would be much appreciated. I won't object to a direct proof of duality either. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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Re: Stone duality for generalized Boolean algebras
by George Janelidze 26 Jan '11

26 Jan '11

Dear Bill, Beautiful! But is not the conclusion exactly opposite?: Let M(X,Y) be the set of all maps X --> Y and let P(X,Y) be the set of partial maps X --> Y. Then P(X,Y) = M(X,Y+1) (where "=" means "natural bijection" of course), and if instead of sets we consider objects in an elementary topos, then - according to your discovery - Y+1 should be replaced with Ytilda (which is actually the definition of Ytilda). So, the whole point is to describe partial maps in terms of maps and not the other way around! I (still) insist that Andrej Bauer's original question has the trivial answer independently given by Fred and me, and that topos-theoretic Ytilda, in spite of its fundamental role in topos theory (e.g. 1tilda = omega), has nothing to do with it. This does not mean that I do not appreciate your nice and deep comments. With best regards, George From: F. William Lawvere Sent: Wednesday, January 26, 2011 7:19 PM To: janelg(a)telkomsa.net ; fejlinton(a)usa.net ; categories Subject: RE: categories: Re: Stone duality for generalized Boolean algebras Dear George You ask would any categorically thinking mathematician say that the category of pointed sets needs further description as the category of sets and partial maps? In 1969-1970, recalling a) the preorigins of sheaf theory a hundred years ago in the still-non-trivial problem of extending of partial maps in analysis and topology, and b) desirous of an instrument for describing sheafication in finitely algebraic terms Myles and I proposed Ytilda->Omega as one of the two axioms for an elementary theory of toposes (the other being the Pi right adjoint to pullback; applying Grothendieck’s method of relativization using any given model U of those axioms, the 2-category of U-Toposes was obtained thus capturing precisely the original SGA4 notion by choice of U). Of course these axioms were soon shown to be deducible from special cases, but the importance of classifying partial maps X..->Y remains. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

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