From: piggy@hilbert.maths.utas.edu.au (La Monte Yarroll) Date: Mon, 4 Jan 93 10:33:36 EST A friend in the US sent me the following quote from the October 1992 issue of Unix Review. > Stan kelly-Bootle's Devil's Advocate column lists the interesting lines: > > I've always wondered if MacLane's "Categories for Working > Mathematicians" has chapters covering "Tenured" and "Non-tenured", > and if there is a version for the unemployed? +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Date: January 4, 1993 From: Bob Rosebrugh (rrosebrugh@mta.ca) This is the routine distribution for the categories mailing list. It is the file routine.dist in [.categories] and was last updated on January 4, 1993. Subscribers should note that the From: field of a categories posting is categories@mta.ca (which is different from some other mailing lists). Thus, an automatic reply is redirected to the entire list (unless another intention is clearly detected by the moderator). Administrative items (address changes etc.) can be sent to categories-request@mta.ca, or directly to the moderator. Usually, items of this sort sent to categories@mta.ca will not be posted. The archives of postings on categories are held at the ftp site macc2.mta.ca (138.73.1.2) in the directory [.categories]. Note that this is a Vax VMS system and so the cd command requires appropriate syntax i.e. cd [.new] moves one node down the directory tree to new cd [-] moves one node up the tree. The postings are filed in yearly subdirectories called [.90], [.91] etc. Within those subdirectories there are monthly files, and an annual list of dates and subjects of postings. In the [.categories] directory there is also a file called ftp.sites with information about ftp sites holding files of interest to subscribers. Several TeX diagram macro packages are in the subdirectory [.macro]. Information about ftp sites with holdings of interest is welcome. This ftp site will also archive files which represent preprints in category theory for persons without access to an ftp archive. It is preferred that these be in TeX, but other submissions will be considered. Submissions should be sent to the moderator by e-mail only, and should include a short abstract suitable for posting to the categories list. Submissions must be ascii files only (so TeX source code in any flavour is fine, but dvi files are not acceptable currently.) If you need detailed instructions on how to use ftp ask anyone knowledgable about the Internet at your site, or write to me. Bob Rosebrugh Phone: +1-506-364-2538 Department of Mathematics and Computer Science Fax: -364-2210 Mount Allison University Sackville, N. B. E0A 3C0 Email: rrosebrugh@mta.ca Canada +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: products of sites Date: Tue, 12 Jan 93 13:47:03 -0500 From: "Allan Adler" Does the category of sites have products? By a morphism X-->Y of sites, one means a functor Y-->X whose composition with any sheaf of sets on X is a sheaf of sets on Y. I found some stuff in Johnstone's book Topos Theory which says that products exist in the category of Grothendieck toposes. So it is natural to wonder if this extends to sites. Allan Adler ara@altdorf.ai.mit.edu P.S. Please reply to me directly since I haven't yet figured out how to subscribe to this list. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: etale cohomology Date: 12 Jan 93 17:27:22 EST From: ------------- I have not really understood the role of toposes in Grothendieck and Deligne's work on the Weil conjectures. Is it roughly right to say: For etale cohomology over a field k you put the etale topology on the category of (finitely presented?) k-algebras and look at the corresponding topos. Schemes over k are objects in that topos, and the etale cohomology of a scheme S is the topos cohomology of the slice of that topos over S. Or is that hopelessly wrong? Colin McLarty +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: address change From: Reinhard Boerger Date: Thu, 14 Jan 1993 12:59:16 -0500 Dear Bob, please send all categories postings for me to the address I used for this message till June 25. Afterwards, please send them to my new Hagen e-mail address "reinhard.boerger@fernuni-hagen.de". I think it would be easy to inform other people about these address changes by the categories network. Why do you object to this? Cordial greetings reinhard +++++++++++++++++++++++++++++++++++++++++++ Note from moderator: But of course address changes are welcome here! +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: re: etale cohomology Date: Thu, 14 Jan 93 13:59:31 EST From: vladimir@math.ias.edu (Vladimir Voevodsky) What you are saying is definitely right except for the fact that you should consider not the category of k-algebras but the opposite category of affine schemes over k with etale (or whatever else) topology. Vladimir Voevodsky. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: site etal\'e Date: Thu, 14 Jan 93 23:37:10 -0500 From: "Allan Adler" Given a sheaf on a topological space, one can construct the espace etal\'e of that sheaf and recover the sheaf as the sheaf of sections. It would be nice if there were a similar construction for sites. Does anyone know how to do it? Allan Adler ara@altdorf.ai.mit.edu +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Horror story Date: 15 Jan 93 15:50:32 EST From: ------------- Miles Reid, in his _Undergraduate Algebraic Geometry_ tells with evident horror that: I actually know of a thesis on the arithmetic of cubic surfaces that was initially not considered because 'the natural context for the construction is over a general locally Noetherian ringed topos'. This is not a joke. p.116 Now, I know little about the arithmetic of surfaces. I do know that classical cubic surfaces are pretty well understood at least in some ways. Are there substantial open questions about their arithmetic? Basically my question is just what is the point here? Is Reid objecting that arithmetic of cubic surfaces is already so hard in the classical case that it is crazy to look at a more general one? Or is he saying it is so easy that it is crazy to make it hard by bringing in toposes? Any help on this will be much appreciated. Colin McLarty +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Horror story Date: Fri, 15 Jan 93 22:47:09 EST From: vladimir@math.ias.edu (Vladimir Voevodsky) Well, there is a lot of problems in the arithmetic of cubical surfaces. There is a book by Yu.I.Manin called "Cubical surfaces" which was written in sixties and as far as I know most of the conjectures are still open (see also recent reviews by Manin & Co.. on the arithmetic of rational varities). It is very nice and hard part of the arithmetic algebraic geometry which is in a way a generalization of the classical arithmetic of elliptic curves. I really doubt that toposes (especially ringed toposes...) can be of much help here mostly because the theory of toposes is much more elementary thing than all these arithmetical problems. Vladimir Voevodsky. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: site etal\'e Date: Mon, 18 Jan 1993 15:30:36 +0100 From: Francois Lamarche There is an equivalent to the espace etale construction for Grothendieck toposes, which only works (unsurprisingly) if the topos in question has enough points. It is a well-kept secret in category theory, mentioned briefly only here and there (a single line in Johnstone's textbook, also van Osdol's article in SLN 235 (I think this is right), the book titled "exact categories and categories of functors", Barr and van Osdol eds.) The idea is that if a topos E defined over a base topos S has enough points (and here we can take E to be an ordinary Grothendieck topos and S the category of sets) then E is comonadic over a certain S/I, I being an object of S. In other words, and if we specialize a bit, let E be a Grothendieck topos that has enough points (that is, such that there is a small set I of inverse image functors E -> Set such that the joint functor E -> S^I is faithful). Then E is equivalent to the category of coalgebras for a certain comonad G defined on S^I = S/I . The comonad G is the called the comonad of GERMS of E. If we specialize even more and take E to be sheaves over a topological space X then I can be taken to be just the set of points of X. Then analysing a coalgebra Ooops, then given a colagebra A for G, it corresponds to a sheaf on X, but out of A we can construct directly the espace etale of the sheaf. The correspondence is good enough to allow one to call the coalgebra A THE espace etale of a sheaf. In particular, a coalgebra is made of a pair (A,h), where A is an object of S/I and h:A -> GA the structural map; A is indeed a set of points defined above I, and the map h is a choice for every point of A of a germ (an equivalence class of sections) which allows to recover the espace etale topology on A. This is what allows us to consider coalgebra structures as espaces etales for any comonad G defined in the right fashion from a Grothendieck topos E. Francois Lamarche +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: site etal\'e From: Reinhard Boerger Date: Tue, 19 Jan 1993 18:23:39 -0500 Doesn't the following construction work for locales, even without points? Take all sections over arbitrary opens as generators. If s is a section over an open U and if U is a join of opens U(i), then demand s to be the join of the restrictions of s to the U(i). This should give a suitable notion of "locale etal" in terms of generators and relations. If not, why not? Greetings Reinhard +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: advertisement From: Richard Wood Date: Wed, 20 Jan 1993 14:18:57 -0400 DALHOUSIE UNIVERSITY: The Department of Mathematics, Statistics and Computing Science invites applications for several postdoctoral positions in pure and applied Mathematics for the academic year 1993/94. The research interests of the applicants should be compatible with those of an individual or research group in the department. While these are primarily research positions there is a teaching component and applicants are encouraged to supply evidence of their teaching ability. Interested recent Ph.D. graduates should send complete resumes and letters of reference to: Dr. R.P. Gupta, Chair Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, N.S. B3H 3J5 CANADA (Fax 902-494-5130, e-mail gupta@cs.dal.ca) to arrive before February 28, 1993. Dalhousie University is an Employment Equity/Affirmative Action Employer. The University encourages applications from qualified women, aboriginal peoples, visible minorities and persons with disabilities. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Coherence for weakly distributive categories Date: Tue, 19 Jan 93 14:26:43 EST From: rags@triples.Math.McGill.CA (Robert A. G. Seely) We wish to announce that the following paper may be obtained by anonymous ftp (details below) or directly from the authors. Natural deduction and coherence for weakly distributive categories by R.F. Blute, J.R.B. Cockett, R.A.G. Seely, T.H. Trimble (The abstract follows, but first let us highlight some points for two specific audiences:) Some points that might interest categorists: * Proof of coherence for weakly distributive categories. (These are a natural generalization of monoidal categories to include a second "interwoven" tensor, the interweaving given by tensorial strength - the "weak distributivities".) * New uses of logical techniques to prove coherence: proof nets and empires (from linear logic). * Conservativity of the extension to *-autonomous categories. (Here we mean that the unit of the adjunction is fully faithful.) Some points that might interest linear logicians: * By studying the weak fragment of linear logic that weakly distributive categories represent, one can better get to the structure of PAR vis a vis TIMES (without duality confusing the issue). * Units are handled in a constructive version of the usual treatment: we introduce "thinning links" to attach units to the net structure so as to preserve the "acyclic and connected" net criterion, and then determine how these links can be moved about without disturbing the essential structure (coherence). * Coherence represents (part of) the structure of the proof theory of logic, namely when do proof structures represent "the same proof". ========================================================================== ABSTRACT We define a two sided notion of proof nets, suitable for categories, like weakly distributive categories, which have the two-tensor structure (TIMES/PAR) of linear logic, but lack a NEGATION operator. These proof nets have a structure more closely parallel to that of traditional natural deduction than Girard's one-sided nets do. In particular, there is no cut, and cut elimination is replaced by normalization. We prove a sequentialization theorem for these nets and the corresponding sequent calculus, and deduce the coherence theorem for weakly distributive categories. We also extend these techniques to cover the case of non-symmetric ("planar") tensors. We further extend the treatment of coherence to include the units for the tensors, giving a characterization of the Lambek equivalence relation on deductions (i.e. equality of morphisms) in terms of the notion of empire. Finally, we derive a conservative extension result for the passage from weakly distributive categories to *-autonomous categories. ========================================================================== To obtain a copy by anonymous ftp, ftp triples.math.mcgill.ca and login as anonymous - use your email address as password; cd to the directory pub/rags/nets and take the appropriate file: either nets.dvi.Z or nets.ps.Z (you will have to uncompress them at your own site). Contact rags@math.mcgill.ca if you have any trouble with the ftp process or with printing the files at your site. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: site etal\'e Date: Thu, 21 Jan 93 08:23:17 EST From: pjf@saul.cis.upenn.edu (Peter Freyd) Reinhard Boerger asks a question of the following sort: This works for spaces why not for locales? To find an answer to all such questions first try whatever it is on a complete atomless boolean algebra. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: site etal\'e Date: Thu, 21 Jan 1993 17:56:14 +0100 From: Francois Lamarche There is a simple notion of what is a local homeomorphism of locales, along with the theorem: given a locale L then the full subcategory of Loc/L whose objects are local homeos is equivalent to Sheaves on L. All this can be found in Joyal-Tierney monograph "An extension of the Galois theory of Grothendieck". The definition of local homeo extends to toposes. But then the theorem becomes trivial: if E->F is a local homeo of toposes then there is an object I of F such that E is equivalent to F/I. If we want results for toposes that are more concrete and less trivial Peter Freyd's suggestion seems the way to go: replace sets by complete boolean algebras^op and use Barr's theorem that says that to any Grothendieck topos is covered (in the sense of having a surjective geometric morphism to it) by a topos of sheaves over a CBA. F Lamarche +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: graphs and spaces Date: Sat, 23 Jan 93 01:56:12 AST From: Vaughan Pratt It seems to me that the essential difference between graphs (including hypergraphs) and topological spaces resides in the difference between the covariant and contravariant power set functors. In both cases an object is a family of subsets of a set, with such subsets traditionally called respectively edges and open sets. But whereas a morphism of hypergraphs is a function for which the image of an edge is an edge, a continuous function is one for which the *inverse* image of an open set is open. Is this contrast discussed somewhere? And just how different does this make the resulting two concepts? (Treat the latter as being about *generalized* topological spaces, which drop the requirement that arbitrary unions and finite intersections of open sets be open.) Hypergraphs are a staple of the combinatorial literature, which seems to prefer small edges, doubletons being the motivating case definitive of the so-called simple graphs (of the undirected kind). Edges *coexist* in the sense that the whole hypergraph is viewed as the result of pasting its edges together. In contrast a topological space is perceived as more geometric than combinatorial. And open sets are not so much coexisting constituents of the space as *alternative* ways of "smoothly" partitioning the space into a closed and an open subset. Open sets are typically larger than edges, witness Euclidean space whose nonempty open subsets are all infinite. Whereas the natural view of an edge is an atomic constituent of the whole hypergraph, the natural view of an open set is as a (smoothly bounded) subspace. This passage from conjunctions of edges to disjunctions of open sets would appear to be a natural correlate of the passage from the covariant to the contravariant power set functor. My interest in this distinction stems from Chu's completion of a closed category to a self-dual closed (= *-autonomous) category with a specified dualizer. The morphisms of Chu's completion are defined contravariantly, so that when applied to Set it yields exactly these generalized topological spaces rather than hypergraphs. -- Vaughan Pratt +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: toposes and foliations Date: Sun, 24 Jan 93 20:13:17 -0500 From: cxm7@po.CWRU.Edu (Colin Mclarty) I found this question on the sci.math bulletin board and thought it seemed interesting enough to send it here. (His first point was to recommend Mac Lane and Moerdijk's book.) > Second, I want to ask if some Topos/Sheaf fanatic have studied > relationship between the generalized spaces etc of Topos theory, > and the non-conmutative "spaces" of Alain Connes. > I'm not expert in the matter, so I could be asking a nonsense. > Anyway, a motivation for the question is that the Kronecker > foliation of the bidimensional torus, which were the main example > of Connes' theory, is also one principal example of Sheaf/Topos > theory when used to calculate De Rham Homology. At least, it is > used by Moerdijk (although not in that book) to show all the > constructios of clasifiyng topos, BG etc... > -Alejandro Rivero > Zaragoza Univ, Spain rivero@cc.unizar.es or rivero@sol.cie.unizar.es +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: graphs and spaces Date: Sun, 24 Jan 93 23:03:59 PST From: Vaughan Pratt In my previous message tracing the distinction between (hyper)graphs and spaces to the covariant and contravariant power set functors respectively, I asked how different this made graphs and spaces. The two differences I claimed to see were that open sets were larger than edges, and that edges had a conjunctive quality (this edge is present AND that) while open sets were disjunctive (make this cut OR that). Allen Knutson objected to my size argument (that the open subsets of Euclidean space are infinite---indeed they have positive measure) on the ground that closed subsets can be finite. I agree, my size argument is neither complete nor terribly convincing, though I do think there is some correlation between size and the more basic differences. So let me have a shot at answering my question, how do these two functors differentiate graphs from spaces? Both functors take sets to complete atomic Boolean algebras (CABA's) and functions to sup-preserving functions, i.e. morphisms of complete sup-semilattices. The difference lies in their choice of sup-preserving maps. The covariant power set functor chooses the atom-preserving ones, the contravariant one picks the complement-preserving (hence inf-preserving) ones, better known as CABA maps. I claim this makes graphs point-oriented and spaces cut-oriented. Graphs care about how they are held together (by this edge AND that), spaces focus on how they can come apart (in this way OR that). Whereas an edge is a subobject, an open set is a congruence class. Now if we were dealing with say Boolean algebras this distinction would be clear "statically", i.e. without considering morphisms, since no proper subset of a Boolean algebra can be both a subalgebra, which must contain 0 and 1, and a congruence class, which cannot contain both 0 and 1. But for *sets* this distinction is completely invisible statically: every subset can be either a subobject or an equivalence class. Only by watching how a given subset behaves as its parent set "moves" (is acted on by a function) can we tell which it is. Under the action of a function, a subobject behaves like an element: it is neither destroyed nor duplicated, but distinct subobjects may merge into one, and new subobjects can join the existing ones. The behavior of a congruence class under a function is exactly complementary to that of an element: a class can be destroyed or duplicated, but it cannot merge with another class, and no new classes may appear. These features of subobjects and congruence classes constitute the essential differences between the edges of a graph and the open sets of a space respectively. One might also look for differences between graphs and spaces in terms of the localic view of the latter. However locales owe their separate existence to the difference between frame morphisms and CABA maps. This would seem to be a rather more subtle difference than those arising from the differences between the two power set functors, which surely get closer to the heart of the difference between graphs and spaces. Naturally I'd be very interested in pointers to other perspectives on the relationship between graphs and spaces. -- Vaughan Pratt +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: graphs ad spaces From: "Allen Knutson" Date: Thu, 28 Jan 93 19:31:28 EST > Allen Knutson objected to my size argument (that the open subsets of Hurray! Always nice to see my name spelled correctly in a (semi)public place. > Whereas an edge is a subobject, an open set is a congruence class. Now Mm? I would've called a closed set a congruence class - putting on an equivalence relation should be like taking a (Hausdorff) quotient, which unifies a closed subset, not an open one. So I still think open sets are a red herring for your purposes. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: The enclosed Date: Thu, 28 Jan 93 15:23:01 EST From: barr@triples.Math.McGill.CA (Michael Barr) This just came to me and I pass it on for those who want to do anything about it. --Michael From zeleny@husc.harvard.edu Thu Jan 28 13:09:53 1993 Date: Thu, 28 Jan 93 13:09:57 -0500 From: zeleny@husc.harvard.edu Message-Id: <9301281809.AA03731@husc10.harvard.edu> To: barr@triples.Math.McGill.CA Subject: Alonzo Church Status: R Dear Professor Barr, Jonathan Seldin suggested that I write to you about the following. I am a former student of Alonzo Church (at UCLA, from 1986 to his retirement in 1990). As you may know, Professor Church, who is currently living in a retirement community in Hudson, Ohio, will turn 90 on June 14 of this year. Accordingly, I have been trying to find an agency interested in hosting, or participating in a birthday symposium for him. At the moment, there are several possibilities of scheduling sessions in his honor, including the LICS and the ALC. I also have several well-known contributors to the planned _Festschrift_, which is the reason I am contacting you. If you happen to have any papers that you think appropriate, please consider submitting them for publication in the said volume. Church is still actively pursuing his research in logic, and currently has two papers in the publication queue at _Nous_, so presumably he could contribute something of his writings. (However, at the moment, he is still unaware of the organizational efforts; I am planning to tell him as soon as the contributions reach the critical mass.) So if you can participate in either of the above sessions, or submit a paper to the _Festschrift_, please let me know. Failing that, perhaps you could tell the logicians of your acquaintance to send anniversary messages in care of Alonzo Church, Jr., 136 Hudson St., Hudson, OH 44236, who will pass them to his father on June 14th. cordially, michael zeleny@husc.harvard.edu 872 Massachusetts Avenue, Apt 707 Cambridge, Massachusetts 02139 (617) 661-8151