January 1993 - Categories - categories.org.au

Re: graphs ad spaces
by Allen Knutson 07 Feb '93

07 Feb '93

> 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. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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The enclosed
by barr＠triples.Math.McGill.CA 29 Jan '93

29 Jan '93

This just came to me and I pass it on for those who want to do anything about it. --Michael >From zeleny(a)husc.harvard.edu Thu Jan 28 13:09:53 1993

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Alonzo Church
by zeleny＠husc.harvard.edu 29 Jan '93

29 Jan '93

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(a)husc.harvard.edu 872 Massachusetts Avenue, Apt 707 Cambridge, Massachusetts 02139 (617) 661-8151

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graphs and spaces
by Vaughan Pratt 25 Jan '93

25 Jan '93

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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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toposes and foliations
by cxm7＠po.CWRU.Edu 25 Jan '93

25 Jan '93

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(a)cc.unizar.es or rivero(a)sol.cie.unizar.es +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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site etal\'e
by Allan Adler 22 Jan '93

22 Jan '93

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(a)altdorf.ai.mit.edu +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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advertisement
by Richard Wood 21 Jan '93

21 Jan '93

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(a)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. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Coherence for weakly distributive categories
by rags＠triples.Math.McGill.CA 20 Jan '93

20 Jan '93

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(a)math.mcgill.ca if you have any trouble with the ftp process or with printing the files at your site. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Horror story
by cxm7＠pop.cwru.edu 16 Jan '93

16 Jan '93

------------- 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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etale cohomology
by cxm7＠pop.cwru.edu 15 Jan '93

15 Jan '93

------------- 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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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