January 1997 - Categories - categories.org.au

Query on w.e.'s
by T.Porter 01 Feb '97

01 Feb '97

Dear All, Can someone give a full reply to this? I was unable to give an adequate one with references etc as I have not been working in this area recently. Thanking you, Tim. ---------- Forwarded message ---------- Date: Fri, 31 Jan 1997 15:21:48 +0000 (GMT) From: Takis Psarogiannakopoulos <takis(a)dpmms.cam.ac.uk> To: t.porter(a)bangor.ac.uk Dear Friend I am sorry that I disturb you with this letter but since nobody here in Cambridge doent seem to know an answer to something (and since I have seen your name in Pursuing Stacks of Grothndieck) I am writing to you with the hope that you probably be able to answer me . In fact what I try to find out if it is true, is a very "easy thing": In the paper of Quillen Higher Algebraic K -Theory there is his theorem about whether a morphism F:C--->B in Cat is a w.e.: if we know that all comma categories F/b are contratible (for any b in B) then F is a w.e. I am wondering if the converse is definitely true , ie: if F:C---->B is a w.e. in Cat (in the sense of Nerve functor) then for every b in B all the comma categories are contractible. In fact what I want to know is: if we have a commutative diagram in Cat as H : C -----> B | | f | | g J = J (ie categories over J) where the map H is a w.e. (ie Ner(H) is a w.e. of simplicial sets) then for every object j of the category J , the "map over j" H/j: f/j ----> g/j is a w.e. Is something like that true? Since the idea of Ouillen in his criterion is that "F/b plays the role of homotopy fibre of the corresponding maps of classifying spaces" it seems to me that the above is true. But the fact that Quillen doesnt refer this explicitly to his paper makes me wondering if there is a simple counterexample where this fails (so there is no reason to sit down and try to write a proof). ( I know that in the case that we define w.e s in Cat through cohomology (ie restricting the w.e.s to the comological ones) the above is true because we actually thinking with the corre- sponding toposes but is this fact remain true for the case of Nerve-w.e s ?) I thank you in advance that you took the time and read this. Sincerely Takis Department of Pure Maths ,Cambridge

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Lax Indexed Functors?
by Alfio Martini 01 Feb '97

01 Feb '97

At the moment we are investigating logical systems and their relations based on the formal concepts of institution (Goguen/Burstall) and entailment system/logic (Meseguer). For our analysis we found appropriate to use concepts like "lax indexed functors" thereby having in mind the corresponding definitions for ordered categories in "Extending properties to categories of partial maps" from Barry Jay [Jay90] (TR ECS-LFCS-90-107). To give an impression about what we are doing, I will give the definition that turns out to be the adequate one for our purposes: A lax indexed functor F from an indexed category C:IND->CAT to an indexed category D:IND->CAT is given by functors F(i):C(i)->D(i) for each i in |IND| and by natural transformations F(g):C(g);F(j)=>F(i);D(g):C(i)->D(j) for each morphism g:i->j in IND such that the following compositionality condition is satisfied for any g:i->j and h:j->k in IND: F(g;h) = (C(g);F(h))*(F(g);D(h)). (We also need the other version where F(g) goes from F(i);D(g) to C(g);F(j).) To get the right feeling and insight we have developed all necessary results by ourselves. Especially we were interested in the generalization of the Grothendieck construction to "lax indexed functors". Now, before fixing these things in a technical report, were are looking for corresponding references of work already done in this direction. Especially we don't want to introduce new names for already known concepts. We need some advice here... Our observation is that we have essentially used for many concepts and results the 2-categorical structure of CAT. Thus we strongly believe that somebody has already defined and investigated "lax functors" and "lax natural transformations" for 2-categories. Thanks for any help. With all best wishes, Alfio Martini.

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anafunctors
by john baez 30 Jan '97

30 Jan '97

What is the locus classicus for "anafunctors"? As far as I know, an anafunctor F: C -> D is a presheaf on C x D^{op} such that F(c,.) is representable for any object c of C. Is this how it's normally defined? Where is composition of anafunctors discussed? Are there other names for these things? John Baez

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finiteness
by carboni＠vmimat.mat.unimi.it 27 Jan '97

27 Jan '97

Regarding the last message of Bill on finiteness and the answer of Vickers, I would like to point out that the 25 years old reference of Bill should be the one at the bottom of page 14 of lesson 3 of 1972 Perugia Notes. In other words, the L he is suggesting should be the internalization of the following description: L consists of those ideals T : N---->Omega such that for all n Tn = Inf{Tm | m > n}. The precise meaning of this definition is explained in the given reference, as well as in Bill's messages. I hope that this is correct. Aurelio Carboni

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PSSL at Bangor
by Ronnie Brown 25 Jan '97

25 Jan '97

63rd Peripatetic Seminar on Sheaves and Logic University of Wales, Bangor March 1, March 2, 1997 As usual, talks will be welcomed on category theory, sheaves, logic and related areas. You are invited, but not required, to offer a talk, and the programme will as usual be decided on the evening of Friday, February 28. Please complete the information required on the booking form and return it by fax or email as soon as possible. Booking Form Name: Title: Home Department: Tel: Fax: E-mail: Time of Arrival (Feb 28): Type of Accommodation required (see attached list): (complete for University accommodation or if you want us to make the arrangements, at your responsibility.) Title of talk: Abstract of talk: Duration: Preferred time: Note: We are asking for support generally and for the social evenings on Friday and Saturday evening. If this is not obtained, we may have to make a charge for these. Details of venues will be given to those coming. All talks will be in the School of Mathematics. Professor R. Brown School of Mathematics University of Wales, Bangor Dean St. Bangor Gwynedd LL57 1UT United Kingdom tel: +44 (0)1248 382474 office: +44 (0)1248 382475 fax: +44 (0)1248 355881 HOW TO GET TO BANGOR Train from London Euston, 3.5 -4 hrs; regular service, often change at Crewe. Trains from Birmingham, 3.3 hrs, from Manchester 3.1 hrs. So the easiest connection by air is via Manchester, where there is a rail connection at the airport. Weather: usually little snow. ACCOMMODATION IN AND AROUND BANGOR *University Accommodation* Telephone Number Bryn Haul, "Oswalds" (01248) 382910 / 383500 1 Double and 3 Twin Bedrooms B&B �17.73 or �12.73 Room Only Comfortable accommodation; inform them of the PSSL, or book through us on a first come first serve basis. About 20 minutes walk from the Department. Student accommodation as follows may be available on a first come first served basis. This should be arranged through the department. Prices are for 2 nights. About 20 minutes walk from the Department. Restaurant on site Bedding supplied, but NO soap, towels, toilet rolls etc �14.96 Self-catering �17.60 Self-catering, En-suite �22.20 En-suite, includes �2.80 per day food credit You may make your own arrangements for the following. We will try to help if there are difficulties. *Guest Houses* (priced from �13.00 - �15.00 per night B & B, five minutes walk) Union Hotel, Garth Road, Bangor 362462 Baytree Lodge, Garth Road, Bangor (01248) 362230 Bro Dawel, Garth Road, Bangor 355242 Dilfan Guest House, Bangor 353030 *Hotels* (priced from �22.50 - �28.95 per night B&B, En-suite, up to twenty minutes walk) British Hotel, High Street, Bangor 364911 Regency Hotel, Holyhead Road, Bangor 370819 Eryl Mor Hotel, Upper Garth Road, Bangor 353789 (overlooking the Menai Straits, 5 min walk) *Hotels* (priced from �22.50 - �30 per night B&B, En-suite, within three miles) Abbeyfield Hotel, Talybont, Nr Bangor 352219 Anglesey Arms Hotel, Menai Bridge 712305 / 712912 Auckland Arms Hotel, Menai Bridge (3 crowns) 712545 / 712453 Country Bumpkin, Llandegai 370477 *Hotels* (priced from �35.00 - �49.50 per night B&B, En-suite, up to 10 minutes walk) Menai Court Hotel, Craig y Don Road, Bangor 354200 *Hotels* (priced from �35.00 - �49.50 per night B&B, En-suite) Carreg Bran Hotel, Llanfairpwllgwyngyll **** 714224 (near the Brittania Bridge) Telford Hotel, Glan Aethwy, Bangor 362971 / 352543 (this side of the Menai Bridge)

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Re: Finiteness in Toposes
by F William Lawvere 23 Jan '97

23 Jan '97

Re: Finiteness in Toposes Jan 17 1997 This concerns the possibility , mentioned in my previous message, of two internal toposes of finite objects. The conjecture that there are two natural internal categories of finite objects is partly supported by the fact that there are two natural natural-numbers objects, the usual one N that parameterizes compositional iteration and another semicontinuous one L with the following features: 0) It is a rig, so receives a homomorphism from N and its elementary arithmetic starts out looking very similar. 1) But unlike N it has a least-number-property in the sense that it is inf-complete and better. 2) It can be constructed internally using truth-valued sheaves on N. 3) Hence it also contains a map from (big) omega, which permits (unlike N) the use of the standard method in finite combinatorics where (for example) a binary relation is considered as a matrix which is valued (not only in a rig where 1+1 = 1, but instead) in a rig in which natural numbers are distinct; the resulting generalized characteristic functions are added, multiplied, infed etc. according to the usual methods of arithmetic and analysis and then translated back into the combinatorics of the original finite structures. Of course, in each case one hopes that the answer to a combinatorial problem might turn out decidable, but that shouldnt require us to stay in the bounds of two-valued subsets in the course of a construction. 4) This internally-defined order-complete rig in E has also an external characterization if E is an S-based topos, namely it is the sheaf of germs of S-geometrical morphisms from E to the topos often called S-sets -through-time (I dont think that depends on any presumption that the N in S ,used to parameterize the transitions through time, coincides with its completion in S). In localic or open set terms, there is in S a (T sub zero) space whose points are N, but whose open sets have the usual order on N as their specialization order; continuous functions from any space E to this space are called semi-continuous and there is in E a sheaf of them. 5) The application to the variable linear algebra over algebraic or complex-analytic spaces needs L too, because dimension of a vector space is a semi-continuous function. More precisely, if A is a good module in a ringed topos E, R then for each X and E there should be a map X--> L which is the fiber-wise dimension of X*A. The basic case is perhaps that where E,R is an algebraic affine scheme, and the conceptual problem is to get at what sort of sets contained in A this dimension function is counting (or bounding). One should not expect that equality of dimension will imply isomorphism. This object L has been discussed for 25 years, but I dont know if anyone published the working-out of its properties and role. Bill

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New share package for crossed modules
by Ronnie Brown 20 Jan '97

20 Jan '97

The following could be of interest to some on the category theory bulletin, since this gives explicit computation of certain finite crossed modules and so of specific kinds of double groupoids, ... Best wishes, Ronnie ---------- Forwarded message ---------- Date: Sat, 18 Jan 97 16:25:41 +0000 (GMT) From: Derek Holt <dfh(a)maths.warwick.ac.uk> To: Multiple recipients of list <GAP-Forum(a)Math.RWTH-Aachen.DE> Subject: New share package for crossed modules A new GAP share package written by Chris Wensley and Murat Alp, for calculating in crossed modules and cat1 groups is now available, and can be collected from the GAP incoming directory in the file xmod131.tar.gz. The GAP web page description is included below. Derek Holt. ****************************************************************************** XMod Authors: Chris Wensley, Murat Alp Language: GAP Operating System: Any Available from: http://www.math.rwth-aachen.de/ftp/pub/incoming/xmod131.tar.gz, Description This package enables construction of and computation with objects within the equivalent categories of crossed modules and cat1-groups, where the groups involved are finite of moderately small order. A crossed module consists of two groups S and R, together with a homomorphism from S to R which essentially commutes with conjugation within S and R. Functions are provided for each of the standard constructions of crossed modules, and for computing with sub- and quotient-structures and homomorphisms. A cat1-group consists of a group G together with two homomorphisms from G to G satisfying certain conditions. It was shown by Loday that there is a natural one-one correspondence between cat1-groups and crossed modules. An important notion is a derivation of a crossed module, which is a map from R to S satisfying certain conditions. It is possible to compute all (or all regular) derivations of a crossed module. There are also functions for computing the actor of a crossed module X, which is a crossed module whose range is the automorphism group of X. Authors Chris Wensley and Murat Alp School of Mathematics University of Wales, Bangor Gwynedd, LL57 1UT GB email: c.d.wensley(a)bangor.ac.uk ~e

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question on finiteness in toposes
by Thomas Streicher 16 Jan '97

16 Jan '97

One knows that for any topos E that the full subcategory of decidable K-finite objects forms a topos itself with 2 = 1+1 as subobject classifier. It is also said that E_kf, the full subcat of E on K-finite objects need not form a topos. That's what I could find out from PTJ's Topos Theory. The counterexample given there is E = Set^2 (where 2 = 0 -> 1). The K-finite objects in Set^2 are the surjective maps between finite sets. It is clear that E_kf is not closed under equalisers taken in E (!). Nevertheless, I think that E_kf itself does have equalisers : if f,g : X -> Y then take the equaliser e_0 of f_0 and g_0 and take the epi-mono-factorisation of x o e_0 : E_0 >---> X_0 | | | epi | x where X = X_0 -> X_1 V V E_1 >---> X_1 this clearly demonstrates that the inclusion E_kf >--> E does not preserve equalisers BUT it does not show that E_kf is not a topos. I would be interested in a reference or example where E_kf really is not a topos. Maybe, E = Set^2 alraedy works but it must have another defect than not being clossed under subobjects w.r.t. E because the decidable K-finite objects have this "defect" as well. Thomas Streicher

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Workshop announcement
by Matthew Hennessy 14 Jan '97

14 Jan '97

EU KIT / INSTITUTE OF SOFTWARE, CAS International Scientific Cooperation ------------------------------------------------ SUMMER SCHOOL AND WORKSHOP -------------------------- Formal Models of Programming and their Applications September 17 - 20, 1997 Beijing China !!!! CALL FOR PARTICIPATION !!!! The aim of the Summer School and Workshop, organised by the EU KIT project SymSem, is to bring together in an informal atmosphere researchers, working in the general area of the Semantic Foundations of Computation. Participation by Chinese researchers and students is particularly encouraged. Roughly half of the meeting will be devoted to expositary seminars given by invited speakers. The remainder will consist of workshop presentations chosen on the basis of submitted abstracts. LIST OF INVITED SPEAKERS: Gerard Boudol, INRIA-Sophia Antipolis, France Zhou CaoChen, IIST, Macau Pierre-Louis Curien, Ecole Normale Superieure, Paris, France Matthew Hennessy, University of Sussex, UK Gerard Huet, INRIA-Rocquencourt, France Colin Stirling, University of Edinburgh SUBMISSIONS: We solicit submissions on original research not published or submitted for publication elsewhere in the form of Extended Abstracts, not to exceed 2500 words (approximately 5 pages). The abstracts must be written in English. The topic of the meeting is to be interpreted in a broad sense, to include semantic and algorithmic aspects of * Programming languages * Verification methods and systems * Program logics * Concurrency: theory and applications * Type theory and applications * Program specification * Formal languages and automata * Rewriting systems It is intended to publish a volume of papers based on the abstracts presented at the meeting. Three copies of abstracts should be sent to Huimin Lin - KIT PO Box 8718 Institute of Software Chinese Academy of Sciences Beijing 100080 China or alternatively electronic submissions (in the form of a uuencoded postscript file) may be sent to kit-submissions(a)cogs.sussex.ac.uk IMPORTANT DATES: Deadline for submission of abstracts: 30/3/97 Notification of Acceptance: 30/4/97 Workshop dates: 17-20/9/97 Information on the workshop will be maintained at the website http://www.cogs.susx.ac.uk/users/huimin/kit-workshop.html Organising Committee: P.L.Curien (France), M. Hennessy (UK), H.Lin (China). Local Organiser: H.Lin (China) Note: TACS is on in Japan on the following week 24 (Wed) -26th Sept. =================

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New Address
by Francisco Marmolejo 08 Jan '97

08 Jan '97

Hello all, I have moved to Mexico, this is my new address Instituto de Matematicas Universidad Nacional Autonoma de Mexico Area de la Investigacion Cinetifica Mexico D. F. 04510 Mexico e-mail quico(a)matem.unam.mx Cheers Francisco.

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