From cat-dist Wed Jan 8 16:31:06 1997 Received: by mailserv.mta.ca; id AA31298; Wed, 8 Jan 1997 16:29:44 -0400 Date: Wed, 8 Jan 1997 16:29:44 -0400 (AST) From: categories To: categories Subject: New Address Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 7 Jan 1997 16:19:29 -0600 (CST) From: Francisco Marmolejo 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@matem.unam.mx Cheers Francisco. From cat-dist Fri Jan 10 12:34:02 1997 Received: by mailserv.mta.ca; id AA26537; Fri, 10 Jan 1997 12:32:35 -0400 Date: Fri, 10 Jan 1997 12:32:35 -0400 (AST) From: categories To: categories Subject: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 10 Jan 1997 12:57:02 MEZ From: Thomas Streicher 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 From cat-dist Sat Jan 11 13:15:55 1997 Received: by mailserv.mta.ca; id AA05923; Sat, 11 Jan 1997 13:15:36 -0400 Date: Sat, 11 Jan 1997 13:15:36 -0400 (AST) From: categories To: categories Subject: Re: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sat, 11 Jan 1997 09:05:39 -0500 (EST) From: Peter Freyd Let me expand. If one bores into just why Set^2_kf can't be boolean and looks for a minimal example of its non-booleaness one inevitably lands on the object 2 -> 1. At first blush its lattice of subobjects does look boolean. Until one notices that there's a monomorphism from 2 -> 2 to 2 -> 1 (where 2 -> 2 is the identity map). Having noticed that, one has a quicker proof that it's not a topos: not every mono-epi is an equalizer. From cat-dist Sat Jan 11 13:15:57 1997 Received: by mailserv.mta.ca; id AA05824; Sat, 11 Jan 1997 13:14:47 -0400 Date: Sat, 11 Jan 1997 13:14:47 -0400 (AST) From: categories To: categories Subject: Re: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sat, 11 Jan 1997 08:38:43 -0500 (EST) From: Peter Freyd The answer that first occures me for Thomas Streicher's question is that in Set^2_kf the terminator generates, hence if it were a topos it would have to be a boolean topos. Which it clearly isn't. Thomas wrote: 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 From cat-dist Sun Jan 12 16:43:17 1997 Received: by mailserv.mta.ca; id AA18931; Sun, 12 Jan 1997 16:42:24 -0400 Date: Sun, 12 Jan 1997 16:42:23 -0400 (AST) From: categories To: categories Subject: RE: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Sun, 12 Jan 97 01:31 EST From: Fred E J Linton <0004142427@mcimail.com> Supplementing Peter's answer to Streicher's K-finiteness question, I recall Prop. 7.4 on p. 97 of SLNM #753, which states, for presheaf topoi E = (C^op, Sets), that, with E_Kf the full subcategory of K-finite E-objects: E_Kf is balanced iff it's a topos iff each K-finite is decidable iff C is a "2-way" category iff ... . Streicher's >--> sure isn't 2-way, hence ... . The rest of that 20 year old report on my student Acun~a's thesis with me is also fun. -- Fred From cat-dist Mon Jan 13 10:27:32 1997 Received: by mailserv.mta.ca; id AA13683; Mon, 13 Jan 1997 10:26:44 -0400 Date: Mon, 13 Jan 1997 10:26:43 -0400 (AST) From: categories To: categories Subject: RE: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sun, 12 Jan 1997 16:53:57 -0500 (EST) From: F William Lawvere Now that the question of finiteness as been reactivated here, may I bring up again the following question ? What concept of finiteness is appropriate for those important mathematical applications in topology for which K/S doesn't seem right ? (For example the equalizer closure of K/S or...??) Especially, a suitably "finite" module should be a vector bundle or a FAC in the sense of Serre so that our simplified topos theory could apply more directly to those things it should. Bill L From cat-dist Mon Jan 13 13:39:46 1997 Received: by mailserv.mta.ca; id AA09767; Mon, 13 Jan 1997 13:39:02 -0400 Date: Mon, 13 Jan 1997 13:39:02 -0400 (AST) From: categories To: categories Subject: Workshop announcement Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 13 Jan 1997 16:51:57 +0000 (GMT) From: Matthew Hennessy 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@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. ================= From cat-dist Tue Jan 14 20:16:21 1997 Received: by mailserv.mta.ca; id AA11885; Tue, 14 Jan 1997 20:14:15 -0400 Date: Tue, 14 Jan 1997 20:14:14 -0400 (AST) From: categories To: categories Subject: RE: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 14 Jan 1997 10:52:38 -0500 (EST) From: F William Lawvere Sorry, I used K/S for an abbreviation of what was called Kuratowski until someone pointed out that it was due to Sierpinski :an object whose mark belongs to the smallest sub-semilattice of its power set which contains the singleton map, or in case there is an NNO an object which in a suitable sense is locally enumerable by the segment under a section of the NNO . While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, still the theory of it in the last 25 years of topos theory seems to mainly be justified by formal analogy and/or independence relative to abstract set theory (=topos with choice). However there are important uses of "finiteness" in algebraic geometry and differential topology (where topos theory after all started) : Consider a ringed topos E,R . For example, the sheaves on an algebraic variety or on a Cinfty manifold. Within the abelian category of R-modules in E, we need to single out two important subcategories FAC (Serre 1955)=coherent sheaves..these tend to be an abelian subcategory and tend to vary covariantly as one E,R is mapped to another E',R' (thus give rise to an extensive K-homology) and vector bundles , which one thinks of as a finite-dimensional vector space varying smoothly over the base space of E ,so they cry out for internalization ; in algebraic geometry these are identified with locally FINITELY free R-modules... they vary contravariantly with E,R (so give rise to K-cohomolgy rigs which act on the FACs,ie intensives acting as densities on the extensives; with further conditions on E,R one can at the level of the riNgs generated by these rigs define a sort of Radon/Nikodym derivative via an alterating sum of Tors , but in general the covariant abelian category FAC and the contravariant tensored category Vect are distinct...The "derived category" of E,R (now allegedly replacing homological algebra in complex analysis and C*-algebra theory) should be the derived category of one of these two linear categories (here I mean dc in the linear sense..nonlinear "derived categories" are more like the stable homotopy of E)) Already the intuitionists speculated about (in effect) subobjects of K/S objects, and it seems we need something of the sort perhaps a category of finites closed under subquotient in order to define the notion of eg finitely-generated R-module in a way which not merely mimics abstract set theory but actually captures the vector bundles . Perhaps it will be easier if E itself satisfies a noetherian condition. It would be best if the desired content could be entirely int- ernalized to E,R but perhaps it is really relative to a base S,K..but perhaps without restriction on S ?? I hope this clarifies the problem. Sincerely Bill From cat-dist Wed Jan 15 10:33:42 1997 Received: by mailserv.mta.ca; id AA23534; Wed, 15 Jan 1997 10:33:21 -0400 Date: Wed, 15 Jan 1997 10:33:21 -0400 (AST) From: categories To: categories Subject: RE: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 15 Jan 97 10:19 GMT From: Dr. P.T. Johnstone Not an answer to Bill's question (which I agree is an important one), but a minor correction. Bill wrote: While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, It isn't, and he didn't. Gavin used finite cardinals to construct the object classifier over an arbitrary base topos with NNO (and I subsequently extended the construction to finitary algebraic theories), but it doesn't work over a topos without NNO (and in particular it can't be made to work using K-finiteness). Andreas Blass showed that the existence of an object classifier for toposes over E implies that E has a NNO. Incidentally, I think it is correct to give credit to Kuratowski for the notion of K-finiteness. It's true that Sierpinski's paper was earlier, but his definition was a "global" one (i.e. he defined the class of all finite sets as the sub-semilattice of the universe generated by he singletons), whereas Kuratowski made the crucial observation that the finiteness of a particular set X can be determined locally (i.e. within the power-set of X), without which the notion could never have been imported into topos theory. Peter Johnstone From cat-dist Wed Jan 15 21:25:46 1997 Received: by mailserv.mta.ca; id AA28134; Wed, 15 Jan 1997 21:25:06 -0400 Date: Wed, 15 Jan 1997 21:25:06 -0400 (AST) From: categories To: categories Subject: RE: question on finiteness in toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 15 Jan 1997 15:22:00 -0500 (EST) From: F William Lawvere Concerning Peter Johnstone's clarification: Of course I didn't mean that the object classifier could be constructed without an internal parameterizer for the finite objects in the base S .... but what exactly are the finite objects ? While the classifier as a topos is determined by the 2-category of bounded S-toposes , the site for it isn't. I was under the impression that an internal category parameterizing the objects which are both K-finite and separable(=decidable) could be used (while internal presheaves on "all" K=finites would presumably be much bigger..what does IT classify ?) Anyway my point was that at any rate no further extension of the notion of finiteness is needed for classifying in that sense the objects or the group objects in S-toposes, whereas by contrast it seems that to give the mathematically correct notion of "vector space for which there exists a finite basis" does need such an extension. In any topos, a subobject of a nonnon sheaf is always separable ; when is the converse true ? Perhaps there is an internal topos object V which is largest with respect to being fully embedded in the given topos E while at the same time having A as its subtopos of internal nonnon sheaves. Here by A is meant the Boolean internal topos mentioned above which parameterizes the separable K-finites of E (Fred recalled Acunya's work showing among other things that it is Boolean) and to say that V "is" fully embedded in E has sense for any internal category with a terminal object , namely we require that the canonical parametrized (="indexed") functor from V to E is an equivalence E(X,V)--> E/X for each X. The latter functor is defined by merely pulling back the fibration 1/V--> V of pointed objects in V. When the answer to the above question is affirmative, Johnstone's locally separable reflection Vsubqd will consist of subquotients and the K-finites may fit in . It seems that the inclusion of A in V will preserve sums but only certain epis. The idea is that V can't be too large since the inverse to the inclusion will enrich it in A. On Wed, 15 Jan 1997, categories wrote: > Date: Wed, 15 Jan 97 10:19 GMT > From: Dr. P.T. Johnstone > > Not an answer to Bill's question (which I agree is an important one), > but a minor correction. Bill wrote: > > While the K/S definition is right for the construction of > the object classifier over an arbitrary base topos (as Gavin > showed) and hence for classifiers for various kinds of > finitary algebras over an arbitrary base topos, > > It isn't, and he didn't. Gavin used finite cardinals to construct > the object classifier over an arbitrary base topos with NNO (and I > subsequently extended the construction to finitary algebraic > theories), but it doesn't work over a topos without NNO (and in > particular it can't be made to work using K-finiteness). Andreas > Blass showed that the existence of an object classifier for toposes > over E implies that E has a NNO. > > Incidentally, I think it is correct to give credit to Kuratowski for > the notion of K-finiteness. It's true that Sierpinski's paper was > earlier, but his definition was a "global" one (i.e. he defined the > class of all finite sets as the sub-semilattice of the universe > generated by he singletons), whereas Kuratowski made the crucial > observation that the finiteness of a particular set X can be determined > locally (i.e. within the power-set of X), without which the notion > could never have been imported into topos theory. > > Peter Johnstone > From cat-dist Fri Jan 17 16:05:56 1997 Received: by mailserv.mta.ca; id AA30853; Fri, 17 Jan 1997 16:04:23 -0400 Date: Fri, 17 Jan 1997 16:04:23 -0400 (AST) From: categories To: categories Subject: RE: Finiteness in Toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST) From: F William Lawvere 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 From cat-dist Mon Jan 20 14:45:34 1997 Received: by mailserv.mta.ca; id AA29893; Mon, 20 Jan 1997 14:43:22 -0400 Date: Mon, 20 Jan 1997 14:43:22 -0400 (AST) From: categories To: categories Subject: New share package for crossed modules Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 20 Jan 1997 10:11:53 +0000 (GMT) From: Ronnie Brown 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 To: Multiple recipients of list 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@bangor.ac.uk ~e From cat-dist Wed Jan 22 14:43:07 1997 Received: by mailserv.mta.ca; id AA31635; Wed, 22 Jan 1997 14:41:21 -0400 Date: Wed, 22 Jan 1997 14:41:21 -0400 (AST) From: categories To: categories Subject: RE: Finiteness in Toposes Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 22 Jan 1997 18:18:53 +0000 From: Steve Vickers >Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST) >From: F William Lawvere > >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: > >... > > This object L has been discussed for 25 years, but I dont know if >anyone published the working-out of its properties and role. Am I right in thinking this to be Idl N, the ideal completion of the natural numbers (with their usual order)? I conjecture that this is a suitable value domain for the ranks of matrices over localic fields such as the reals: rank^-1{n} is not open, but rank^-1{n, n+1, n+2, ...} is. Then rank A is the set of natural numbers n such that we can find enough apartnesses to prove linear independence of n rows of A, and this is an ideal of N - the definition also smoothly incorporates infinite matrices. (Perhaps this is just one of the things that have have been discussed for 25 years and I'm reiventing it.) Anyway, I have investigated Idl N as a fixpoint object (in the sense of Crole and Pitts) in the category of Grothendieck toposes (modulo 2-categorical niceties that I didn't investigate too closely) in a paper "Topical Categories of Domains". Steve Vickers. From cat-dist Thu Jan 23 14:50:48 1997 Received: by mailserv.mta.ca; id AA00818; Thu, 23 Jan 1997 14:49:55 -0400 Date: Thu, 23 Jan 1997 14:49:55 -0400 (AST) From: categories To: categories Subject: finiteness Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Thu, 23 Jan 1997 19:43:40 EST From: carboni@vmimat.mat.unimi.it 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 From cat-dist Fri Jan 24 10:55:16 1997 Received: by mailserv.mta.ca; id AA22685; Fri, 24 Jan 1997 10:54:41 -0400 Date: Fri, 24 Jan 1997 10:54:41 -0400 (AST) From: categories To: categories Subject: anafunctors Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Thu, 23 Jan 1997 13:26:33 -0800 From: john baez 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 From cat-dist Fri Jan 24 10:55:39 1997 Received: by mailserv.mta.ca; id AA24353; Fri, 24 Jan 1997 10:55:32 -0400 Date: Fri, 24 Jan 1997 10:55:32 -0400 (AST) From: categories To: categories Subject: Re: finiteness Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 24 Jan 1997 10:54:59 +0000 From: Steve Vickers >From: carboni@vmimat.mat.unimi.it > >... L consists of those ideals T : N---->Omega such that for all n >Tn = Inf{Tm | m > n}. I understand this as saying that n is in the ideal iff every greater m is in the ideal (but I think the inequality m > n has to be non-strict to make sense of this). Hence it's really a filter of N. If that's correct, then my suggestion was wrong. L would be not Idl N, but Idl(N^op). That makes sense regarding dimensions, for if a real vector space is finitely presented using an mxn matrix A (presenting R^n/Im A) then its dimension is n-rank(A), so if rank(A) is in Idl(N), the dimension should be in Idl(N^op). (By the way, what's a full reference for the "Perugia Notes"?) Steve. From cat-dist Fri Jan 24 15:38:23 1997 Received: by mailserv.mta.ca; id AA04283; Fri, 24 Jan 1997 15:36:40 -0400 Date: Fri, 24 Jan 1997 15:36:38 -0400 (AST) From: categories To: categories Subject: PSSL at Bangor Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: QUOTED-PRINTABLE Status: O X-Status: Date: Fri, 24 Jan 1997 15:04:12 +0000 (GMT) From: Ronnie Brown 63rd Peripatetic Seminar on Sheaves and Logic=20 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,=20 to offer a talk, and the programme=20 will as usual be decided on the evening of Friday, February 28.=20 Please complete the information required on the booking form and return=20 it by fax or email as soon as possible. =20 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:=20 Preferred time:=20 Note: We are asking for support generally and for the=20 social evenings on Friday and Saturday evening. If this is=20 not obtained, we=20 may have to make a charge for these.=20 Details of venues will be given to those coming.=20 All talks will be in the School of Mathematics.=20 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,=20 often change at Crewe.=20 Trains from Birmingham, 3.3 hrs, from Manchester 3.1 hrs.=20 So the easiest connection by air is via Manchester,=20 where there is a rail connection at the airport.=20 Weather: usually little snow.=20 ACCOMMODATION IN AND AROUND BANGOR *University Accommodation* Telephone Number Bryn Haul, "Oswalds" (01248) 382910 / 383500 1 Double and 3 Twin Bedrooms B&B =A317.73 or =A312.73 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=20 the Department.=20 Student accommodation as follows may be available on a first come first=20 served basis. This should be arranged through the department. =20 Prices are for 2 nights. About 20 minutes walk from=20 the Department.=20 Restaurant on site Bedding supplied, but NO soap, towels, toilet rolls etc =A314.96 Self-catering =20 =A317.60 Self-catering, En-suite =A322.20 En-suite, includes =A32.80 per day food credit You may make your own arrangements for the following.=20 We will try to help if there are difficulties.=20 *Guest Houses* (priced from =A313.00 - =A315.00 per night B & B,=20 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 =A322.50 - =A328.95 per night B&B, En-suite, up to= =20 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 =A322.50 - =A330 per night B&B, En-suite, within=20 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=20 *Hotels* (priced from =A335.00 - =A349.50 per night B&B, En-suite, up to= =20 10 minutes walk) Menai Court Hotel, Craig y Don Road, Bangor 354200 *Hotels* (priced from =A335.00 - =A349.50 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)=20 From cat-dist Mon Jan 27 13:18:56 1997 Received: by mailserv.mta.ca; id AA15614; Mon, 27 Jan 1997 13:15:55 -0400 Date: Mon, 27 Jan 1997 13:15:55 -0400 (AST) From: categories To: categories Subject: finiteness Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sun, 26 Jan 1997 18:16:28 EST From: carboni@vmimat.mat.unimi.it Regarding my last message on finiteness, I should have said `functors N---->Omega' instead of `ideals N--->Omega'. I repeat that I `think' that this is what Bill wanted to say, but I am not sure that I am correct. As for the reference of Bill's Perugia Notes, they are an internal publication of Perugia University in 1972 of the lectures given by Bill Lawvere when he was visiting that University. They should be available there (write to prof. L. Stramaccia, Dipartimento di Matematica, Universita' di Perugia, via Vanvitelli 1, 06123 Perugia, Italy, email: stra@gauss.dipmat.unipg.it). Also, they were quite spread out, so that you should be able to find somebody nearby you who has them. Other possibilities are asking the author himself and eventually myself. Aurelio Carboni. From cat-dist Wed Jan 29 16:05:44 1997 Received: by mailserv.mta.ca; id AA24779; Wed, 29 Jan 1997 16:05:38 -0400 Date: Wed, 29 Jan 1997 16:05:38 -0400 (AST) From: categories To: categories Subject: Lax Indexed Functors? Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Wed, 29 Jan 1997 18:58:44 +0100 (MET) From: Alfio Martini 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. From cat-dist Wed Jan 29 16:05:47 1997 Received: by mailserv.mta.ca; id AA14984; Wed, 29 Jan 1997 16:04:29 -0400 Date: Wed, 29 Jan 1997 16:04:28 -0400 (AST) From: categories To: categories Subject: anafunctors Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 29 Jan 1997 14:56:30 -0500 (EST) From: Michael Makkai John Baez asked, on January 24, about anafunctors. As far as I know, the notion was first explicitly introduced in my paper "Avoiding the axiom of choice in general category theory", in JPAA 108 (1996), 109-173. The term was suggested by Dusko Pavlovic. Precursors occur in the work of Max Kelly, and Andre Joyal, as I explain in the paper. The concept John gives is equivalent to "saturated anafunctor" in the paper; plain anafunctor is something that generalizes "functor". The wording of the definition of "saturated anafunctor" is different from John's definition, but the equivalence is fairly straightforward. I should mention that John`s definition is a very useful formulation, especially when one wants to generalize things to higher dimensional categories, as I came to realize some time after I started studying the John Baez/James Dolan announcement on weak n-categories. In addition to the paper mentioned above, there is reference to anafunctors in "First Order Logic with Dependent Sorts", a monograph that will appear in Springer's Lecture Notes in Logic as soon as I manage to complete the necessary revisions; it is available electronically from the TRIPLES and HYPATIA (?) sites. From cat-dist Fri Jan 31 14:15:58 1997 Received: by mailserv.mta.ca; id AA19383; Fri, 31 Jan 1997 14:14:33 -0400 Date: Fri, 31 Jan 1997 14:14:32 -0400 (AST) From: categories To: categories Subject: Re: Lax Indexed Functors? Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 31 Jan 1997 14:48:30 GMT From: Sandro Fusco On Jan 29, 4:05pm, categories wrote: > Subject: Lax Indexed Functors? > Date: Wed, 29 Jan 1997 18:58:44 +0100 (MET) > From: Alfio Martini > > 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) > 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)). --------(1) > > > 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". > > Thanks for any help. > > With all best wishes, > > Alfio Martini. > > >-- End of excerpt from categories Greetings! Concerning Alfio Martini's message, I would like to point out that I have been using a similar notion of what I also called "lax indexed functors". The only difference is that instead of condition (1) I have the following: (\Phi_g,h ; F(k)) * F(g;h) = (C(g);F(h)) * (F(g);D(h)) * (F(i) ; \Phi'_g,h) where \Phi_g,h:C(g);C(h)->C(g;h) is the natural isomorphism associated with the indexed category C and \Phi'_g,h is the natural isomorphism associated with D (the reason being that my indexed categories C, D: IND->CAT are basically pseudofunctors). As shown in my thesis abstract below, a generalized Grothendieck construction is established. I expect to have copies of my thesis available by the end of April 1997. Yours truly, Sandro Fusco ------------- Doctoral Thesis Title: Stable Functors and the Grothendieck Construction. Abstract: In classical domain theory, Scott-continuous functions of partially ordered sets (posets) are used to model approximation processes. When replacing posets by (the more general) categories, the so called "stable functors" take on the role of Scott-continuous functions. Different notions of stable functors were studied intensively by various researchers during the past ten years (notably by Paul Taylor of Imperial College, London, and Walter Tholen of York University, Toronto). These notions are closely related to generalizations of two fundamental notions of category theory, adjoint functors, and fibrations, with no apparent link between the two approaches. In this thesis, we 1. introduce and investigate a satisfactory notion and theory of stable functors based on "factorizations relative to a functor" (as given by both, generalized adjoints and fibrations); 2. establish the Grothendieck construction as a 2-functor \Gamma whose domain is the 2-category of \{cal X}-indexed categories and whose target is the suitably defined 2-category of stable functors with codomain \{cal X}, in such a way that \Gamma is part of a higher-dimensional adjunction. This latter part can be exploited at various levels of generality, yielding in particular the well-known equivalence between \{cal X}-indexed categories and cloven fibrations. -- Sandro Fusco Dept. of Mathematics and Statistics York University North York, Ontario Canada M3J 1P3 Tel: (416) 736-2100 Ext. 40617 Fax: (416) 736-5757 From cat-dist Fri Jan 31 14:16:51 1997 Received: by mailserv.mta.ca; id AA17192; Fri, 31 Jan 1997 14:16:44 -0400 Date: Fri, 31 Jan 1997 14:16:44 -0400 (AST) From: categories To: categories Subject: Query on w.e.'s Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 31 Jan 1997 17:19:24 +0000 (GMT) From: T.Porter 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 To: t.porter@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