Date: Wed, 1 Mar 1995 04:29:48 -0400 (AST) Subject: Composition as a relation? Date: Tue, 28 Feb 1995 15:12:04 -0500 From: David Espinosa Has there been any work on "categories" with composition as a relation instead of a function? We would allow several "composites" of two arrows but would likely require unique composition with the identity: f o id = {f} David Date: Wed, 1 Mar 1995 04:31:16 -0400 (AST) Subject: Re: APCS questionnaire form Date: Wed, 1 Mar 95 12:12:13 +1100 From: Max Kelly OK, Peter - you know from Tours that I too detest their questionnaire, and I am happy that you have stirred me into action. I shall write now. Regards, Max. Date: Sat, 4 Mar 1995 00:29:34 -0400 (AST) Subject: Re: APCS questionnaire form Date: Tue, 28 Feb 95 07:56:00 EST From: Michael Barr I am replying to Peter's letter about Kluwer. I had actually intended to reply to him, but of course replies go to the list. First off, let me say that the simplest solution to the point he raised is to discard the form and write your own report. Beggars can't be choosers and they are begging you. A similar problem with the referee forms used by NSERC, the Canadian granting agency. The forms are pretty useless for mathematics. Many referees follow them anyway and you have to wade through the irrelevancy to get to the meat, if any, of the report. Others ignore the form and simply write a report and those are the most useful. However, I will not referee a report for ACS under any circumstances and I guess those forms are indirectly responsible. I submitted a paper to them on fuzzy models of linear logic. Briefly, it showed that there was a *-autonomous category whose objects were sets and arrows were fuzzy relations. Now, I won't go into the other flaws in the paper as originally submitted, which might well have justified rejection, but the actual reason given for rejecting it was that it was not within the scope of the journal. Since it was squarely an application of category theory, I have concluded that I have no idea what the scope of the journal is and have therefore refused to referee papers for them. I believe that the real reason for the journal's problems are that the publisher deals with the refereeing process instead of having editors for that purpose. It is probably unimportant, since I don't really expect most print journals to be around in ten years. Michael Date: Sat, 4 Mar 1995 00:35:06 -0400 (AST) Subject: Re: Composition as a relation? Date: Wed, 1 Mar 1995 09:20:42 +0100 From: mas013@clvax.bangor.ac.uk Dear All, re categories with relational composition: David Espinosa asked: "Has there been any work on "categories" with composition as a relation instead of a function? We would allow several "composites" of two arrows but would likely require unique composition with the identity: f o id = {f} Such objects naturally occur in some studies of homotopy coherence in abstract homotopy theory. There is a set of possible composites so a choice of composite is made to make life easy, but in this context, any two choices are related by higher dimensional data. Of course associativity does not work but does up to higher diemnsional data. This is linked to some ideas in bicategories. Perhaps the fact that the resulting THING is constructed rather like a Kleisli category but with a choice of "multiplication" on the "monad" is significant. If anyone is interested I can expand further on this. Tim Tim Porter, Bangor, e-mail: mas013@clss1.bangor.ac.uk Date: Sat, 4 Mar 1995 00:40:24 -0400 (AST) Subject: Re: Composition as a relation? Date: Wed, 1 Mar 1995 20:37:52 -0500 From: David Espinosa Do you have any examples in mind? Abstractions with no examples have an unfortunate track record. Jim, Thanks for your suggestions, and sorry to omit the examples. I'm considering a "category" whose objects are type constructors, given as endofunctions on a base category, and whose arrows are _situated monads_ relating the type constructors. There's no _function_ for monad composition, but perhaps we can obtain a _relation_ by describing conditions for a monad to be a composition of two others. Does that sound reasonable (or like anything else you've seen)? After working out the details, I find that under a relational definition of monad composition, M = M o id but M not= id o M In other words, there exists a situated monad N (not equal to M) such that N = id o M For "composition as a relation" to be useful, I would guess that both identities would have to come out exactly (as well as associativity). After all, it's not much fun to compose the identity and get something else back! But, in general, I like the idea of relational composition and would be interested to see some applications. Thanks again for your help. The full development follows, in case you're interested. David Situated Monads --------------- A _situated monad_ PQ is a triple PQ : im(P) -> im(Q) unitPQ : P(A) -> Q(A) bindPQ : (P(A) -> Q(B)) -> (Q(A) -> Q(B)) obeying the usual monad laws and also PQ(P(A)) = Q(A). PQ is the object part of a functor between full subcategories im(P) and im(Q) whose arrow part is mapPQ : (P(A) -> P(B)) -> (Q(A) -> Q(B)) mapPQ(f) = bindPQ(unitPQ o f) UnitPQ is a natural transformation from i_P : im(P) -> C to i_Q o PQ : im(P) -> C where i_P and i_Q are the inclusion functors of im(P) and im(Q) into the base category C. BindPQ (as usual) seems to have a categorical description solely as a map between hom-sets. Join (monadic eta) seems inappropriate because PQ is not an endofunctor, and thus join is ill-typed. Monad composition as a relation ------------------------------- If we want to say that (as situated monads) PR = PQ o QR we can postulate unitPR = unitPQ o unitQR bindPR(unitQR o f) = mapQR(bindPQ(f)) for f : P(A) -> Q(B). The latter obtains by pushing f both ways through the composition triangle. These imply mapPR = mapPQ o mapQR So far, so good. Now let's see what happens if PQ = id or QR = id. Does it follow that "composition" with the identity is the identity? With QR = id, we find unitPR = unitPQ bindPR = bindPQ but with PQ = id, we have unitPR = unitQR mapPR = mapQR which is much weaker. We could add another law bindQR(f) = bindPR(f o unitPQ) for f : Q(A) -> R(B), but this implies for QR = id that f = bindPQ(f o unitPQ) which is false for monads (take PQ as the list monad and f as reverse). So left composition with the identity doesn't yield exactly the same monad back (under the above definition of composition). We have PQ = PQ o id but QR not= id o QR This asymmetry doesn't surprise me, since Kleisli composition isn't symmetric (the arrows to be composed don't have symmetric types if we look at its most general typing). Unfortunately, I would expect that both identities (and associativity as well) would need to hold for composition to make sense, even as a relation. David Date: Sat, 4 Mar 1995 00:44:20 -0400 (AST) Subject: Re: Composition as a relation? Date: Thu, 2 Mar 1995 00:50:16 -0500 From: Jim Otto Dear David, Thanks for your suggestions, and sorry to omit the examples. I'm considering a "category" whose objects are type constructors, given as endofunctions on a base category, and whose arrows are _situated monads_ relating the type constructors. There's no _function_ for monad composition, but perhaps we can obtain a _relation_ by describing conditions for a monad to be a composition of two others. Does that sound reasonable (or like anything else you've seen)? To me this gives your question an entirely different flavor. You might want to post your question again with the above target example added. There is a lot of work on combinig triples, e.g. Manes. My impression is that it is not obvious what it means to combine 2 triples. (One way is the stacking below.) There is some mention of combining triples with distribute laws in Barr, Wells, toposes, triples, and theories, '85, Springer Bicategories may be relevant. MacLane, Pare, coherence for bicategories and indexed categories, JPAA (Journ. of Pure and Applied Alg.) '85 There is a paper by Palmquist, in I think Springer LNM 195, on the double category of adjoint pairs. Double categories are categories internal to the category of small categories. 2-categories are a special case of both bicategories and of double categories. Monoidal categories are a special case of bicategories. Borceux, handbook of categorical algebra, '94, Cambridge may be helpful on some of this. Finitary triples are those commuting with filtered colimits. One can stack them by starting with set to a discrete power, taking a finitary triple on that, forming the category of algebras, taking a finitary triple on that, forming the category of algebras, ... The net result of a finite such stack can also be obtained with just 2 special layers. The 1st has as algebras the sketches in the sense of my `tensor and linear time' and the 2nd follows from the reflector following from the orthogonality. The full development follows, in case you're interested. Maybe I'll read it sometime. The sad truth is that it's hard to write and even harder to get anyone to read it. (However I have found one reader for my writing. Myself. It's amazing what one forgets.) Regards, Jim Date: Sat, 4 Mar 1995 00:47:22 -0400 (AST) Subject: A couple of simplicial questions Date: Fri, 3 Mar 1995 18:50:19 -0500 From: Todd Wilson 1. A question about notation: In accounts of internal category theory, one uses d_0, d_1 : C_1 -> C_0 as, respectively, domain and codomain, in analogy (I suppose) with simplicial objects. But then I would have expected it to be the other way around: in line with the analogy of an arrow f:A->B as an oriented one-simplex between vertices A and B , domain should rather be the face operator d_1 ("delete 1") and codomain d_0 ("delete 0"). Am I backwards? 2. The category Grp of groups has the property that every simplicial object in Grp satisfies the extension condition (i.e., is a so-called Kan complex). Is there a characterization of the categories with this property? Are there interesting uses of homotopy in these (other) categories? --Todd Wilson Date: Sun, 5 Mar 1995 23:48:57 -0400 (AST) Subject: Re: Composition as a relation? Date: Sat, 4 Mar 1995 00:16:18 -0500 From: Jim Otto Dear People, >Dear David, > > ... > I'm considering a "category" whose objects are type constructors, > given as endofunctions on a base category, and whose arrows are > _situated monads_ relating the type constructors. > ... > >To me this gives your question an entirely different flavor. You Whoops. I had meant that to be private. I guess I may as well add a few more references in public mode. triples: E. Manes, Algebraic Theories, 1976, Springer-Verlag the monoidal case of bicategories: A. Joyal and R. Street, The geometry of tensor calculus, I, 1991, Advances in Mathematics some implicit but usually unmentioned coherence trickery: G. Kelly, On MacLane's condition of coherence of natural associativities, commutativities, etc., 1964, J. Algebra Oh well, Jim Date: Sun, 5 Mar 1995 23:51:48 -0400 (AST) Subject: Re: A couple of simplicial questions Date: Sat, 4 Mar 95 14:44:07 EST From: Michael Barr I have no opinion on the notational question. I suppose the standard notation is backward, but I am not going to change now. I use d^0 and d^1, BTW, reserving the lower index for the dimension, which in this case is 1. If an equational category has a Mal'cev operator (a ternary operation t with t(x,x,y) = t(y,x,x) = y), then every simplicial object is Kan. This is essentially clear from John Moore's proof of the group case, in which forms xy^{-1}z appear repeatedly. For an equational category, the converse is also true, since a special case of every simplicial object being Kan is that every reflexive relation is an equivalence relation, a well-known characterization of Mal'cev. Most familiar Mal'cev categories have a group structure, so the Kan condition follows from the case for groups. Just about the only familiar Mal'cev category that I can think of that lacks a group op is Heyting algebras and I don't know any application of simplicial Heyting algebras. --Michael Barr Date: Sun, 5 Mar 1995 23:58:09 -0400 (AST) Subject: APCS questionnaire form Date: Sun, 5 Mar 95 15:07 GMT From: Dr. P.T. Johnstone This is a response to Mike Barr's message of 28 February. I agree with him that the Kluwer form used by `Applied Categorical Structures' is no worse than that used by grant-giving bodies such as NSERC---the British EPSRC forms are every bit as bad as those used by NSERC, believe me--- but there is an important difference: we don't have any `muscle' in arguments with the grant-giving bodies because we are all, to a greater or lesser extent, beholden to them ourselves, whereas (thanks largely to the Internet) referees do now have collective muscle which we can deploy against journal editors and publishers (as Mike rightly points out, they are dinosaurs under imminent threat of extinction in any case), and in my view we should use it if by so doing we can improve the conditions under which we operate. Incidentally, to correct a misunderstanding of my original message which some people seem to have made: I wasn't simply saying that in future I will ignore the APCS questionnaire (I have been doing that since the journal started, in any case), but rather that I will ignore any papers which are sent to me accompanied by the questionnaire---I will only referee papers which are sent without the questionnaire. If we all agree to do that, then Kluwer will very quickly be forced to abandon the wretched thing. Peter Johnstone Date: Tue, 7 Mar 1995 21:07:24 -0400 (AST) Subject: Re: APCS questionnaire form Date: Mon, 06 Mar 1995 11:27:42 -0500 (EST) From: MTHISBEL@ubvms.cc.buffalo.edu I have no speech. It just begins to seem relevant to mention that Peter's original message stirred me to send Kluwer the same thing. John Isbell Date: Tue, 7 Mar 1995 21:09:35 -0400 (AST) Subject: Questionnaire Date: Tue, 7 Mar 1995 00:34:34 -0400 (AST) From: Bob Rosebrugh While not wishing to cut off discussion, I feel that if someone wishes to act on Dusko's suggestion that would be fine (and will be glad to circulate a proposed letter for signatures), and that otherwise the subject might be better handled now by direct communication with Ms. Sonneveld. regards to all, Bob Rosebrugh Date: Tue, 7 Mar 1995 21:03:34 -0400 (AST) Subject: Re: APCS questionnaire form Date: Mon, 6 Mar 1995 10:38:45 +0000 (GMT) From: Dusko Pavlovic According to categories: > > Date: Sun, 5 Mar 95 15:07 GMT > From: Dr. P.T. Johnstone > or lesser extent, beholden to them ourselves, whereas (thanks largely > to the Internet) referees do now have collective muscle which we can > deploy against journal editors and publishers (as Mike rightly points The case against the form is fairly clear (although, so far, I thought it was easier to complete than to discuss). The only question is which muscle should we use: * leftist: flood Mrs. Sonneveld with angry messages, signed something like )8-\ * rightist: dispatch a joint letter, signed something like Internetional Academy of Applied Categorical Referees I doubt that this network will provide for a real flood. For an effect, we'll probably have to act rightist... I mean, seriously, if we all agree, a joint statement will probably do it. Otherwise, I can contribute a message to Mrs. Sonneveld, but it will probably be one of five or so. Regards to all, -- Dusko Pavlovic Date: Thu, 9 Mar 1995 01:25:52 -0400 (AST) Subject: Re: A couple of simplicial questions Date: Thu, 9 Mar 95 14:47:51 +1000 From: Max Kelly Todd Wilson asked on 3 Mar: The category Grp of groups has the property that every simplicial object in Grp satisfies the extension condition (i.e., is a so-called Kan complex). Is there a characterization of the categories with this property? Are there interesting uses of homotopy in these (other) categories? An answer may be found in A. Carboni, G.M. Kelly, and M.C. Pedicchio, Some remarks on Maltsev and Goursat categories, Applied Categorical Structures 1 (1993), 385-421. The slogan is "Kan = Maltsev". Max Kelly. Date: Fri, 10 Mar 1995 02:45:00 -0400 (AST) Subject: some simplicial questions Date: Thu, 09 Mar 1995 12:09:04 +0000 From: Prof R. Brown The nerve of a groupoid (not just a group) is also a Kan complex in the strong sense of a simplicial T-complex (Keith Dakin's definition, 1975). A simplicial T-complex (K,T) has sets T_n in K_n of elements called thin with the property that: 1) any horn has a unique thin filler 2) degenerate implies thin 3) if all faces but one of a thin element are thin, so is the last face. This is of rank less than or equal to n if all simplices in dimensions greater than n are thin. Simplicial T-complexes of rank 1 are equivalent to groupoids (Dakin). Simplicial T-complexes in general are equivalent to crossed complexes (Ashley, see Diss Math 265 (1988) Simplicial T-complexes and crossed complexes; Nan Tie, JPAA 56 (1989) 195-209), to infinity-groupoids, and to other things. The notion of cubical T-com plex is crucial in the proof by Brown-Higgins of an n-dimensional Van Kampen Theorem (JPAA 22 (1981) 11-41), because of the technical point that it easily enables the handling of multiple compositions of homotopy addition lemmas (the boundary of a simplex or cube is the "sum" of its faces) without writing down formulae. For polyhedral versions, see D W Jones, Diss Math 266 (1988) A general theory of polyhedral sets and the corresponding T-complexes.) Part of the point is that the singular complex SX of a space X is a Kan complex but the fillers come from the models, so ought, morally, to be canonical and to satisfy relations. But these relations are up to homotopy, it seems. So it is difficult to be more precise. The T-complex condition is very strong, but is nice in that it is easy to see how to weaken it. These weakenings need lots more investigation. A further point is that a filler of a horn of a triangle determines a product, as if it were a `computation'. Analogous notions for categories are studied by Ross Street and by Dominic Verity. Ronnie Brown Prof R. Brown Tel: (direct) +44 248 382474 School of Mathematics (office) +44 248 382475 Dean St Fax: +44 248 355881 University of Wales email: mas010@bangor.ac.uk Bangor wwweb for maths: http: //www.bangor.ac.uk/ma Gwynedd LL57 1UT UK Date: Sat, 11 Mar 1995 01:19:35 -0400 (AST) Subject: Re: A couple of simplicial questions Date: Fri, 10 Mar 1995 17:02:09 -0500 (EST) From: MTHDUSKN@ubvms.cc.buffalo.edu On Sat, 4 Mar 1995, categories wrote: > Date: Fri, 3 Mar 1995 18:50:19 -0500 > From: Todd Wilson > > 1. A question about notation: In accounts of internal category > theory, one uses d_0, d_1 : C_1 -> C_0 as, respectively, domain and > codomain, in analogy (I suppose) with simplicial objects. But then I > would have expected it to be the other way around: in line with the > analogy of an arrow f:A->B as an oriented one-simplex between > vertices A and B , domain should rather be the face operator d_1 > ("delete 1") and codomain d_0 ("delete 0"). Am I backwards? > > 2. The category Grp of groups has the property that every > simplicial object in Grp satisfies the extension condition (i.e., is > a so-called Kan complex). Is there a characterization of the > categories with this property? Are there interesting uses of homotopy > in these (other) categories? > > --Todd Wilson > > Re 1:No,You are absolutely correct if one wishes to use the usual simplicial conventions of "face opposite" in describing the simplices of t the simplicial object associated with the category : its "Nerve".which provides a most convenient numbering of the projections and compositions which occur there.However, it seems almost impossible to get people to give up what seems to them an illogical convention for arrows which must go from 0 to 1! Re 2: Barr characterised these categories as those which satisfy Malcev's condition.I have never seen his proof or know whether he ever published it.I have one of my own since it really is not too difficult. The observation was the ingenious part! Regards, Jack Duskin Date: Tue, 14 Mar 1995 22:26:12 -0400 (AST) Subject: Re: A couple of simplicial questions Date: Mon, 13 Mar 1995 09:27:43 -0500 (EST) From: James Stasheff category theorists are well known to be of the opposite handedness/orientation Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 On Sat, 4 Mar 1995, categories wrote: > Date: Fri, 3 Mar 1995 18:50:19 -0500 > From: Todd Wilson > > 1. A question about notation: In accounts of internal category > theory, one uses d_0, d_1 : C_1 -> C_0 as, respectively, domain and > codomain, in analogy (I suppose) with simplicial objects. But then I > would have expected it to be the other way around: in line with the > analogy of an arrow f:A->B as an oriented one-simplex between > vertices A and B , domain should rather be the face operator d_1 > ("delete 1") and codomain d_0 ("delete 0"). Am I backwards? > > 2. The category Grp of groups has the property that every > simplicial object in Grp satisfies the extension condition (i.e., is > a so-called Kan complex). Is there a characterization of the > categories with this property? Are there interesting uses of homotopy > in these (other) categories? > > --Todd Wilson > > Date: Tue, 14 Mar 1995 22:39:38 -0400 (AST) Subject: Kan and Maltsev Date: Tue, 14 Mar 95 14:31:15 +1000 From: Max Kelly Subject: Kan and Maltsev Recently, Todd Wilson asked for a characterization of those categories wherein every simplicial object is Kan; and Michael Barr replied that this is so for a variety if and only if it admits a Maltsev operation, whereupon it suffices to imitate John Moore's original proof for the category of groups. Jack Duskin also recalls this unpublished observation of Michael's, remarking that he has his own proof. I noted that a complete proof has been published in [A.Carboni, G.M.Kelly, and M.C.Pedicchio, Some remarks on Maltsev and Goursat categories, Applied Categorical Structures 1(1993), 385-421]. I mention this again, since I failed to point out that our paper shows the equivalence Kan = Maltsev for any REGULAR category, where "Maltsev" now means that equivalence relations commute. The proof is pretty; it does follow the lines of John Moore's, but replaces his calculations with elements by reasonings about equivalence relations. Max Kelly. Date: Wed, 15 Mar 1995 05:00:53 -0400 (AST) Subject: reversing the Grothendeick ring construction? Date: Tue, 14 Mar 95 12:21:01 CST From: David Yetter Does anyone know of any work on "reversing the Grothendieck ring construction" in the sense of constructing (and classifying) k-linear monoidal categories with a given Grothendieck ring? The only results I know in this direction are results of Kazhdan and Kerler which show that any C-linear tortile category with the same Grothendieck ring as Rep(U(sl_2)) must be C-linear monoidally equivalent to Rep(U_q(sl_2)) for some q, and a result (probably old, though I know no source other than my own derivation) that shows that semisimple C-linear monoidal categories with Grothendieck ring Z[G] for a finite group G are in 1-1 correspondence with pairs (\alpha, \phi) where \alpha is a U(1)-valued 3-cocycle, and \phi is a complex number (the multiple of Id_1 which give the component at 1 of the right- and left- identity transformations). (This latter shows that "twisted" Dijkgraaf-Witten theory is a TQFT of Turaev-Viro type.) Surely there is a citation in the literature for "my" result cited above, but where? Are there any other results along these lines? --David Yetter Date: Fri, 17 Mar 1995 03:34:15 -0400 (AST) Subject: Reference about bimodules seen as generalized machines Date: Wed, 15 Mar 1995 17:21:11 +0100 (MET) From: Sebastiano Vigna The title says it all: I'm searching for references of papers which study bimodules as generalized machines with product, composition and feedback. I know of a series of papers by Bainbridge in the seventies, and I've been (vaguely) told about the existence of something written by Bill Lawvere about this. Any help would be appreciated. Thank you! seba Date: Fri, 17 Mar 1995 03:35:45 -0400 (AST) Subject: Research Studentship Date: Wed, 15 Mar 1995 17:27:25 +0000 From: Prof R. Brown EPSRC Earmarked Research Studentship Mathematics Programme Identities among relations for monoids and categories Start: October, 1995 I would be grateful if this studentship could be drawn to the attention of any appropriate person. The full funding would be given to students with UK residential qualifications; EC citizens do qualify, and get fees, but no grant, so it seems unlikely to be appropriate more generally. The work will apply 2-categories and 2-groupoids in areas as in the title, and in areas related to aspects of computer science. It is expected to involve symbolic computation with AXIOM, GAP, MAGMA. The School of Mathematics has a networked Research Laboratory with a DEC Alpha 2100 server, model 4/200, supported by EPSRC and University grants. Fuller details, including the case for the proposal, are on wweb http: //www.bangor.ac.uk/ma/epsrc.html Ronnie Brown Prof R. Brown Tel: (direct) +44 248 382474 School of Mathematics (office) +44 248 382475 Dean St Fax: +44 248 355881 University of Wales email: mas010@bangor.ac.uk Bangor wwweb for maths: http: //www.bangor.ac.uk/ma Gwynedd LL57 1UT UK Date: Fri, 17 Mar 1995 03:37:42 -0400 (AST) Subject: address change Date: Thu, 16 Mar 1995 10:46:06 +0200 From: Dr. Reinhard Boerger (Prof. Dr. Pumpluen) >From now on I am accessile by e-mail again under " reinhard.boerger@fernuni-hagen.de". I was told that this address will remain valid even if our system will be changed. Please use this address in the future. Thanks Reinhard Boerger Date: Fri, 17 Mar 1995 03:39:14 -0400 (AST) Subject: caen 94 proceedings Date: Thu, 16 Mar 1995 15:45:01 --100 From: Pierre Ageron All copies of the proceedings of CAEN 94 that have already been ordered, as well as the authors' copies, will be sent before April 7 : I apologize for the delay. You can still order your own copy (it's free !) Let me recall that the only way to do that is by postal mail. Pierre Ageron Departement de Mathematiques Universite de Caen 14032 CAEN Cedex Date: Tue, 21 Mar 1995 04:06:15 -0400 (AST) Subject: Linear Lauchli Semantics: paper available Date: Mon, 20 Mar 95 23:42:24 EST From: SCPSG@acadvm1.uottawa.ca The paper below is available by anonymous ftp from the following sites: triples.math.mcgill.ca, in the directory: pub/blute, theory.doc.ic.ac.uk, in the directory: papers/Scott. ftp.csi.uottawa.ca , in the directory: pub/papers/PhilScott The file is called: lauchli.ps.Z. Any comments would be greatly appreciated. Cheers, Philip Scott P. S. Of course, you may also contact either of the authors for a hard copy: R. F. Blute & P. J. Scott Dept. of Mathematics University of Ottawa 585 King Edward Ottawa, Ont. K1N 6N5 Canada ---------------------------------------------- LINEAR LAUCHLI SEMANTICS R. F. Blute P. J. Scott We introduce a linear analogue of Lauchli's semantics for intuitionistic logic. In fact, our result is a strengthening of Lauchli's work to the level of proofs, rather than provability. This is obtained by considering continuous actions of the additive group of integers on a category of topological vector spaces. The semantics, based on functorial polymorphism, consists of dinatural transformations which are equivariant with respect to all such actions. Such dinatural transformations are called uniform. To any sequent in Multiplicative Linear Logic (MLL), we associate a vector space of ``diadditive'' uniform transformations. We then show that this space is generated by denotations of cut-free proofs of the sequent in the theory MLL+MIX. Thus we obtain a full completeness theorem in the sense of Abramsky and Jagadeesan, although our result differs from theirs in the use of dinatural transformations. As corollaries, we show that these dinatural transformations compose, and obtain a conservativity result: diadditive dinatural transformations which are uniform with respect to actions of the additive group of integers are also uniform with respect to the actions of arbitrary cocommutative Hopf algebras. Finally, we discuss several possible extensions of this work to noncommutative logic. It is well known that the intuitionistic version of Lauchli's semantics is a special case of the theory of logical relations, due to Plotkin and Statman. Thus, our work can also be viewed as a first step towards developing a theory of logical relations for linear logic and concurrency. Date: Wed, 22 Mar 1995 05:26:00 -0400 (AST) Subject: Not too categorical, this. Date: Wed, 15 Mar 95 16:11:51 +0100 From: F. Lamarche Fellow category theorists This one is not really about category theory, although I have seen discussions on closely related matters on this very list some time ago. I have just applied for a bunch of jobs in France and one particular tentacle of the French administrative hydra ha s asked me for a translation of my McGill PhD vellum. Thus it goes UNIVERSITAS McGILL ad Montem regium in Canada Omnibus ad quos hae litterae pervenerint salutem nos universitatis gubernatores rector socii testamur nos FRANCOIS LAMARCHE cum curriculum studiorum praescritptum cum industria conferecerit et omnes exercitationes quae ei sint iniunctae rite peregerit, creavisse DOCTOREM PHILOSOPHIAE atque ei omnes honores iura beneficia quae ad illum gradum pertineant concessisse. Quod ad confirmandum has litteras sigillo universitatis consignandas et nomina eorum qui res academicas administrant subscribenda curavimus. Datae die VI Mensis Junii Anni MCMLXXXIX It is rather obvious what the general meaning is, but I need a close, professional looking translation. I know there are some good Latinists out there! It is only now that I regret not having paid any attention during the four years of Latin I had to su ffer through, more than 25 years ago "What? Me, learn a dead language?". Francois Lamarche As you can see I am now in Genoa (for two months). My old Imperial College email address still applies, too. Date: Wed, 22 Mar 1995 05:27:34 -0400 (AST) Subject: Whitehead Date: Tue, 21 Mar 1995 14:25:18 +0500 From: James Stasheff What is the maximum generality known for a homology equivalence of CW complexes to imply an honest homotopy equivalence? ?e.g. if the fundamental groups act trivially on the higher homtopy groups?? Date: Sat, 25 Mar 1995 21:00:26 -0400 (AST) Subject: Readership at Queen Mary & Westfield College, University of Lo(fwd) Date: Thu, 23 Mar 95 11:59:23 GMT From: David.Pym@dcs.qmw.ac.uk Department of Computer Science Queen Mary & Westfield College University of London READERSHIP in the Department of Computer Science The Department of Computer Science has strong well-resourced research groups in several mainstream areas of computer science including artificial intelligence; computer graphics; virtual reality; distributed and parallel systems; human computer interaction; programming and the *theory of computation*. The Department was rated 4A in the last research assessment exercise; and we are strongly committed to greater research excellence. The Reader will be expected to have a distinguished research reputation with demonstrable experience of research direction and leadership, and to be an enthusiastic teacher. Applicants in any area of computer science are encouraged to apply, although we particularly welcome those with research relating to the work of the Department that combines theory and application. The Department has a number of modular degree programmes in the University of London degree system. There is an MSc in Advanced Methods in Computer Science, with four streams based on the research expertise within the Department. The Department also runs a conversion MSc in Information Technology jointly with the Department of Electronic Engineering. Informal enquiries can be made to Mel Slater (Head of Department) on 0171-975 5242 (email mel@dcs.qmw.ac.uk). Further information can be obtained on URL: http://www.dcs.qmw.ac.uk/. For an application form and further details of this post, please telephone 0171-975 5171, quoting reference number 95044. Applications should be returned by 28th April 1995 to the Recruitment Coordinator, Queen Mary and Westfield College, London E1 4NS. Date: Sat, 25 Mar 1995 21:04:36 -0400 (AST) Subject: Fukaya Date: Fri, 24 Mar 1995 08:04:54 +0500 From: James Stasheff Have any category theorists looked at Fukaya's ``Floer homology for 3-manifolds with boundary''? Might could have overlooked the fact that he deals with A_{\infty}-cats with one strong motivating example. Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 Date: Mon, 27 Mar 1995 20:29:36 -0400 (AST) Subject: the thesis, Complexity Doctrines, by web or ftp Date: Mon, 27 Mar 1995 17:14:24 -0500 From: Jim Otto Dear People, The thesis, Complexity Doctrines (14+121 pages), contains the chapters
  • Tensor and Linear Time
  • V-Comprehensions and P Space
  • Dependent Products and Church Numerals
  • 3-Comprehensions and Kalmar Elementary
and was submitted 3-28-95.
It is available by web from either of ftp://triples.math.mcgill.ca/ctrc.html ftp://triples.math.mcgill.ca/pub/otto/otto.html or by ftp from triples.math.mcgill.ca /pub/otto/thesis.ps.gz (E.g. use gunzip and ghostview.) Bon Soir, Jim Otto Date: Fri, 31 Mar 1995 12:03:38 -0400 (AST) Subject: query Date: Wed, 29 Mar 1995 08:25:07 -0500 (EST) From: James Stasheff Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 ---------- Forwarded message ---------- Date: Wed, 29 Mar 1995 12:08:48 +0200 From: Johannes Huebschmann To: G.Segal@pmms.cam.ac.uk, dupont@mpim-bonn.mpg.de, jds@math.unc.edu Cc: huebschm@mpim-bonn.mpg.de Subject: query Dear colleagues Consider a simplicial group K. Its realization |K| is a topological group (at least when K is countable). The realization of the W-construction yields a universal |K|-bundle over |overline W K|. THere should be a canonical map from the latter to the bar construction B|K| on |K|. Has this been worked out somewhere in the literature? Many thanks in advance. Sincerely yours Johannes Johannes Huebschmann Max Planck Institut fuer Mathematik Gottfried Claren Str. 26 D-53 225 BONN Date: Fri, 31 Mar 1995 12:06:15 -0400 (AST) Subject: change of address Date: Wed, 29 Mar 95 19:26:18 MET DST From: Koslowski Dear Categorists, After nearly 2.5 years without a real job I'm back! Here is my new address: J"urgen Koslowski Institut f"ur theoretische Informatik TU Braunschweig Postfach 3329 D-38023 Braunschweig Germany email: koslowj@iti.cs.tu-bs.de Best regards, -- J"urgen Date: Fri, 31 Mar 1995 12:09:21 -0400 (AST) Subject: change of address Date: Wed, 29 Mar 95 19:26:18 MET DST From: Koslowski Dear Categorists, After nearly 2.5 years without a real job I'm back! Here is my new address: J"urgen Koslowski Institut f"ur theoretische Informatik TU Braunschweig Postfach 3329 D-38023 Braunschweig Germany email: koslowj@iti.cs.tu-bs.de Best regards, -- J"urgen Date: Fri, 31 Mar 1995 12:10:25 -0400 (AST) Subject: trivial ext Date: Thu, 30 Mar 1995 13:36:54 +0500 From: James Stasheff Griffiths et al have a LNM volume on Trivial extensions *of* abelian cats any work be done to extend it to non-trivial extensions??