Subject: change of address Date: Mon, 4 Oct 1993 02:44:00 -0400 From: Please note my change of address: Dr. Michel Hebert Department of Science Mathematics Unit American University in Cairo 113, Sharia Kasr El Aini P.O.Box 113 Cairo 11511 EGYPT E-Mail: mhebert@egaucacs.BITNET ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: cubical monads Date: Fri, 1 Oct 93 18:21:35 +0100 From: grandis@cartesio.dima.unige.it. (Marco Grandis) The following preprint is available Cubical monads and their symmetries Marco Grandis Abstract. This work is concerned with a setting for homotopical algebra based on a cylinder endofunctor I, and more precisely on the notion of diad (I, d-, d+, e, g-, g+), or cubical monad, or I-category with connections, possibly enriched with symmetries, reversion r: I -> I and interchange s: I^2 -> I^2. Generalised symmetries, relative to involutive endofunctors R, S and applying for instance to cubical objects (with connections) or differential graded algebras, are also considered. A few basic developments of this setting are given here. In particular: The associated monads (C-, d, g), (C+, d, g) obtained by collapsing one base of the cylinder and called lower- and upper-cone; their isomophism, in the presence of a reversion; their homotopical invariance, in the presence of an interchange; and the extension of these results to generalised symmetries. A second paper will show how the Puppe sequence of a map works in this setting, as a consequence of connections and interchange. Marco Grandis Dipartimento di Matematica Universita' di Genova Via L.B. Alberti 4, Italia grandis@cartesio.dima.unige.it ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI (2 posts) Date: Sat, 2 Oct 93 21:00:38 EDT From: barr@triples.Math.McGill.CA (Michael Barr) I have just had a chance to read what Saunders wrote. I think there are a number of good observations there. I just wanted to add something on the question of empty models. I have tried to get UAists to explain the reason to me and there was only reason that had any force and that was that an ultraproduct ``should'' be empty iff the set of coordinates in which it is empty is large. But the usual definition as a quotient of a product does not have that property since it is empty as soon as one coordinate is. But there is another definition that is equivalent, UNLESS some coordinate is empty and that is as a (filtered) colimit of products, taken over large sets of indices. This works in all cases and has the desired properties when some factors are empty. --Michael ++++++++++++++++++++++++++++++++++++++++ Date: Mon, 4 Oct 1993 11:33:56 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) Saunders MacLane wrote that - >The "empty algebra" has been proscribed in UA. Since there is now occasion to >consider models of algebras not just in sets (possibly in topoi or in regular >categories) this early proscription seems too narrow. Hear, hear! Even in classical sets, the exclusion of empty carriers gets remarkably messy when you deal with many-sorted algebras. There are well-known paradoxes in the logic (e.g. apparent failures of the transitive law) when empty carriers are allowed, but the constructivists have shown us how to reason cleanly by labelling each sequent or equation with a list of the free variables that are being used (if there are any, then the algebra must be inhabited by the interpretations of those free variables). In our first year logic course we teach natural deduction in a form in which for forall-elimination and exists-introduction, the terms used cannot contain freely invented variables - they must be made up from those posed in the problem or introduced for forall-intro or exists-elim. (We use nested boxes to mark the scopes of these introduced variables.) This is sound even for empty carriers, and it's my belief that the restricted rules are easier to understand than the unrestricted ones. Steve Vickers --------------------------------------------------------------------------- President, Imperial Society for the Prevention of Cruelty to Empty Sets. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI Date: Tue, 5 Oct 1993 10:19:57 +0100 From: poigne@gmd.de (Axel Poigne) Steve Vickers wrote > >Hear, hear! Even in classical sets, the exclusion of empty carriers gets >remarkably messy when you deal with many-sorted algebras. There are >well-known paradoxes in the logic (e.g. apparent failures of the transitive >law) when empty carriers are allowed, but the constructivists have shown us >how to reason cleanly by labelling each sequent or equation with a list of >the free variables that are being used (if there are any, then the algebra >must be inhabited by the interpretations of those free variables). > I fully subscribe. UAs are just sloppy about language. Writing a term means to introduce a name. An "entity" may have several names, and a name may fail to denote an "entity". A theory of identity cares for the first phenemenon, a theory of existence for the second. (A look at : D.S.Scott, Identity and Existence in Intutionistic Logic, In: Applications of Sheaves, Proc. Durham, LNiMath 753, 1979, is recommended) Now UA (if not concerned with partial algebra) usually relate introduction of names to an existence statement; whenever they write 'a' there is something which is denoted by 'a', in case of emergency "a" itself (meaning free generation). They stick to this firm belief when considering variables. This is why proofs of the kind b = f(x) = b' (where f : A -> B) go wrong if the interpretation of A does not provide denotations for x.. (Due to the peculiarities of universal quantification not even b = b' is sound). Of course, we may abandon the idea that names denote. Then, please, use a different logic (see Scott). Otherwise, keep a record of the variables, you want to use as suggested by Steve and many other people. Axel Poigne >Imperial Society for the Prevention of Cruelty to Empty Sets. I subscribe to the society. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI - Empty Carriers From: John C. Mitchell Date: Tue, 05 Oct 93 21:13:00 -0700 I'm glad to see this discussion. In the book that I am (still) writing, I went to some length to do all the universal algebra with explicit lists of variables. At times, I have wondered whether this was syntactic overkill. However, it makes for clean completeness theorems and existence of initial algebras. In addition, it meshes well with later chapters on typed lambda calculi, where contexts (lists of variables and their types) are needed. One minor question, while I am thinking of it. I used "sort" in the chapter on algebra, and "type" in the chapters on lambda calculi, etc. What do the UA-literate think about using "type" uniformly throughout? Would it be offensive to simplify terminology this way? BTW, there is not much math in the chapter on UA, just soundness, completeness, homomorphisms, initiality, and a number of CS examples (stacks, queues, trees, ...), ending with a brief overview of rewrite systems. John Mitchell ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI - Empty Carriers Date: Thu, 7 Oct 1993 08:09:50 +0100 From: poigne@gmd.de (Axel Poigne) John Mitchell wrote > >I'm glad to see this discussion. In the book that I am (still) >writing, I went to some length to do all the universal algebra >with explicit lists of variables. At times, I have wondered whether >this was syntactic overkill. However, it makes for clean completeness >theorems and existence of initial algebras. In addition, it meshes >well with later chapters on typed lambda calculi, where contexts >(lists of variables and their types) are needed. You might have a look at my chapter on "Basic Category Theory" in the "Handbook of Logic in Computer Science (Abramsky, Gabbay, maibaum eds.), OUP, 92. There I follow your program to use an explicit list of free variables which gives a reasonable account how composition in categories relates to substitution on free variables (as opposed to application where bound variables are substituted) (mainly in sections 1,2, 8). >One minor question, while I am thinking of it. I used "sort" >in the chapter on algebra, and "type" in the chapters on >lambda calculi, etc. What do the UA-literate think about using >"type" uniformly throughout? Would it be offensive to simplify >terminology this way? in principle not, but it seems that UAs use sorts for the syntactic entities and types for the semantic ones. In a way, type theory talks about both levels. Axel Poigne ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI - Empty Carriers Date: Wed, 6 Oct 93 20:56:01 -0700 From: rowan@garnet.berkeley.edu I, too, have been frustrated for many years at the proscription of empty algebras. However, I feel that there may be some movement in people's positions from time to time, which may continue if we all try to be diplomatic. I would add that I don't think universal algebraists are the only ones that have this particular opinion. The logicians tend to require structures to be nonempty. I believe this goes back to a decision by Tarski, although I certainly wasn't there. Given the strong allegiance some universal algebraists have to logic, I think that it would be important to try to have a dialog with the logicians about this. Thus, I was very interested in Dr. Barr's remarks concerning a better definition of ultraproducts. About _types_ and _sorts_: I think they are different. To me, a _type_ is an N-tuple of sets S_n, where elements of S_n are n-ary basic operations. There is a free functor from types to clones (or theories), and a forgetful functor the other way. However, in this situation, there would only be one _sort_. For what it's worth, I have a half-written paper that allows any category of sorts, with an additional structure of operations from limits of diagrams in the sort category (which limit might not exist!) back to objects in the sort category. I sent in an abstract about this to the UACT conference. Unfortunately, I seem to have put that project on the back burner. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: universal algebra aand partial algebra Date: Fri, 8 Oct 93 9:07 GMT From: MAS010@BANGOR.AC.UK Re the discussion on empty algebras: Is it naive to ask that universal algebra theory should also include partial algebras, and in particular groupoids, categories, multiple groupoids, multiple categories (of various kinds)? In algebraic work with these objects, the empty object is considered as a matter of course, since one wants the theory of groupoids, say, to be an extension in some sense of the theory of sets. I expect that such a notion of extension is standard in universal algebra. Readers will be interested in the following quotation from p.4 of Ch I of A Connes "Non commutative geometry" (an English translation of "G\'eometrie non commutative", with many additions, about three times as long, to be published by Academic Press). "It is fashionable among mathematicians to despise groupoids [not among readers of this bulletin!] and consider that only groups have an authentic mathematical status, probably because of the pejorative suffix oid. To remove this prjudice we start Chapter I by Heisenberg's discovery of quantum mechanics. The reader will hopefully realise there how the experimental results of spectroscopy forced Heisenberg to replace the classical frequency group of the system by the groupoid of quantum transitions. Imitating for this groupoid the construction of the group convolution algebra Heisenberg rediscovered matrix multiplication and invented Quantum mechanics." Ronnie Brown ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: On UA-CAT Workshop at MSRI - Empty Carriers Date: Mon, 11 Oct 1993 10:26:48 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) Bill Rowan says that - >...To me, a _type_ is >an N-tuple of sets S_n, where elements of S_n are n-ary basic operations. "Signature" is used for this notion (or many-sorted generalizations) often enough that I feel it could be standard. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: universal algebra and partial algebra Date: Mon, 11 Oct 1993 10:36:56 +0000 From: sjv@doc.ic.ac.uk (Steven Vickers) Ronnie Brown asks - >Is it naive to ask that universal algebra theory should also include >partial algebras, and in particular groupoids, categories, multiple >groupoids, multiple categories (of various kinds)? It seems that where the methods of universal algebra generalize well to cover partial algebras is in _essentially_ algebraic theories, i.e. those in which there is a hierarchy of operators such that the domain of definition of each is defined by a conjunction of equations involving more primitive operators. This certainly covers categories and groupoids - anyone who knows Phillip Higgins' book can see how smoothly universal algebra works for them. As for multiple groupoids and multiple categories, I can't say because I'm not familiar with them. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: universal algebra and partial algebra Date: Mon, 11 Oct 93 23:35:00 +0100 From: Thorsten Altenkirch >Is it naive to ask that universal algebra theory should also include >partial algebras, and in particular groupoids, categories, multiple >groupoids, multiple categories (of various kinds)? It seems that where the methods of universal algebra generalize well to cover partial algebras is in _essentially_ algebraic theories, i.e. those in which there is a hierarchy of operators such that the domain of definition of each is defined by a conjunction of equations involving more primitive operators. This certainly covers categories and groupoids - anyone who knows Phillip Higgins' book can see how smoothly universal algebra works for them. As for multiple groupoids and multiple categories, I can't say because I'm not familiar with them. As far as the partial case is concerned: why not generalized algebraic theories ala Cartmell? They seem to have essentially the same basic semantic interpretation (categories with finite limits) and are applicable to similar examples. I prefer generalized algebraic theories because they are essentially a first order subset of Type Theory (ala Martin L"of) which is used by many people in Computer Science to formalize mathematics and to verify algorithms. I personally also find generalized algebra more natural, e.g. when formalizing catgeories (why not call them monoidoids ?) one introduces homsets and composition as indexed by the objects instead of having cod and dom and composition as a partial operation. Thorsten Altenkirch Host: muppet77.cs.chalmers.se Office: 2438 snail: Dep. of Computer Sciences; Chalmers U. of Technology; 412 96 Gothenburg phone: (+ 46) 31 772 1079 [office], (+ 46) 31 55 54 40 [home] Sweden ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: universal algebra and partial algebra Date: Wed, 13 Oct 93 11:55:34 EDT From: barr@triples.Math.McGill.CA (Michael Barr) I just wanted to add that it is unneccessary, in defining essentially algebraic theories to worry about a hierarchy of types in which the domain of each is defined equationally in terms of simpler types. I realized this because I once tried to show that the theories classified by essentially algebraic theories are essentially weaker than those classified by finite limit sketches. They are not because if you had two types, the domain of each of which was described equationally in terms of the other (or a type whose domain was described equationally in terms of itself), you could add new dummy types at the lowest level of the hierarchy, use them to define the domains of the type(s) in question and then add new unary operations of each of the dmmy types taking values as the other and finally equations that forced those operations to be isomorphisms. Nonetheless, I think that FL sketches are the way to go and the definition of essentially algebraic a kludge. Michael ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: chu is cofree From: pavlovic@triples.Math.McGill.CA (Dusko Pavlovic) Date: Wed, 13 Oct 93 13:15:54 EDT A paper on THE CHU CONSTRUCTION AND COFREE MODELS OF CLASSICAL LINEAR LOGIC by Dusko Pavlovic is available by anonymous FTP from triples.math.mcgill.ca. The files chu-{A4,US}.{dvi,ps}.Z are the directory pub/pavlovic. Each of them contains the whole paper, but in various formats. This version supersedes the previous ones, that I had sent to some people. Abstract We describe a couniversal property of the Chu construction. It induces a comonad on a category of autonomous categories. *-Autonomous categories are exactly the coalgebras for this comonad. The models of full classical linear logic (with the exponentials) are then obtained as the coalgebras on the models of intuitionistic linear logic --- this time for a comonad derived from the Chu construction. In view of the computational interpretations of linear logic, these results suggest an interesting connection of the functional and the concurrent programming. For this reason, some effort has been spent to make all the constructions effective. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: my new address Date: Thu, 14 Oct 93 07:28:08 ADT From: "Marek Zawadowski" Marek Zawadowski Instytut Matematyki Uniwersytet Warszawaski 00-913 Warszawa ul. S.Banacha 2 Poland e-mail: zawado@mimuw.edu.pl Please send all correspondence there, Marek Zawadowski ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Universal Algebra v. Category Theory Date: Tue, 19 Oct 93 09:53:18 EDT From: es@math.mcgill.ca (Elaine Swan) Here are some remarks in connection with the Universal Algebra versus Category Theory debate initiated by Saunders MacLane. (i) There is yet another way to describe a (multisorted) algebraic theory, alias clone, namely as a Cartesian multicategory, that is, a Gentzen style deductive system with appropriate equations between deductions. Deductions f:A_1...A_n ---> B should be viewed as operations, in the spirit of the formulas as types paradigm. (ii) Why anyone would deny that the subalgebras of an algebra form a complete lattice is beyond me. The subobject lattice and congruence lattice are both subsumed in the lattice of subcongruences, which I found myself in my paper "Goursat's theorem and the Zassenhaus lemma", Can. J. Math. 10(1957), 45-56. Subcongruences have recently been resurrected as partial equivalence relations; they can of course be defined in any category. (iii) If one studies algebras on graphs instead of sets, as pioneered by Burroni, one may view many structured categories, even toposes, as algebras. (iv) To avoid problems with empty types, e.g. lack of transitivity, one should declare variables both in equations and deductions. Phil Scott and I did so in our book of 1986, by placing a subscript X={x_1,...x_n} on the equality and entailment symbols. Unfortunately, some other books did not follow this practice. Jim Lambek ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Universal Algebra v. Category Theory Date: Sat, 23 Oct 93 19:14:48 BST From: Joseph.Goguen@prg.oxford.ac.uk Some people may be interested in reading what Jose Meseguer and I did with variables in equations and deduction, starting around 1981. The final 1985 version discusses some connections with categorical approaches; for example, we point out a flaw in Benabou's work. All this was motivated by the need to reason about abstract data types in Computer Science. @article(complas, title = "Completeness of Many-sorted Equational Logic", author = "Joseph Goguen and Jos\'e Meseguer", year = 1985, journal = "Houston Journal of Mathematics", volume = 11, number = 3, pages = "307--334", note = "Preliminary versions have appeared in: {\it SIGPLAN Notices}, July 1981, Volume 16, Number 7, pages 24--37; SRI Computer Science Lab, Report CSL-135, May 1982; and Report CSLI-84-15, Center for the Study of Language and Information, Stanford University, September 1984") &&&&&&&&&&&&&&&&&&&&&&&&&&&&& Signature File &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Joseph A. Goguen, Professor of Computing Science, Programming Research Group, Oxford University Computing Lab, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom. email: Joseph.Goguen@prg.ox.ac.uk [internet] -- usually also works in the UK, but if not, try Joseph.Goguen@uk.ac.ox.prg phone: 283504 [my office]; 283505 [secy]; 273838 [PRG office]; 273839 [FAX]. From USA, dial 011-44-865-...; from UK, dial (0865)-... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: FYI Date: Tue, 26 Oct 93 09:27:36 EDT From: James Stasheff ------------------------------------------------------------------------------ send mail only to hep-th@xxx.lanl.gov, do not reply to no-reply@xxx.lanl.gov . rigorously identical files also available from hep-th@babbage.sissa.it send complaints regarding untexable papers directly to submitter. use a single `get' to request multiple papers, `list macros' for available macro packages, and `help' for a list of available commands and other info. ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ received from Mon Oct 25 00:06:08 MDT 1993 to Tue Oct 26 00:06:04 MDT 1993 ------------------------------------------------------------------------------ \\ Paper: hep-th/9310164 From: "Dr John W. Barrett" Date: Mon, 25 Oct 93 16:45:14 +0100 Spherical Categories, by John W. Barrett and Bruce W. Westbury, 16 pages, amstex \\ This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple. \\ (hep-th/9310164 , 100kb) %-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%-%- ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Localization Date: Wed, 27 Oct 93 16:40:23 CET From: Marek Golasinski Dear Colleagues, It is well known that one of possible model for the homotopy category there is the category Cat of small categories. Does anybody know a construction of a P-localization (P is a set of primes) using the categorical structure only? Many thanks in advance for your reply. With my best wishes, Marek G. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: KZ-terminology From: pavlovic@triples.Math.McGill.CA (Dusko Pavlovic) Date: Wed, 27 Oct 93 8:55:19 EDT In a paper recently announced on this network ("The Chu construction and the cofree models of classical linear logic", available from triples.math.mcgill.ca), I remarked that the Chu construction was a KZ-codoctrine. The term KZ-doctrine was coined, I believe, by Ross Street in his "Fibrations in bicategories" (Cahiers XXI-2) --- to abbreviate the term Kock-Z\"oberlein-doctrine. Now I have been warned from several sides that this abbreviation awakes in German-speaking people associations to concentration camps. I was going to shrug my shoulders and say that people called Dick usually don't pay any attention to the association their name may awake (especially in children), but then I remembered that a guy I know, whose name happens to be Adolf, had a serious fight with his father about this. History seems to be more serious than anatomy. So what do you say: Should we change the terminology or not? Regards to all, Dusko Pavlovic ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: KZ-terminology Date: Thu, 28 Oct 93 16:11:10 +1000 From: kelly_m@maths.su.oz.au (Max Kelly) Dusko Pavlovic sees problems with the name " K-Z doctrine ", used by Street to abbreviate "Kock-Zoeberlein doctrine". I dislike both names, and in the paper (with R. Blackwell and A.J. Power) "Two-dimensional monad theory", J. Pure Appl. Algebra 59 (1989), 1-41, I wrote "quasi-idempotent monad" and "co-quasi-idempotent monad"; the first (resp the second) includes those whose algebras are categories with certain limits (resp. colimits). Max Kelly. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: KZ-terminology Date: Thu, 28 Oct 93 15:17:57 +1000 From: street@macadam.mpce.mq.edu.au To Kock and Z\"oberlein we owe the concept. I plead guilty to the abbreviation, yet ignorant of the connotation. Who would be shocked by Diwmilatu [resp. Diwmiratu] (doctrines-in-which-multiplication-is-left-[resp. right] -adjoint-to-unit? These could well be Javan swearwords. On this topic, Anders Kock has two recent nice papers: "Monads for which structures are adjoint to units" and "Generators and relations for delta as a monoidal 2-category". Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Name for this concept? Date: Thu, 28 Oct 1993 21:30:45 -0700 From: "William H. Rowan" What do you call it if you have a category C, and you have a class X of arrows of C such that if x in X, then gxf in X for all composable isomorphisms f and g. A functor C->C' which is one leg of an equivalence takes such a set to another one X' in C', and any functor which is the other leg of the equivalence takes X' back to X. Bill Rowan ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Subject: Re: Name for this concept? Date: Sun, 31 Oct 1993 14:17:25 -0400 (EDT) From: MTHISBEL@ubvms.cc.buffalo.edu I do not remember seeing a name for this. There would be a fair chance of finding one, most likely in some paper discussing various stronger/weaker kinds of factorization systems. Trouble is, nobody would remember the name, so you have to explain every time you use it. John Isbell