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January 1994

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Fuzzy +
by categories 14 Jan '94

14 Jan '94

Date: Fri, 14 Jan 1994 09:05:25 +0100 From: Axel Poigne <poigne(a)gmd.de> I yesterday attended a talk about fuzzy logic (I know this to be ``degoutant'', but...) where a ``Lukaciewicz norm'' was discussed as a t-norm. If I recollect correctly, a t-norm is a has a binary operator _\wedge_ which is associative, commutative, and monotonic, the latter being a mystery, the order being due to the real interval [0,1]. Moreover it seems that a negation \neg is assumed to exist, since an operator \vee is defined by de Morgan law. Quite clearly, min determines a norm as well as the multiplication. It seems to be an assumption that negation is always \neg a = 1-a. Now the Lukaciewicz norm is of the form a \wedge b = min{a + b, 1}. As consequence, a \vee b = max{a + b - 1, 0}. This norm satisfies a \wegde \neg a = 0 and a \vee \neg a = 1, but \vee and \wedge are not distributive, which is true for the other norms. (I hope this to be a correct recollection of what I heard) Trying to make head and tail of this, I wonder whether one really should say that one has a lower semi-lattice for the order, or even an Heyting algebra (in fact the \sqcap and \bigsqcup is about in all the arguments), and just add a binary monotonic, etc operator _ \otimes _ (replacing the \wedge in the t-norm). This structure rather looks like a quantale (units are available as well). I have no idea how negation fits the picture, but it reminds me of classical linear logic. Does this ring a bell ? I am just puzzled, having no idea about fuzzy logic, and little knowlege about linear logic. I know that Michael Barr has written a paper on Fuzzy sets as toposes but he uses only geometric logic, meaning a Heyting algebra. Axel A related question : these people seem to use \bigsqcup in general to compute suprema. It appears to be more consistent to use \bigoplus on occasions. How would this be defined in linear logic ? (Sorry, my linear logic is very poor) >From cat-dist(a)mta.ca Ukn Jan 14 13:18:16 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA08758; Fri, 14 Jan 1994 13:18:15 +0400 Date: Fri, 14 Jan 1994 12:56:25 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: I like my coffee crisp To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401141225.G25806-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Fri, 14 Jan 94 9:21:59 EST From: Al Vilcius <vilcius(a)mathstat.yorku.ca> This is an appeal for help to my friends and acquaintances on CATEGORIES: I have been enlisted to give a talk on "Fuzzy vs. Probability Theory" to an audience in finance whose backgrounds include applied mathematics, physics, engineering, and finance. The audience is not familiar with categories, no less toposes, which means that subtleties such as adding fuzzy equality to yield variable sets (sheaves) would be lost. Nevertheless, they are intrigued by the "fuzzy stuff" that is currently popular. My predicament is then to choose between: (1) torturing my conscience by giving an insubstantial and superficial talk on memberships vs distribution functions; (2) torturing the underlying mathematics into layman's prose. I am hoping that some of the learned readers of CATEGORIES may have already performed torture # 2 in a humane fashion (either in public or in private) and have some material and/or suggestions on how best to commit this heinous act. My preferred approach would be a la M. Barr via variable sets and sheaves, combined with the description of fuzzy and probabilistic algebraic theories given by E. Manes. I am already aware of many other fine (and some not so fine) works on fuzzy sets and fuzzy logic, however, I don't know how to make these understandable to non-categorists. I may well have to resort to torture # 1, but would like to avoid doing so if possible. All comments and suggestions, either privately to me at vilcius(a)clid.yorku.ca or publicly on CATEGORIES, on how I could have my "coffee crisp" would be most welcome. Thank you ............................... Al Vilcius, Toronto /\ / / \ / /--->\ / / \/ >From cat-dist(a)mta.ca Ukn Jan 14 22:00:51 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA02115; Fri, 14 Jan 1994 22:00:50 +0400 Date: Fri, 14 Jan 1994 21:53:59 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: simple characterization of weak cartesian closedness To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401142159.F22242-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 14 Jan 94 18:55:12 +0100 From: Thomas Streicher <streiche(a)informatik.uni-muenchen.de> I wonder whether the following trivial observation is generally known : a category C with finite products is WEAKLY CARTESIAN CLOSED iff for all objects A , B in C the functor C( _ x A , B) is a retract of a representable functor. (The embedding part of the retraction gives a choice of functional abstraction which is stable under substitution) and the projection part gives evaluation). Especially this entails that if C has splitting of idempotents then the notions of cartesian closedness and weakly cartesian closedness are equivalent. I don't think that the remark above is a deep insight !! BUT usually people refer to the quite heavy machinery of Hayashi's semifunctors when they speak about the categorical semantics of typed lambda calculus without eta-rule. Thomas Streicher >From cat-dist(a)mta.ca Ukn Jan 14 22:13:22 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA06984; Fri, 14 Jan 1994 22:13:21 +0400 Date: Fri, 14 Jan 1994 22:04:18 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: cantor-bernstein To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401142218.H22242-c100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 14 Jan 94 15:44:32 EST From: Peter Freyd <pjf(a)saul.cis.upenn.edu> Pierre Ageron asked (way back on 1 Dec): The statement and the AC-free proof of Cantor-Schroeder-Bernstein's theorem have obviously some categorical content. Has this been already investigated ? The property, when it holds, is an important property on the category. It is, however, a rare property. There are, as usual, different ways to interpret the property in general categories. I would opt for the following: CANTOR-SCHROEDER-BERNSTEIN PROPERTY: If two objects be retracts of each other they are necessarily isomorphic. I trust that CSB holds for any boolean topos. (Anyone want to confirm?) Kaplansky in his booklet on infinite abelian groups pointed out that CSB holds in the category of countable torsion abelian groups (as a consequence of the Ulm invariants). He raised it as one of three test problems for advances in the theory of abelian groups. Does CSB continue to hold, for example, if countablility is dropped? (Kaplansky did not, of course, talk about retracts. He talked about two groups appearing as direct summands of each other.) Someone found a counterexample in the latter 50's. (Anybody know who?) If _A_ and _B_ are categories, _A_ a retract of _B_, it is routine that a counterexample for CSB in _A_ is transported to a countexample in _B_. _Abelian_Groups_ is a retract of _Topological_Spaces_ (via Moore spaces and homology) hence there are a pair of spaces which appear as retracts of each other but are different enough to have different homology groups. That fact became better known in the late 50's than the fact about abelian groups. (And in the late 50's it was damned difficult to explain why it should be viewed as a trivial corollary.) There's a stronger property: if two objects be retracts of each other the retraction maps are isomorphisms. The two most immediate examples are the categories of finite sets and of finite dimensional vectors spaces. But note that any category that is locally finite (i.e. all hom-sets are finite)--or any linear category that is locally finite dimensional--immediately inherits the property. By moving to a 2-category setting one may state the obvious general theorem of which these are special cases. I am not sure if any of this should be viewed as having "categorical content." best thoughts, peter >From cat-dist(a)mta.ca Ukn Jan 14 22:19:20 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA09262; Fri, 14 Jan 1994 22:19:19 +0400 Date: Fri, 14 Jan 1994 22:12:16 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: syntactic criterion for join? To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401142216.P22242-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 14 Jan 1994 16:47:01 -0500 From: David Espinosa <espinosa(a)cs.columbia.edu> Does anyone know a syntactic criterion for the existence of a natural transformation join : TTA -> TA for a given endofunction T built from +, *, -> ? There is a well-known (correct me if I'm wrong) syntactic criterion for covariance which determines whether T can be extended to an endofunctor. Can this criterion be extended to the existence of join? Also, does anyone know a reference for the covariance criterion? David >From cat-dist(a)mta.ca Ukn Jan 14 22:23:22 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA11191; Fri, 14 Jan 1994 22:23:21 +0400 Date: Fri, 14 Jan 1994 22:15:51 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: Fuzzy + To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401142251.U22242-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 14 Jan 94 13:18:44 EST From: Michael Barr <barr(a)triples.Math.McGill.CA> One thing to say is that I did not write a paper on fuzzy sets as toposes, but someone named Eytan did. I wrote a paper on fuzzy sets as non-toposes and it differs from Eytan's in being correct. On the other hand, fuzzy sets are a quasi topos, which means they do have a first order logic. That said, it has to admitted that the first order logic is probably not what they really had in mind as fuzzy logic and what they did have in mind (using operators like truncated sum and negations like - minus is closer to linear logic than to classical, even intuitionistic classical, logic. I once started to write a paper on this, but have not completed it it; maybe one day I will. And, BTW, Andy Pitts, unbeknownst to me, also once wrote a paper on fuzzy sets as a non-topos. His is also correct. Michael >From cat-dist(a)mta.ca Ukn Jan 14 22:27:53 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA13408; Fri, 14 Jan 1994 22:27:52 +0400 Date: Fri, 14 Jan 1994 22:19:28 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: I like my coffee crisp To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401142227.Z22242-c100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Fri, 14 Jan 94 12:28:42 PST From: "Michael J. Healy (206) 865-3123" <mjhealy(a)espresso.rt.cs.boeing.com> > Date: Fri, 14 Jan 94 9:21:59 EST > From: Al Vilcius <vilcius(a)mathstat.yorku.ca> > > This is an appeal for help to my friends and acquaintances on > CATEGORIES: > > I have been enlisted to give a talk on "Fuzzy vs. Probability > Theory" to an audience in finance whose backgrounds include > applied mathematics, physics, engineering, and finance. The > audience is not familiar with categories, no less toposes, which > means that subtleties such as adding fuzzy equality to yield > variable sets (sheaves) would be lost. Nevertheless, they are > intrigued by the "fuzzy stuff" that is currently popular. > > My predicament is then to choose between: > > (1) torturing my conscience by giving an insubstantial and > superficial talk on memberships vs distribution functions; > > (2) torturing the underlying mathematics into layman's prose. > > I am hoping that some of the learned readers of CATEGORIES may > have already performed torture # 2 in a humane fashion (either in > public or in private) and have some material and/or suggestions > on how best to commit this heinous act. > > My preferred approach would be a la M. Barr via variable sets and > sheaves, combined with the description of fuzzy and probabilistic > algebraic theories given by E. Manes. I am already aware of many > other fine (and some not so fine) works on fuzzy sets and fuzzy > logic, however, I don't know how to make these understandable to > non-categorists. I may well have to resort to torture # 1, but > would like to avoid doing so if possible. > > All comments and suggestions, either privately to me at > vilcius(a)clid.yorku.ca or publicly on CATEGORIES, on how I could > have my "coffee crisp" would be most welcome. > > Thank you ............................... Al Vilcius, Toronto > > /\ / > / \ / > /--->\ / > / \/ > > I have a related predicament. I'm an industrial mathematician with a need to learn what I can as soon as possible about a mathematical background for fuzzy logic. I am also furiously learning what I can about category theory and logic in connection with some work in formal methods for software engineering and machine learning (neural networks). So I really need to find an appropriate, no-nonsense (i.e., mathematical) formalism that meets all these requirements; given that, I can afford to invest considerable effort learning it. My current choice is to study categorical or category-related theories, and am currently reading up on Steven Vickers' work on topological systems as well as Goguen and Burstalls' work on institutions. If anybody has information that might help, or could elaborate on your reply to Al Vilcius so that a categorical novice might understand as well, I would be most grateful. I did study topology and algebra in grad school many years ago. Thank you, Mike Healy mjhealy(a)atc.boeing.com >From cat-dist(a)mta.ca Ukn Jan 16 12:52:30 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA08390; Sun, 16 Jan 1994 12:52:29 +0400 Date: Sun, 16 Jan 1994 12:37:25 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Acyclic models To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401161225.E25298-c100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Sat, 15 Jan 94 11:43:33 EST From: Michael Barr <barr(a)triples.Math.McGill.CA> There have been two apparently quite different categorical versions of acyclic models. The first, as found for example in Barr-Beck, COCA, 1966, says that if K = {Kn}, augmented over K(-1) and similarly L were chain complex functors and G is a cotriple such that (Kn)epsilon: (Kn)G --> Kn has a natural splitting when n >= 0 and if the complex LG --> L(-1)G --> 0 has a natural contracting homotopy, then any natural transformation K(-1) --> L(-1) can be extended to a unique up to homotopy map K --> L. In many cases the required naturality is too hard to verify (or false) and so a second form of the theorem is used. Here we simply suppose that the complex (Kn)G* --> Kn --> 0 is acyclic (an easy consequence of the splitting above), that LG --> L(-1)G --> 0 is acyclic and that K(-1) is isomorphic to L(-1) and conclude that H(K) is isomorphic to H(L). (Kn)G* stands for the standard powers-of-G resolution coming from eps. This version is easy to apply, but suffers from three defects. First, it works only in the case of isomorphism, not arbitrary maps. Second, it does not in itself give naturality, although that could probably be remedied by using a category of relations. Third, and probably most important, it gives no uniqueness. This means, for example, that although you can use it (in conjunction with an argument involving simplicial subdivision) to show that singular and simplicial homology of triangulated spaces are isomorphic, you cannot show this way that the isomorphism is induced by the inclusion of the simplicial chains into the singular ones. I have recently discovered a version of acyclic models that repairs all three defects. Moreover, it gives a single proof of both forms as well as third form involving what I will call weak homotopy equivalence. (This is not a Quillen model category in general, although there would appear to be considerable overlap.) Let C be the category of chain complexes of functors from some category X to an abelian category A. Say that an arrow K --> L in C is a weak homotopy equivalence if for each object x of X, Kx --> Lx is a homotopy equivalence (has a homotopy inverse and homotopies, etc., but not assumed natural). Let Sigma stand for one of the classes: (a) homotopy equivalences (b) weak homotopy equivalences (c) homology isomorphisms and let D denote the category of fractions gotten from C by inverting Sigma. Let (G,eps) be a pair consisting of an endofunctor on X and a natural transformation G --> Id. Say that the augmented object K --> K(-1) --> 0 of C is Sigma-trivial if the 0 endomorphism is in Sigma. Say that the object K of C is G presentable (w.r. to Sigma) if for each n >= 0, the chain complex (Kn)G* --> Kn --> 0 is Sigma-trivial and K is G acyclic (w.r. to Sigma) if KG --> K(-1)G --> 0 is Sigma-trivial. Then Theorem: If K is G presentable and L is G acyclic, both w.r. to Sigma, then any natural transformation K(-1) --> L(-1) can be extended in D to an arrow, unique in D, K --> L. In case (a), this is the theorem of Barr-Beck, 1966 and in case (c), this repairs the three defects cited above, while case (b) appears to be genuinely new. The proof is embarrassingly easy. Consider the diagram (alpha K)G* K(eps*) K(-1)G* <----------- KG* ---------> K | | | | v (alpha L)G* L(eps*) L(-1)G* <----------- LG* ---------> L alpha K and alpha L are the augmentation arrows. The G-presentability implies that K(eps*) is in Sigma and the G-acyclicity that (alpha L)G* is. When these are inverted, we get the desired map K --> L as the composite. A paper on the subject will be posted in the usual ftp location within a week or two. >From cat-dist(a)mta.ca Ukn Jan 16 12:53:10 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA08568; Sun, 16 Jan 1994 12:53:09 +0400 Date: Sun, 16 Jan 1994 12:44:32 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: cantor-bernstein To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401161232.N25298-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sat, 15 Jan 94 11:36:22 EST From: Michael Barr <barr(a)triples.Math.McGill.CA> One additional example and you don't even need retracts. In the category of finitely generated modules over a commutative ring, all epis are isos. As a result, if you have epis in both directions, they are isos. So the dual category category is S-B. This is fairly easy if the ring has ACC, but there is a trick that works for any ring to reduce it to that case. Since f.d. vector spaces are self-dual, this example encompasses that one. Michael >From cat-dist(a)mta.ca Ukn Jan 17 09:36:47 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA22458; Mon, 17 Jan 1994 09:36:46 +0400 Date: Mon, 17 Jan 1994 09:26:04 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: simple characterization of weak cartesian closedness To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401170904.F22295-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 94 9:44:41 MET From: Simone Martini <martini(a)di.unipi.it> Thomas Streicher asks >> whether the following trivial observation is generally known: >> >>a category C with finite products is WEAKLY CARTESIAN CLOSED iff >>for all objects A , B in C the functor C( _ x A , B) is a >>retract of a representable functor. I cannot say about "generally known", but.. this property it is quoted as one of the elementary characterizations of wCCC in a paper of mine (Categorical Models for non-extensional lambda-calculi, Mathematical Structures in Computer Science (1992), vol 2, pag 327--357). The paper, which has a definite didactic pace, discusses also the case where there is only an epy natural transformation from C(_,A=>B) to C( _ x A , B), which gives models of typed, non extensional, Combinatory Logic; and it gives conditions on the existence of models of type-free lambda-calculus as reflexive objects in wCCCs. Simone Martini Universit\`a di Pisa, Dipartimento di Informatica. >From cat-dist(a)mta.ca Ukn Jan 17 09:40:11 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA22486; Mon, 17 Jan 1994 09:40:10 +0400 Date: Mon, 17 Jan 1994 09:29:33 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: simple characterization of weak cartesian closedness To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401170933.K22295-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 1994 11:37:33 +0100 (MET) From: Raymond Hoofman <raymond(a)fwi.uva.nl> Quoting Thomas Streicher, > I wonder whether the following trivial observation is generally known > : > > a category C with finite products is WEAKLY CARTESIAN CLOSED iff > for all objects A , B in C the functor C( _ x A , B) is a > retract of a representable functor. Yes, this is the "degenerate" case of a semi-adjunction between a functor G and a semi-functor F: the semi-isomorphism D(F(-), ...) \cong_{s} C(-, G(...)) becomes a retraction (see [1], also [2]). > I don't think that the remark above is a deep insight !! > BUT usually people refer to the quite heavy machinery of Hayashi's > semifunctors when they speak about the categorical semantics of typed > lambda calculus without eta-rule. However, if the products of your typed lambda calculus also do not satisfy the eta-rule, the semi-isomorphism above does not degenerate, and it is less obvious how to give a simple characterization without semi-functors (apart from saying that the Karoubi envelope of the category is Cartesian closed). [1] The theory of Semifunctors, R. Hoofman, MSCS 3 [2] Collapsing Graph models by preorders, R. Hoofman & H. Schellinx, LNCS 530 With kind regards, Raymond Hoofman. >From cat-dist(a)mta.ca Ukn Jan 17 16:49:30 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA23804; Mon, 17 Jan 1994 16:49:29 +0400 Date: Mon, 17 Jan 1994 16:18:00 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Fuzzy references To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401171600.A15244-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 17 Jan 1994 12:57:49 -0400 From: Mike Wendt <wendt(a)cs.dal.ca> ========================================================================== Hi Al: I'm not sure if this is what you want but here are a couple of references I've noticed recently (my search for categorical-measure-theory-type stuff): Bandemer, H., Nather, W, "FUZZA DATA ANALYSIS," Theory and Decision Library, Kluwer Academic Press, Series B, Vol. 20 (Norwell, Mass., 1992). Rodabaugh, S., Klement, E., Hoehle, U. (eds.), "APPLICATIONS OF CATEGORY THEORY TO FUZZY SUBSETS," Kluwer Academic Press, Series B, Vol. 14 (Norwell, Mass., 1992). I'm sorry, I can't give you a review of these books yet. I have only peeked in the first one. It seems interesting enough and is at an introductory level. Regards, -Mike Wendt ========================================================================== >From cat-dist(a)mta.ca Ukn Jan 17 16:49:47 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA23830; Mon, 17 Jan 1994 16:49:47 +0400 Date: Mon, 17 Jan 1994 16:24:16 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: Fuzzy + To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401171616.F15244-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 94 15:36:35 +0100 From: Pierre Ageron <ageron(a)univ-caen.fr> In my thesis "Structure des logiques et logique des structures", I tried (very shortly) to understand what the algebraic and categorical counterparts of fuzzy logic are. There are different proposals in the literature, the reason is that there are plenty of interesting operations on the interval [0,1] and that it is very difficult to tell which ones are relevant for fuzzy logic. The most general axiomatization was given by Rene Guitart in his 1979 thesis (or a 1982 paper in the Cahiers). He considered complete ordered abelian monoids (in their 1990 book, Barr and Wells restricted to complete Heyting algebras). I observed that every complete ordered abelian monoid has a canonical Lafont algebra structure: this means that (this) fuzzy logic is the extension of intuitionistic linear logic with infinitary versions of the additive connectives "plus" and "with". Guitart defined the notion of "algebraic universe": essentially a category equipped with a monad P looking like the monad of subsets on Ens (I mean Set !). This notion subsumes the notion of elementary topos and allows to give higher order semantics for logics other than intuitionistic logic. In the case of fuzzy logic, the point is that every complete ordered abelian monoid defines such a structure on Ens. The Kleisli category of P is the category of fuzzy relations. All this is explained in my thesis using the notations of linear logic. All that framework gives Tarskian semantics for (propositional or higher order) fuzzy logic. It is not clear whether there are Heytingian semantics for fuzzy logic, i.e. a proof theory. The difficulty is that every small complete category is a poset (but this result by Freyd uses AC, so hope remains...) Pierre AGERON >From cat-dist(a)mta.ca Ukn Jan 17 16:50:40 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA24033; Mon, 17 Jan 1994 16:50:39 +0400 Date: Mon, 17 Jan 1994 16:26:39 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: cantor-bernstein To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401171639.K15244-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 94 10:13:02 EST From: Peter Freyd <pjf(a)saul.cis.upenn.edu> Mike Barr writes: One additional example and you don't even need retracts. In the category of finitely generated modules over a commutative ring, all epis are isos. As a result, if you have epis in both directions, they are isos. So the dual category category is S-B. This is fairly easy if the ring has ACC, but there is a trick that works for any ring to reduce it to that case. Wonderful thought: all epis are isos. Anyway, I see a proof that any epi endo on a finitely presented module over a commutative ring is iso, but finitely generated? There's a metaprinciple that says that a result like this should generalize from commutative to PI rings (that is, rings that satisfy some non-trivial Polynomial Identity). Can anyone confirm? A corollary would be that in any additive category if two objects each appear as retracts of the other, and if the ring of endomorphisms of one of them is a PI ring then the retractions are isos. best thoughts, peter >From cat-dist(a)mta.ca Ukn Jan 17 22:08:34 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA17367; Mon, 17 Jan 1994 22:08:34 +0400 Date: Mon, 17 Jan 1994 21:46:07 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: mathematics made hard To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401172107.B10909-c100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 94 17:49:08 EST From: Peter Freyd <pjf(a)saul.cis.upenn.edu> This is about the opposite of category theory. I'm going to give a soft proof of something and ask how to get the hard proof. All of this because of Mike Barr's note about modules of commutative rings. Let M be a finitely presented module over a commutative ring R and f:M -> M an epimorphic endomorphism. We will show that f is necessarily an isomorphism. First specialize to the case that R is Noetherian. The kernels of the powers of f form an ascending chain of submodules of M, hence must stabalize. That is, there k+1 k k+1 is a natural number k such that Ker(f )= Ker(f ). Since f is epi, it is a cokernel for its kernel and there must exist g:M -> M k+1 k such that f g = f . (I'm composing maps in the diagramatic order.) Using for the second time that f is epi we may cancel to obtain fg = 1. Since fgf = f1 we cancel once more (using that f is epi for the third time) to obtain gf = 1. Now, let r and n be natural numbers and r n R -> R -> M -> O an exact sequence. There must be an rxn matrix K, an rxr matrix A' an nxn matrix A, another nxn matrix B, and an nxr matrix C such that KA = A'K BA + CK = I. (K describes the map r n n from R to R that defines M, A describes the endomorphism on R r that "lifts" f, A' describes the endomorphsim on R . Since f is n+r n epi the map R -> R obtained by stacking A and K is also n n+r epi, hence it has a left-inverse (B,C):R -> R .) Specialize to the case that R is the the "generic ring", that is the ring generated by the 2nn+2nr+rr entries of K,A,A',B,C with nr+nn equations. We may infer that there is an rxr matrix X and an nxr matrix Y such that KB = XK AB + YK = I. The entries of X and Y are necessarily given by polynomials in the generating "variables" and the last two matrix equations must result in rr+nr equations that are direct consequences of the nr+nn defining equations. Hence the original theorem works for any finitely presented module over any commutative ring. Now for the hard part: what are these polynomials? In the case n = 1 its easy (and reveals quickly the need for commutativity). Try it for n=2, r=1. Given: ac+be = ga ad+bf = gb hc+ie+la = 1 hd+if+lb = 0 jc+ke+ma = 0 jd+kf+mb = 1 find, for a start, a polynomial on these variables, x, such that ah+bj = xa ai+bk = xb. >From cat-dist(a)mta.ca Ukn Jan 17 22:18:57 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA21023; Mon, 17 Jan 1994 22:18:56 +0400 Date: Mon, 17 Jan 1994 22:05:50 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: cantor-bernstein To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401172250.B16680-c100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 17 Jan 94 20:15:09 EST From: Michael Barr <barr(a)triples.Math.McGill.CA> > Mike Barr writes: > > One additional example and you don't even need retracts. In the category > of finitely generated modules over a commutative ring, all epis are > isos. As a result, if you have epis in both directions, they are > isos. So the dual category is S-B. This is fairly > easy if the ring has ACC, but there is a trick that works for any > ring to reduce it to that case. > > Wonderful thought: all epis are isos. Anyway, I see a proof that any > epi endo on a finitely presented module over a commutative ring is > iso, but finitely generated? > > There's a metaprinciple that says that a result like this should > generalize from commutative to PI rings (that is, rings that satisfy > some non-trivial Polynomial Identity). Can anyone confirm? A corollary > would be that in any additive category if two objects each appear as > retracts of the other, and if the ring of endomorphisms of one of them > is a PI ring then the retractions are isos. > > best thoughts, > peter > > I will try to recall the argument (on-line). Given an epi-endomorphism f, look at the ascending chain ker(f), ker(f^2), ker(f^3),.... In the noetherian case, this stabilizes so that ker(f^n) = ker(f^{n+1}). Assume thatn is as small as possible, so that ker(f^{n-1}) < ker(f^n). Choose an element x in the ker of f^n, not in the lesser one. x = f(y) for some y, since f is onto. 0 = f^n(x) = f^{n+1}(y), so that 0 = f^{n}(y) = f^{n-1}(x), a contradiction. That takes care of the noetherian case and doesn't even use commutativity, it would seem. For the general case, suppose R is the ring, M the module, f: M --> M the endomorphism and x an element with f(x) = 0. Now pick a set of generators for M, say y_1,...,y_n. What you have to do is to find a suitable finite subset of R, with just the right elements in it to express all the f(y_i), x and at least one preimage of each y_i as linear combinations of the y_i using coefficients from that subset. Now let S be the subring of R generated by that finite set of elements and N be the least S-submodule of M containing all the y_i. If I have left anything required out of S, add that too. Anyway, S is noetherian (this does use commutativity, I believe) and f induces a counter-example on N. I believe this argument is due to one of the Rutgers people like Faith or Osofsky, but I am far from certain of that. It will be true for PI rings if affine PI rings have acc on left ideals. For commutative rings it is essentially the Hilbert basis theorem. Michael >From cat-dist(a)mta.ca Ukn Jan 19 22:39:16 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA04251; Wed, 19 Jan 1994 22:39:15 +0400 Date: Wed, 19 Jan 1994 22:19:51 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: RE Fuzzy + To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401192251.B26571-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 19 Jan 1994 11:45:07 -0600 From: Lawerce Neff Stout <lnstout(a)uxh.cso.uiuc.edu> I've done quite a lot of work on categories of fuzzy sets. The main paper is in the volume edited by H\"ohle and Rodabaugh referred to earlier. Barr is correst that fuzzy sets form a quasitopos, but the logic of that quasitopos is that of the underlying set category, hence not interesting as a place to do fuzzy mathematics. The fuzzy connectives come from a second monoidal closed structure obtainable from, for example, the t-norms usually referred to in the fuzzy literature. This gives a very satisfactory logic if one uses what I called unballanced subobjects (the map involved is both monic and epic). There is a weak representor for these subobjects (representation is not unique though there is an ordering on maps which allows a canonical choice of representative to be made) allowing an internal representation of a large fragment of higher order fuzzy logic. I have a more recent paper (to appear in the proceedings of the 1992 Linz seminar, being published by Kluwer sometime later this year) in which I look at categories of fuzzy sets with values in a Quantale or Projectale. That paper is available from me by e-mail (I don't have ftp facilities available). It includes a characterization of categories of fuzzy sets in terms of the representability of the logic and the property of being topological over Sets. Larry Stout >From cat-dist(a)mta.ca Ukn Jan 21 09:51:34 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA04197; Fri, 21 Jan 1994 09:51:33 +0400 Date: Fri, 21 Jan 1994 09:30:23 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: categorical treatment of F_omega? To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401210923.A4077-9100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Thu, 20 Jan 1994 17:24:00 -0500 From: David Espinosa <espinosa(a)cs.columbia.edu> 1. Could someone send me a good reference for a categorical treatment of the Girard / Reynolds F_2 polymorphic type system? That is, polymorphic functions as (some form of) natural transformations? 2. More importantly, has there been a categorical treatment of Girard's F_omega type system? David >From cat-dist(a)mta.ca Ukn Jan 22 13:58:28 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA01235; Sat, 22 Jan 1994 13:58:27 +0400 Date: Sat, 22 Jan 1994 13:49:21 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Terminology question To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401221320.B26593-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Sat, 22 Jan 1994 2:44:17 -0500 (EST) From: D_FELDMAN(a)UNHH.UNH.EDU Is there a standard terminology for the following sort of gadget or something very similar? Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite sets and bijections, satisfying\\ (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism $\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\ (ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively computable) finite set. David Feldman >From cat-dist(a)mta.ca Ukn Jan 24 13:43:31 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA08176; Mon, 24 Jan 1994 13:43:30 +0400 Date: Mon, 24 Jan 1994 13:15:21 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: Terminology question To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401241321.E23999-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Mon, 24 Jan 1994 16:13:07 +0000 (GMT) From: Edmund Robinson <edmundr(a)cogs.susx.ac.uk> > > Date: Sat, 22 Jan 1994 2:44:17 -0500 (EST) > From: D_FELDMAN(a)UNHH.UNH.EDU > > Is there a standard terminology for the following sort of gadget or something > very similar? > > Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and > a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite > sets and bijections, satisfying\\ > (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a > unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism > $\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\ > (ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively > computable) finite set. > > David Feldman > > I think this would traditionally be described as a "finite discrete opfibration (over Bij)". The functors corresponding to condition (i) are discrete opfibrations, and the finite comes from condition (ii). Neither of these uses any special property of Bij (such as the fact that it is a groupoid). It might be more modern to use "cofibration" instead of "opfibration". See Barr & Wells "Toposes, Triples and Theories" p231 ex [OPF] for more conventional definitions, and perhaps Benabou "Fibred categories and the foundations of naive category theory" (J. Symbolic Logic (50) No. 1, 1985, 10-37) for more of an indication of why these sorts of structures are so common. Another way of looking at the structure would be to turn it around and say that you have a functor G: Bij -> FiniteSet given on objects by G(t) = F^{-1}(t). best wishes, Edmund Robinson >From cat-dist(a)mta.ca Ukn Jan 25 07:26:29 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA26948; Tue, 25 Jan 1994 07:23:14 +0400 Date: Tue, 25 Jan 1994 07:21:14 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: Terminology question To: categories <categories(a)mta.ca> Cc: cdl2 -- France Dacar <France.Dacar(a)ijs.si>, Robert Dawson <rdawson(a)husky1.stmarys.ca>, "Oege de.Moor" <Oege.de.Moor(a)prg.ox.ac.uk>, "Valeria de.Paiva" <vcvp(a)cl.cam.ac.uk>, "Ruy de.Queiroz" <rjq(a)doc.ic.ac.uk>, "Fer-Jan De.Vries" <ferjan(a)cwi.nl>, Kyung-Goo Doh <kg-doh(a)u-aizu.ac.jp>, James Dolan <jdolan(a)ucrmath.ucr.edu>, Xiaomin Dong <xdong(a)clid.yorku.ca>, Winfried Drecmann <mas031(a)bangor.ac.uk>, Dominic Duggan <dduggan(a)watmsg.waterloo.edu>, Gerald Dunn <gdunn(a)nmsu.edu>, Hans Dybkjaer <dybkjaer(a)ruc.dk>, Abbas Edalat <ae(a)doc.ic.ac.uk>, David Espinosa <dae(a)martigny.ai.mit.edu>, Michel Eytan <eytan(a)dpt-info.u-strasbg.fr>, Joe Fasel <jhf(a)c3.lanl.gov>, David Feldman <d_feldman(a)unhh.unh.edu>, Zbigniew Fiedorowicz <zigf(a)mps.ohio-state.edu>, Juarez Muylaert Filho <jamf(a)doc.ic.ac.uk>, Stacy Finkelstein <stacy(a)saul.cis.upenn.edu>, Kathleen Fisher <kfisher(a)cs.stanford.edu>, Maria Frade <mjf(a)di.uminho.pt>, Peter Freyd <pjf(a)saul.cis.upenn.edu>, Tom Fukushima <fukushim(a)cpsc.ucalgary.ca>, Jonathan Funk <jfunk(a)morgan.ucs.mun.ca>, Fabio Gadducci <gadducci(a)di.unipi.it>, Vijay Gehlot <gehlot(a)saul.cis.upenn.edu>, Wolfgang Gehrke <wgehrke(a)risc.uni-linz.ac.at>, Silvio Ghilardi <ghilardi(a)vmimat.mat.unimi.it>, Paul Glenn <glenn(a)cua.edu>, Joseph Goguen <Joseph.Goguen(a)prg.ox.ac.uk>, Marek Golasinski <mg001(a)vm.cc.uni.torun.pl>, Al Goodloe <agoodloe(a)mason1.gmu.edu>, Bob Gordon <gordon(a)euclid.math.temple.edu>, Francoise Grandjean <grandjean(a)agel.ucl.ac.be>, John Gray <gray(a)math.uiuc.edu>, Luzius Grunenfelder <luzius(a)cs.dal.ca>, Stefano Guerrini <guerrini(a)di.unipi.it>, Alessio Guglielmi <guglielm(a)di.unipi.it>, James Harland <jah(a)cs.mu.oz.au>, Robert Harper <rwh(a)cs.cmu.edu>, Magne Haveraaen <magne(a)eik.ii.uib.no>, "Michael J. Healy" <mjhealy(a)espresso.rt.cs.boeing.com>, Michel Hebert <mhebert(a)egaucacs.bitnet>, Murray Heggie <heggie(a)cad.uccb.ns.ca>, Luis Javier Hernandez <zl(a)cc.unizar.es>, Walt Hill <whill(a)netcom.com>, SATO Hiroyuki <schuko(a)sun4.cc.kyushu-u.ac.jp>, Bernard Hodgson <bhodgson(a)mat.ulaval.ca> Message-Id: <Pine.3.05.9401250714.C24955-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Date: Tue, 25 Jan 1994 10:18:25 +0000 From: Steven Vickers <sjv(a)doc.ic.ac.uk> Do others suffer the same heartsink as I do when confronted with a posting like this? >Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and >a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite >sets and bijections, satisfying\\ > (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a >unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism >$\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\ > (ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively >computable) finite set. For human readers (and after all, is this message _ever_ going to be presented to a Latex interpreter?) most of the $'s and \'s here are not only completely unnecessary, but, worse, a positive barrier to understanding. I would expect - but this is something that can be put to the test - that even people completely familiar with Latex would find it easier to read the following version. It certainly involves less typing. >Define a ?????? to be a pair (S,F) consisting of a category S and >a functor F from S to Bij, the category of finite >sets and bijections, satisfying - > (i) If F(s_1)=t_1 and tau:t_1 -> t_2, then there exists a >unique object s_2 in S and a unique S-morphism >sigma: s_1 -> s_2 such that F(sigma)=tau. > (ii) For t in Bij, F^{-1}(t) is an (effectively >computable) finite set. (I have ignored the puzzle of whether {\bf ...} is mathematically meaningful - in the original $S$ turns into ${\bf S}$. If it _is_ mathematically meaningful, then in Latex it should be macroized.) Steve Vickers. p.s. Having rephrased the question, I still don't know the answer - sorry. >From cat-dist(a)mta.ca Ukn Jan 25 12:17:27 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA28518; Tue, 25 Jan 1994 12:17:27 +0400 Date: Tue, 25 Jan 1994 11:47:26 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: Re: cantor-bernstein To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401251126.C22422-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 19 Jan 1994 09:37:14 -0500 From: Stephen Chase <chase(a)math.cornell.edu> With regard to the remarks of Barr and Freyd on surjective endomorphisms of finitely generated modules: I haven't digested those remarks, but here is a slick proof of the result, communicated to me years ago by Bill Waterhouse (instead of reduction to the Noetherian case, it uses localization): It is enough to prove bijectivity at all localizations, so we can assume that the commutative ring A is local with maximal ideal m. Given a surjective endomorphism f of a finitely generated A-module M, let F be a finitely generated free A-module mapping onto M so that the mapping induces an isomorphism F/mF ----> M/mM. Let K = Ker(F ---> M). f then lifts to an endomorphism g of F, which is an isomorphism because it is so mod m. Then g(K) is contained in K, and to prove f is bijective we need only show g'(K) is likewise contained in K (with g' the inverse of g). But g satisfies its characteristic polynomial, which has invertible constant term det(g) (up to sign); thus g' is a polynomial in g and so maps K into itself. I haven't seen this proof in the literature. However, the following related reference might be of interest: M. Orzech, L. Ribes, "Residual finiteness and the Hopf property in rings", J. Algebra 15 (1970), 81-88. Sincerely, Steve Chase >From cat-dist(a)mta.ca Ukn Jan 25 12:22:48 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA28645; Tue, 25 Jan 1994 12:22:48 +0400 Date: Tue, 25 Jan 1994 11:53:49 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: terminology To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401251149.J22422-b100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: [Note from moderator: 1. The two posts following are being forwarded, but I don't feel that this list is really the place for a long discussion of suitable notation for e-mail, so I hope that any discussion will be short] Date: Tue, 25 Jan 94 08:37:53 EST From: Peter Freyd <pjf(a)saul.cis.upenn.edu> I certainly agree with Steve's point. But I would go further: Define a ?????? to be a pair (*S*, F) consisting of a category *S* and a functor F from *S* to *Bij* , the category of finite sets and bijections, satisfying - (i) If F(S ) = T and f:T -> T' , then there exists a unique object S' in *S* and a unique morphism g: S -> S' such that F(g) = f. (ii) For T in *Bij*, F (T) is an (effectively computable) finite set. But: I must confess that I also experience a little "heartsink" when I see a list of addresses as long as that above. best thoughts, peter ++++++++++++++++++++++++++++++++++++++++++++ Date: Tue, 25 Jan 94 9:51:20 EST From: Al Vilcius <vilcius(a)mathstat.yorku.ca> Referring to the "rephrased" question of Steve Vickers: > > Do others suffer the same heartsink as I do when confronted with a posting > like this? > > >Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and > >a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite ...... Yes, I certainly do, and much prefer the "humanized" alternative: > > >Define a ?????? to be a pair (S,F) consisting of a category S and > >a functor F from S to Bij, the category of finite ...... > -- /\ / Al Vilcius, Toronto / \ / /--->\ / / \/ >From cat-dist(a)mta.ca Ukn Jan 27 12:30:09 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA27146; Thu, 27 Jan 1994 12:30:08 +0400 Date: Thu, 27 Jan 1994 11:53:21 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: RE: terminology To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401271121.E11880-9100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Wed, 26 Jan 1994 2:35:50 -0500 (EST) From: D_FELDMAN(a)UNHH.UNH.EDU Thank you to all those who responded, including those who pointed out my e-faux pas. Incidently, the complaint about S versus {\bf S} alerted me to a typo in a paper under preparation (these should have been the same) and so I am especially grateful for that. David Feldman >From cat-dist(a)mta.ca Ukn Jan 31 16:29:46 1994 Received: from bigmac.mta.ca by nimble.mta.ca (5.65/DEC-Ultrix/4.3) id AA01906; Mon, 31 Jan 1994 16:29:45 +0400 Date: Mon, 31 Jan 1994 15:59:00 +0400 (GMT+4:00) From: categories <cat-dist(a)mta.ca> Subject: New address of Fer-Jan de Vries To: categories <categories(a)mta.ca> Message-Id: <Pine.3.05.9401311500.E23091-a100000(a)nimble.mta.ca> Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Date: Mon, 31 Jan 1994 20:17:52 +0100 From: F.J.de.Vries(a)cwi.nl CWI, January 31, 1994 Dear Colleague. The coming year I will live and work in Japan. My addresses will be the following: Office: from March 1st, 94, onwards NNT, Communication Science Laboratories Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Phone +81-7749-5-1841, Facsimile +81-7749-5-1851 Email ferjan(a)progn.kecl.ntt.jp Home: from Feb 1st, 94, onwards Seresu-Gakuenmae 305 Gakuen-Naka 1-1542-190 Nara-shi, Nara 631 Phone: yet unkown... Sayonara, Fer-Jan de Vries.

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