December 1993 - Categories - categories.org.au

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by dyetter＠math.ksu.edu 28 Dec '93

28 Dec '93

I have not seen a name for the structure R. Loader inquires about. It may be a lax version of a structure which has been considered in the context of "quantum topology" (cf. my paper in Topology '90): Let C be a monoidal category (e.g. category with (chosen) products), let X be category equipped with an "action" of C, i.e. a functor a:C x X ----> X, satisfying the "obvious" coherence conditions. X is then a "C-module" and a functor F:X--->X such that for all c in C, x in X such that there is a natural isomorphism from F(a(c,x)) to a(x,F(x)) again satisfying "obvious" coherence conditions is a "C-module functor". In the case in question C = X, tensor = a = cartesian product. I think, however, that one wants a natural transformation, not a natural isomorphism. Best Thoughts, David Yetter

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None
by kock＠mi.aau.dk 28 Dec '93

28 Dec '93

The extra structure on the endofunctor F given by hom(X x Y,Z) ---> hom(X x FY,FZ) is called a tensorial strength, for reasons deriving from enriched category theory, cf Kock: Strong Functors and Monoidal Monads, Archiv der Math. (Basel) 23 (1972), 113-120. In loc.cit., it is given in terms of X x FY ---> F(X x Y), but such data is equivalent to data of the above kind, by a Yoneda-like argument (put Z = X x Y, and apply to the identity map of X x Y). A kind of structure that implies such monoidal strength is the data of F being an indexed endofunctor (assuming the category in question has pull-backs, thus being indexed over itself). This seems to be the case for any endofunctor F on a topos, provided F is described by logical means. Anders Kock +++++++++++++++++++++++++++++++

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by Fred E J Linton 28 Dec '93

28 Dec '93

Data of the form F*(X,Y,Z): hom(XxY, Z) --> hom(XxFY, FZ) appear in the notion of a multilinear functor. See my Multilinear Yoneda Lemmas article in an ancient Springer Lecture Notes in Mathematics. -- Fred +++++++++++++++++++++++++++++

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Revised report on sketches
by cfw2＠po.CWRU.Edu 24 Dec '93

24 Dec '93

The final version of the document "Sketches: an outline with references" is now available by FTP. The present version corrects errors in the document distributed in July, 1993, and contains many new references. You can use anonymous FTP to get the paper from ftp.cwru.edu. The DVI file of the paper is SKETCH.DVI, and the BiBTeX source of the bibliography is SKETCH.BIB. The latter is an ASCII file which requires no knowledge of TeX or BiBTeX to read. Both are in the directory math/wells. The file README.TXT contains information about other documents in the directory. I will be glad to send a printed copy of "Sketches: an outline with references" to anyone who cannot use FTP or DVI files. I am sending this announcement to two mailing lists; my apologies if you get it twice. -- Charles Wells, Department of Mathematics, Case Western Reserve University 10900 Euclid Avenue, Cleveland OH 44106-7058, USA Phone 216 368 2880 or 216 774 1926 FAX 216 368 5163 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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None
by MTHFWL＠ubvms.cc.buffalo.edu 24 Dec '93

24 Dec '93

<< Let C be a category with finite products, and let F : C -> C be a functor, along with a natural transformation F*(X,Y,Z) : hom(X x Y, Z) --> hom(X x FY, FZ) that preserves composition and identities in the appropriate manner. (The idea is that this gives a behaviour for F on collections of arrows from Y to Z indexed by X.) >> It seems that Ralph Loader has rediscovered an important case of what are called Enriched Functors, see eg Max Kelly's book. He also notes that these are distinct from Internal Functors. Internal and Enriched are both included under and interact within the "indexed" (I would prefer "parameterized") theory treated in another important book: Springer LNM 661 ++++++++++++++++++++++++++++++

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seeking Michael Wright
by GARY ALEXANDER 24 Dec '93

24 Dec '93

At the suggestion of Martin Hyland we are trying to make contact with Michael Wr ight, who Hyland thought was living in London. He was in Cleveland, Ohio in 1990/ 91 where he helped Colin McLarty with his book "Elementary Categories, Elementary Toposes". Colin McLarty, we believe, is active on email. Can anyone tell us how to contact Michael Wright (or are you on this mailing list , Michael)? Please reply to Gary Alexander at the Open University, as g.r.alexander(a)open.ac.uk. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Re: Has anybody got a name for this(several posts)
by Richard Wood 24 Dec '93

24 Dec '93

If V_ is a symmetric monoidal category but not closed there is stilll a sense in which V_ itself is a V_-category. After all, V_ "acts" on itself via tensor:V_xV_--->V_ and it is useful to think of an arrow X ten Y --->Z as providing an X-indexed family of arrows from Y to Z. If A_ and B_ are categories equipped with actions of V_ one can ask for the relevant notion of functor from A_ to B_ that respects such actions and the condition is the one that you enquire about, your specific case involving endofunctors on the base which having binary products can be seen as a symmetric monoidal category. If the idea of A_ having families of arrows indexed by the objects of V_ is taken as primitive we are led to consider categories A_ equipped with a functor A_^op x V_^op x A_ ---> SET that is suitably compatible with the ordinary hom functor for A_. The relevant conditions ultimately ensure that A_ has the structure of a SET^(V_^op) category. This last is considered to have the monoidal ((bi)-closed) structure introduced by Brian Day. If the data for such is presented as I have it in the last paragraph then the idea is that for A and B in A_ and X in V_ the value of the structure functor at (A,X,B) provides a SET of X-indexed arrows from A to B. This structure admits the possibility of representability in each of the 3 variables so that SET^(V_^op)-category includes the notions of V_-category, "V_-tensored category" and "V_-cotensored category". If A_ and B_ are SET^(V_^op) categories then the notion of SET^(V_^op) functor F between them is clear but there are 3x3=9 special cases that bear inspection in case both A_ and B_ are any of the special cases mentioned above. In each case "the effect of F on homs" (I believe that is the term you want) admits a compact description. In my 1976 thesis there is a table which displays them. Best regards RJ +++++++++++++++++++++++++++++++++++++++++

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Has anybody got a name for this
by loader＠maths.ox.ac.uk 21 Dec '93

21 Dec '93

Let C be a category with finite products, and let F : C -> C be a functor, along with a natural transformation F*(X,Y,Z) : hom(X x Y, Z) --> hom(X x FY, FZ) that preserves composition and identities in the appropriate manner. (The idea is that this gives a behaviour for F on collections of arrows from Y to Z indexed by X.) Does anyone have a name for a functor with such a natural transformation? Note that if C is cartesian closed, then this is the same as having a natural transformation from Y^Z to (FY)^(FZ) that preserves composition and identities. This sort of functor arises in logic as follows: Let A[X] be a formula with a positively occuring propositional variable X. Then B |- C ------------ A[B] |- A[C] is a derived rule, making A[-] into a functor. But we can also do this with additional side formulae: D, B |- C --------------- D, A[B] |- A[C] giving a natural transformation as described as above. I'd use the term "internal functor", except that this is already used for a functor between two internal categories. Ralph Loader. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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EUROFOCS Ad
by gdp＠dcs.ed.ac.uk 16 Dec '93

16 Dec '93

EUROFOCS European Institute in the Logical Foundations of Computer Science Fellowship Programme (Closing date for applications - 5th January 1994) The European Institute in the Logical Foundations of Computer Science invites applications for PostDoctoral Fellowships. These can be held for periods ranging from six months to one year. The goal of the work in the Institute is to achieve a unified and applicable theory of the semantics and logic of languages used to describe, design and program computing systems. The research of the Insititute is characterised by a unified approach to foundational problems that emphasises the combination of ideas arising in design and practice with ideas originating in logic and allied mathematical areas. The applicability of the ideas developed is demonstrated by producing experimental systems for specification, verification and development. The research has both national and EC support, with involvement in several ESPRIT Basic Research Actions, including CONFER, CONCUR, COMPASS, Categorical Logic in Computer Science and Types for Proofs and Programs. Research Topics of Interest include: Semantics of Programming Languages; Logic Programming Theory; Object-Oriented Programming; Functional Programming Theory; Lambda-Calculus; Formal Development of Programs and Systems; Concurrency Theory. The Institute sites and contact points are: CWI, Amsterdam Jaco de Baaker <jaco(a)cwi.nl> (+31) 20 592 4136 The University of Cambridge; Andrew Pitts <ap(a)cl.cam.ac.uk> (+44) 223 334621 The University of Edinburgh Rod Burstall <rb(a)lfcs.ed.ac.uk> (+44) 31 650 5156 Imperial College, London Samson Abramsky <sa(a)doc.ic.ac.uk> (+44) 71 589 5111 x5005 ENS and INRIA, Paris Giuseppi Longo <longo(a)dmi.ens.fr> (+33) 14 432 3328 The University of Pisa. Ugo Montanari <ugo(a)di.unipi.it> (+39) 50 510 221 INRIA, Sophia Antipolis; Gilles Kahn <gilles.kahn(a)sophia.inria.fr> (+33) 93 65 78 01 APPLICATIONS must be sent to the prospective host site. More general information on the Institute or the fellowship programme may be requested from Gordon Plotkin <gdp(a)lfcs.ed.ac.uk> (scientific information), or George Cleland <g.cleland(a)lfcs.ed.ac.uk> (administrative information) ALL APLICATIONS MUST BE RECEIVED BY THE HOST SITE BY 5TH JANUARY 1994 The initial application should include: 1. Name, Address, e-mail,telephone & fax nos. 2. Academic Qualifications 3. Post-doctoral positions 4. Names, addresses, e-mail and phone numbers of at least two referees 5. Publications 6. A summary of research achievments to date 7. Proposed research programme (3-400 words, written in third person) 8. Expected outcomes of research (<200 words) 9. Reason for choice of host institution (<200 words) Selection of candidates will be made on a number of factors, including quality of research proposed, the potential of the candidate, and the contribution of the proposed research to both the host site and the Institute. Salary and payment details vary from site to site and should be discussed with the host site. Those candidates selected will be required to work with the host site to produce a full fellowship document set which is required by the EC. This a non-trivial task for the candidate and will require the selected candidate to be available and in communication for the last two weeks of January. Prospective candidates are particularly asked to note the ELIGIBILITY CRITERIA for the fellowships. These conditions, imposed by the European Commission, are: i) The applicant must be a national of one of the Member States of the European Community or EFTA, or be resident in the European Community. ii) The applicant must be a national of a country other than that in which the host institution is established and must not have carried out their normal activity in that country for more than two years prior to the date of submission of the application. (EC nationals presently resident in a non-EC country are eligible providing the proposed fellowship is in a country other than their country of origin.) iii) Applicants must be young researchers, without other income, having at least six years' higher education and who hold a doctoral or equivalent degree, or, if not, have had two years' research experience following a post-graduate course. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Research area of objects and agent behavior
by pinj＠di.uminho.pt 15 Dec '93

15 Dec '93

Hi, I'm beginning to research in the area of objects and agent behavior (observational properties) and I would like to read something about their categorical and topological definition. I'm also interested in agent hierarchies, behavior inheritance and parametrization of agent systems. Some questions: 1- Can you send me references of papers that you think I should start reading ? 2- If possible, can you please send me information about Computer Science or Mathematic groups who are working in this area, so that I can learn something with them. 3- Is there any topological mailing list ? Thank you, Jose Miranda ___ ____ / / / /\ /\ / /--- / \/ \ \-/ / / \ Jose E. Pina Miranda | EMail: pinj(a)di.uminho.pt Assistant Lecturer | X.500: c=PT@o=Universidade do Minho Departamento de Informatica | @ou=Departamento de Informatica Universidade do Minho | @cn= /* not available yet */ Campus de Gualtar | WWW URL: http://s700.uminho.pt/~pinj 4700 Braga | Voice: +351 (53) 617 021 Portugal | Fax: +351 (53) 612 954 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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