February 1992 - Categories - categories.org.au

New home telephone number for Max Kelly
by kelly_m＠maths.su.oz.au 29 Feb '92

29 Feb '92

>From Mon 2 Mar this will be: (-61-2)- 983-9985. Regards to all, Max - 28 Feb.

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Re: The ``monoid endofunctors of some categories
by dyetter＠math.ksu.edu 28 Feb '92

28 Feb '92

By way of comment on David Feldman's question. One can trivially produce lots of endofunctors of n-MAN. Since n-MAN had coproducts, so does any functor category targetted in n-MAN, in particular END(n-MAN). Second, note that given any endofunctor and a functor from n-MAN to SET, one can take a copower of the endofunctor by the set-valued functor. (Feldman's "functors of set-type" being an example of this in the case where the endofunctor is the identity, but one could just as easily take say X |----> pi_0(X) x A for a fixed n-manifold, A.) It might be better to stick to connected n-manifolds, in which case I'm not sure how to construct any endofunctors other than the constant ones and the identity. ---David Yetter =========================================================================

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Temporary Lectureships
by Michael Fourman 28 Feb '92

28 Feb '92

Please circulate the following announcement, apologies to those who have already seen this elsewhere. We hope to make *two* appointments. - - ---- Start of forwarded text ---- > Received: from gruna.dcs.ed.ac.uk by dcs.ed.ac.uk id aa00780; > 30 Jan 92 16:02 GMT > Date: Thu, 30 Jan 92 16:02:01 GMT > Message-Id: <16831.9201301602(a)gruna.dcs.ed.ac.uk> > From: h o d <hod(a)uk.ac.ed.dcs> > Subject: Advert > To: lmp(a)uk.ac.ed.dcs > Status: RO > > > University of Edinburgh > Department of Computer Science > > > Temporary Lectureship > > Applications are invited for a temporary (five year) lectureship available > from October 1992. Applicants should be qualified to Ph.D. level and should > be able to teach across a range of topics and levels (including first-year) > within the subject. > > The successful candidate will be expected to make a strong contribution to > research in the Department which currently has major interests in parallel > computing through its links with the Edinburgh Parallel Computing Centre > (EPCC) and in theoretical topics through the work of the Laboratory for > Foundations of Computer Science (LFCS). Candidates with excellent records > in other areas are also encouraged to apply. > > The Department has a very high reputation for the quality of both its > teaching and research, and has excellent facilities which include over > 200 workstations and access to a Connection Machine and a Meiko Computing > Surface in EPCC. The existing staff complement consists of 28 lecturing > staff (including 5 professors) and over 20 research workers, supported by > computing officers and technical and secretarial staff. > > Initial salary will be on the Lecturer A scale #12,860 - 17,827 > > with placement according to age, qualifications and experience. > > Further particulars may be obtained by writing to: > > The Personnel Office > University of Edinburgh > 1 Roxburgh Street > Edinburgh EH8 9TB > > to whom applications should be sent before the closing date of 31 March 1992, > or by e-mail from Ms Laura Paterson <lmp(a)dcs.ed.co.uk>. > > - - ---- End of forwarded text ---- - ------- End of Forwarded Message =========================================================================

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alternative name for stable functors
by Paul Taylor 27 Feb '92

27 Feb '92

A stable functor has a left adjoint on each slice (multiajoint a` gauche, to quote Yves Diers). The word stable was first used in this sense by Gerard Berry (I believe), but it's most unfortunate as it already has at least one generic meaning in category theory, namely preservation by pullbacks. Can anyone suggest a better name? Here are some possibilities. 1) Stable endofunctors of the category of sets have power series expansions, so "analytic" (after Joyal) is a possibility. It goes nicely with continuous (in the sense of Dana Scott) although I don't want it to include preservation of filtered colimits. Also, as used by Andre' Joyal and Franc,ois Lamarche, analytic functors need only send pullbacks to weak pullbacks. 2) My definition of a functor U:M->C being stable is that every map X->UC in C factor as a *candidate* (diagonally universal map in Diers' terminology) followed by Uh for some h:A->B in M. cf factorisation systems. This suggests "quotate" or "democratic" (ie there are enough candidates). 3) Much of my work on stable functors as morphisms of domains is based on a technique of considering slices. This suggests "laminated functor" and similarly laminated category, laminated ccc, laminated topos for the corresponding well-behaved categories (the last for a category every slice of which is a topos, not necessarily with a terminal object). 4) Since they're functors with (non-chosen) adjoints on each slice, "slice-adjunctible" is another possibility. 5) I'm also interested in functors which just preserve (binary) pullbacks (between categories with them). I'd like to use the word "cartesian" to refer to anything concerned with pullback squares (and not just products), and call a pullback-preserving functor "cartesian" Any thoughts? Paul Taylor =========================================================================

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Union Conference
by NIEFIELD, SUSAN 27 Feb '92

27 Feb '92

UNION COLLEGE MATHEMATICS CONFERENCES April 25 and 26, 1992 CATEGORY THEORY DYNAMICAL SYSTEMS NUMBER THEORY Keynote Speaker JOHN MILNOR The Mathematics Department of Union College is pleased to announce its eighth occasional spring conference. The meeting will feature a keynote address by John Milnor, as well as parallel sessions in Category Theory, Dynamical Systems, and Number Theory. ------------------------------------------------------------- Invited Speakers for the Category Theory section: F. WILLIAM LAWVERE and JOAN PELLETIER Invited Speakers for the Number Theory Section: KARL RUBIN "Rational points on elliptic curves with complex multiplication" JOSEPH SILVERMAN "Canonical heights on varieties with morphisms" Invited Speakers for the Dynamical Systems section: To be announced ------------------------------------------------------------- CONTRIBUTED TALKS Please contact one of the organizers of the parallel session in which you would like to give a talk. Such talks should be 20-30 minutes long. Conference Organizers for Category Theory: Susan Niefield niefiels(a)union.bitnet or niefiels(a)gar.union.edu Kimmo Rosenthal rosenthk(a)union.bitnet or rosenthk(a)gar.union.edu Conference Organizer for Dynamical Systems Michael Frame framem(a)union.bitnet or framem(a)gar.union.edu Conference Organizers for Number Theory: William F. Hammond (SUNY Albany) hammond(a)csc.albany.edu Karl Zimmermann zimmermk(a)union.bitnet or zimmermk(a)gar.union.edu Department of Mathematics, Union College, Schenectady, NY 12308-2311 Telephone (518) 370-6246 FAX (518) 370-6789 ------------------------------------------------------------- DETAILED CONFERENCE INFORMATION Department of Mathematics, Union College, Schenectady, NY 12308-2311 Telephone (518) 370-6246 FAX (518) 370-6789 Each session will include invited lectures and shorter contributed talks. Your $25 registration fee will cover the cost of a reception Friday night, and a cocktail hour, buffet dinner, and party on Saturday. The fee for graduate students is $15. -------------------------------------------------------------- ACCOMMODATIONS To obtain a room at one of the rates listed below, mention Union College when you call. A block of rooms has been set aside at the Days Inn. All of the hotels are within easy walking distance of campus. 1. Days Inn (518) 370-0851 $49 single $49 double (includes continental breakfast) 2. Holiday Inn (518) 393-4141 $58 single $63 double 3. Ramada Inn (518) 370-7151 $60 single $70 double ---------------------------------------------------------------- SCHEDULE Friday Evening, April 24 9:00-11:00 Reception / 2nd floor, Bailey Hall Saturday, April 25 9:30-10:00 Registration, coffee & donuts / 2nd floor, Bailey Hall 10:00-12:00 Parallel sessions / 2nd floor, Bailey Hall 12:00- 1:30 Lunch / Hale House 1:30- 3:30 Parallel sessions / 2nd floor, Bailey Hall 3:30- 4:00 Coffee / Social Sciences 016 4:00- 5:00 Keynote Address by John Milnor / Social Sciences 016 5:30- 6:30 Cocktails / Hale House Lower Lounge 6:30- 8:00 Buffet Dinner / Hale House 8:00-11:00 Dessert and Party / Milano Lounge Sunday, April 26 9:00- 9:30 Coffee and donuts / 2nd floor, Bailey Hall 9:30-12:00 Parallel sessions / 2nd floor, Bailey Hall 12:00- 1:30 Lunch / Hale House 1:30- 3:30 Parallel sessions / 2nd floor, Bailey Hall The parallel sessions will take place in Bailey Hall, 2nd floor. A schedule and room assignments will be posted outside the Math lounge. For more information contact one of the organizers or contact: Department of Mathematics, Union College, Schenectady, NY 12308-2311 Telephone (518) 370-6246 FAX (518) 370-6789 ------------------------------------------------------------ REGISTRATION INFORMATION Registration Fee: $25 (Graduate Students: $15) Participants are urged to register in advance so that necessary arrangements can be made. If this is impossible, you may register on Saturday morning at the conference, but please contact us in advance via e-mail or phone. If you have any questions or need assistance with your travel arrangements, please call one of the organizers at (518) 370-6246. ------------------------------------------------------------ REGISTRATION FORM For advance registration please send this information by any convenient method to one of the organizers or to the Mathematics Department, Union College at the address above. Name: Institution: Address: City: State: Zip : Phone: Email address: Check here: if you are interested in giving a contributed talk. Session: Title: =========================================================================

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The ``monoid'' of endofunctors of some categories
by D_FELDMAN＠UNHH.UNH.EDU 27 Feb '92

27 Feb '92

I would like to pose a question, first in the concrete situation where it arose, and then more abstractly. Let n-MAN be the category of (topological) n-manifolds and continuous maps. My question is, what are the endofunctors of n-MAN ? To start with, there is no shortage. Any functor F: n-MAN --> Set gives rise to an endofunctor F': n-MAN --> n-MAN as follows. On objects M \in n-MAN , F'(M)=M x F(M) (where F(M) is given the discrete topology) and for morphisms f \in n-MAN , F'(f)(p,x)=(p,f(x)) . Let us call these endofunctors of set type. The endofunctors of set type form a proper class, since for any cardinal k there is a set valued functor taking M to the underlying set of M x k. So one may ask if there are any others? If there are, then one can ask a more refined question. The endofunctors of n-MAN form a (proper class based) monoid. It still makes sense to consider the monoid ideal generated by the class of endofunctors of set type. One might then try to form the quotient monoid and as a measure of how close the class of set type endofunctors comes to exhausting the class of all endofunctors. Is this construction familiar? If one considers instead the category of pointed n-manifolds, then one has the universal covering space endofunctor, which is not of set type; there the quotient monoid will be non-trivial. A few more thoughts: The construction above makes sense in any category with arbitrary coproducts, or even just copowers, if that is the right term. If instead of the category of n-manifolds, I had considered the category of connected n-manifolds (and continuous maps), then the set type endofunctors are no longer available - so is the monoid of endomorphism of this category related the quotient monoid alluded to above (same question for pointed connected n-manifolds.) In the case of the category of n-manifolds (but not pointed n-manifolds) there is a large supply of constant maps. Given an n-manifold M, the lattice of subobjects of M. Fixing another manifold N, consider all the constant maps from N to M or from N to a subobject of N. If an endofunctor F preserves constant maps (and nontrivial endofunctors of set type don't) then considering all this structure, I don't expect that it will be difficult to show that F is trivial. I would appreciate any reactions or references to similar considerations in the literature. Thanks, David Feldman University of New Hampshire =========================================================================

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regular monos, epis and categories
by Paul Taylor 22 Feb '92

22 Feb '92

I am a bit confused about what is or should be the correct definitions of regular monos, epis and categories. In the traditional examples from algebra there are plenty of limits and colimits around, so it doesn't much matter whether you say they're the (co)equalisers of arbitrary pairs or their (co)kernel pairs. Has anyone made use of these concepts in categories which have only the bare minimum (whatever that may be) of limits and colimits? Paul Taylor. =========================================================================

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Paul Taylor's question
by kelly_m＠maths.su.oz.au 21 Feb '92

21 Feb '92

I thought I had sent an answer to Paul Taylor's question, but - since I have not seen it on the bulletin board - I suppose I slipped up somewhere in transmitting it. Anyway, as Ross Street has pointed out in his message of today, the whole matter was discussed in detail in my old paper "Mono- morphisms, epimorphisms, and pull-backs", J.Austral. Math. Soc. 9(1969),124-142 - which contains some interesting results, besides being couched in the generality that Paul seeks. Were I writing it today, I should change it in places; certainly, I should speak of ARBITRARY intersections of monos, or of strong monos, or of regular monos, rather than of SMALL intersections coupled with WELLPOWEREDNESS. There are good things in Freyd-Kelly, "Categories of continuous functors I", J.Pure Appl. Algebra 2(1972),169-191,but we chose there to speak not of STRONG epis but of EXTREMAL ones, these coin- ciding under our hypotheses there. In several later papers I have refined various points from that very early paper - but it contains the guts of the matter. Max Kelly, 21 Feb. =========================================================================

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Algebraically complete categories
by mxh%dcs.edinburgh.ac.UK＠QUCDN.QueensU.CA 19 Feb '92

19 Feb '92

Is it known which functions from N to N are representable in an algebraically closed ccc? (A ccc in which initial algebras for "all" endofunctors exist. "all" includes at least those definable by x and => and those arising from initial algebras like e.g. the List functor. More precisely are there functions representable in F, which aren't in an algebraically closed category? -Martin Hofmann =========================================================================

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jobad
by MAS010＠vaxa.bangor.ac.uk 18 Feb '92

18 Feb '92

UNIVERSITY OF WALES FELLOWSHIP Applications are invited for a Fellowship for two years from October 1, 1992, open to graduates of any University, and tenable in the School of Mathematics, the University of Wales, Bangor. The Fellowship is primarily for research, and candidates should possess a research degree. Preference is likely to be given to candidates with expertise related to one or more of Algebraic Topology, Category Theory, Theoretical Computer Science. The stipend will normally be (sterling) 12,129 (first year) rising to 12,860 (second year) (under review). The Fellowship is sponsored by the University's Validation Board, and further details and application forms may be obtained from the Secretary, University of Wales Validation Board, 22 Park Place, Cardiff CF1 3DQ, United Kingdom, to whom completed forms should be returned not later than March 20, 1992. FAX: International (+44)-222-230820 UK: (0222) 230820 EMAIL for School of Mathematics, Bangor.: MAS006(a)UK.AC.BANGOR FAX for School of Mathematics, Bangor: (248)355881 =========================================================================

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