Papers in Honour of Bernhard Banaschewski: Proceedings of the BB Fest 96, a Conference Held at the University of Cape Town, 15–20 July 1996, on Category Theory and its Applications to Topology, Order and Algebra PDF
Preview Papers in Honour of Bernhard Banaschewski: Proceedings of the BB Fest 96, a Conference Held at the University of Cape Town, 15–20 July 1996, on Category Theory and its Applications to Topology, Order and Algebra
PAPERS IN HONOUR OF BERNHARD BANASCHEWSKI PAPERS IN HONOUR OF BERNHARD BANASCHEWSKI Proceedings of the BB Fest 96, a Conference Held at the University of Cape Town, 15-20 July 1996, on Category Theory and its Applications to Topology, Order and Algebra Edited by GUILLAUME BRUMMER AND CHRISTOPHER GILMOUR Department ofM athematics and Applied Mathematics, University of Cape Town, South Africa Reprinted from Applied Categorical Structures, Vol. 8, Nos. 1-2 (2000), with additional materials SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y. A.C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-5540-8 ISBN 978-94-017-2529-3 (eBook) DOI 10.1007/978-94-017-2529-3 Printed on acid-free paper AlI Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part ofthe material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Table of Contents Preface JOHN ISBELL / Relatively Parametrizable Subalgebras 3 PETER JOHNSTONE / An 'Unsitely' Result on Atomic Morphisms 7 TILL PLEWE / Quotient Maps of Locales 17 PAUL BANKSTON / Some Applications of the Ultrapower Theorem to the Theory of Compacta 45 H. HERRLICH and A. PULTR / Nearness, Subfitness and Sequential Regularity 67 DOMINIQUE BOURN / The Structural Nature of the Nerve Functor for n-Groupoids 81 MARCEL ERNE / Prime Ideal Theory for General Algebras 115 KARL H. HOFMANN / An Illustration of the Power of Structure Theory 145 GUNTHER RICHTER / Coreflectivity of E-Monads and Algebraic Hulls 161 DIKRAN DIKRANJAN and HANS-PETER KUNZI / Separation and Epimorphisms in Quasi-Uniform Spaces 175 K. A. HARDIE, K. H. KAMPS and R. W. KIEBOOM / A Homotopy 2-Groupoid of a Hausdorff Space 209 R. LOWEN and P. WUYTS / A Complete Classification of the Si multaneously Bireflective and Bicoreflective Subconstructs of Approach Spaces 235 E. LOWEN-COLEBUNDERS and C. VERBEECK / Exponential Ob- jects in Corefiective or Quotient Reflective Subconstructs: A Comparison 247 HELMUT ROHRL / Completeness and Cocompleteness of RSmod 257 lN SUNG SA HONG and YOUNG KYOUNG KIM / Cauchy Complete Nearness Spaces 271 ROMAINE JAYEWARDENE and OSWALD WYLER / Categories of Relations and Functional Relations 279 RENATO BETTI / Weak Equivalence of Internal Categories 307 TEMPLE H. FAY and STEPHAN V. JOUBERT / Isolated Submodules and Skew Fields 317 A. TOZZI and V. TRNKOVA / Clone Segments of the Tychonoff Modification of Space 327 XIAOMIN DONG and WALTER THOLEN / Representation of Rela- tions by Partial Maps 339 JORGE PICADO / Structured Frames by Weil Entourages 351 JURGEN REINHOLD / Finite Intervals in the Lattice of Topologies 367 D. BABOOLAL / Local Connectedness Made Uniform 377 PAUL CHERENACK / Frolicher versus Differential Spaces: A Prelude to Cosmology 391 TEMPLE H. FAY / Weakly Hereditary Initial Closure Operators 415 D. N. GEORGIOU and B. K. PAPADOPOULOS / Strongly lJ-Con- tinuous Functions and Topologies on Function Spaces 433 K. R. NAILANA / (Strongly) Zero-Dimensional Partially Ordered Spaces 445 SHU-HAO SUN / Structure Sheaves for Additive Categories 457 VESKO VALOV and DUMISANI YUMA / Lindelof Degree and Function Spaces 475 STEFAN VELDS MAN / Connectednesses and Disconnectednesses of Petri Nets 485 Applied Categorical Structures 8: 1, 2000. G. BrUmmer & C. Gilmour ( eds), Papers in Honour ofB ernhard Banaschewski. 1 © 2000 Kluwer Academic Publishers. Preface In July 1996 a conference, named 'BB Fest 96', was held at the University of Cape Town in honour of the seventieth birthday of Professor Bernhard Banaschewski of McMaster University, Canada. The University of Cape Town hosted the conference in recognition of Banaschewski's inspiring contribution over many years to the Cape Town school of categorical topology. The proceedings of the BB Fest 96 are taking the twofold shape of a selection of papers in Applied Categorical Structures as well as a larger selection in a hard cover book published by Kluwer Academic Publishers. Both selections bear the title 'Papers in Honour of Bernhard Banaschewski'. We are grateful to Bob Lowen, Editor-in-Chief of Applied Categorical Structures, for inviting us to act as guest editors. Likewise we are grateful to all the authors and the many referees for their work, and to all members of the Cape Town group for their help in organising the BB Fest 96. All papers appearing here have been subjected to the regular refereeing proce dures of Applied Categorical Structures. GUILLAUME BROMMER CHRISTOPHER GILMOUR Applied Categorical Structures 8: 3--6, 2000. G. Briimmer & C. Gilmour (eds), Papers in Honour ofB ernhard Banaschewski. 3 © 2000 Kluwer Academic Publishers. Relatively Parametrizable Subalgebras Dedicated to Bernhard Banaschewski for his 70th birthday JOHN ISBELL Department of Mathematics, State University ofN ew York, Buffalo, N.Y.14214 U.S.A. (Received: 1 October 1996; accepted: 3 August 1997) Abstract. m-parametrizable algebras are characterized by a projectiveness property. Mathematics Subject Classification (2000): 08A99. Key words: parametrizable, relatively parametrizable, projective, k-projective, projectively filtered. Introduction An algebra A, of any type, is called parametnzable (respectively m-parametri zable) if for all k (for k = m), the solution set of any system of algebraic equations in Ak is a union of parametric sets. The basic characterization theorem is [3] that A is parametrizable if and only if in the smallest variety of algebras V A to which A belongs, A is projective. The central result of this paper is a localized characterization; oversimplifying drastically, a particular solution set is a union of parametric sets if and only if it is analyzable by mappings into projectives. The main application is to m-parametrizability of A, which is characterized as follows: for every m-generated subalgebra S of A, there exists a factorization of i : SeA across a projective algebra P of VA, S ~ P ~ A. The key definition: a subalgebra S of A is relatively parametrizable if for each point p = (Pa) of Sk (k any cardinal) there is a parametric mapping Jr: Aj ~ Ak representing p (p E Jr(Aj» and identically satisfying all algebraic equations satisfied by p (each Jr(x) satisfies them). Precise statement of the main result: S is relatively parametrizable if and only if i : SeA factors across a projective algebra P of V A. That does not quite tell us when a particular solution set is a union of parametrized sets, but (a) this is where the theorem is: the natural general ization of "parametrizable = projective in V A'" and (b) it has at least the indicated application to m-parametrizability. Note, relative parametrizability is very much a positional property. It is neither necessary nor sufficient that S be parametrizable. Indeed, being parametrizable and generic in V A is sufficient, as that makes S projective in V A. Every subalgebra of a relatively parametrizable subalgebra is relatively parametrizable. 4 JOHN ISBELL Of course k-parametrizability is related to K-freeness in abelian groups, cf. [4]. It is more nearly like K-projective filtration in Boolean algebras [1]. I do not know if they are equivalent (k-parametrizability to k+ -projective filtration). The filtra tion version seems much more restrictive; it requires projective subalgebras, and there must be a neatly arranged family of them. But a counterexample would be desirable. Anyway, the evident one-way relation gives two k-parametrizability corollaries of theorems of S. Fuchino and S. Shelah [1]. (1) For Boolean algebras, w2-parametrizability does not imply U>:3-parametrizability. (2) If V = L and there are no weakly compact cardinals, then for each regular aleph k there are Boolean algebras which are m-parametrizable if and only if m < k. In [3] I stressed that the results apply to infinitary algebras, even such badly behaved ones as complete Boolean algebras. Globally, complete Boolean algebras have a proper class of w-ary operations, but our viewpoint is never global. At most, we work with a variety, often the smallest variety that includes a given algebra, which is no problem for complete Boolean algebras (or worse, such as complete closure algebras [2]). This applies equally to results in [4]. Results The algebraic envelope of a point u E Ak is defined [4] as the set E(u) of all points v which satisfy every equation satisfied by u. A parametric analysis of u is defined as a parametric mapping </J: Aj ---+ Ak which takes u as a value and has </J(Aj) c E(u). So the new definition of a relatively parametrizable subalgebra S of A, given above, says just that every point u of Sk, considered as a point of A k , has a parametric analysis. We shall use Lemma 1.2 from [4]: For every point p E pk over a projective algebra in a variety V there is a k tuple in a V-free algebra on generators xp (f3 < m), ¢ = (¢OI(Xp»OI<b such that the parametric mapping <1>: pm ---+ pk defined <I>«up)p<m) = (¢OI(Up»OI<k takes p as a value, and every algebraic equation satisfied by p is satisfied by ¢. THEOREM 1 The following properties of a subalgebra SeA are equivalent. (a) S is relatively parametrizable. (b) Some point p E sn whose coordinates generate S has a parametric analysis in An. (c) The inclusion i: SeA factors across a projective object of V A. Proof (a) :::} (b); if every point over S has a parametric analysis, a generating point p does. Assume (b), with a parametric analysis n: Am ---+ An of a generating p E sn. Choose a point t of Am for which net) = p; more fully, p = (POl)OI<n, n = (nOl)' t = (tj) j<m, and nOl(t) = POI' The free algebra F of V A on m generators Zj (j < m) can be described as a subset of the set of all functions from Am to A, viz. those expressible as (composite) operations. We define homomorphisms IT: S ---+ F and T: F ---+ A. At each point s of S, IT(s) is defined by expressing s anyhow RELATIVELY PARAMETRIZABLE SUBALGEBRAS 5 as w«Pa» and taking n(s) = w«1l'a(Zj)j<m)a<n). If s can be expressed also as v«Pa», then E(p) is contained in the algebraic set where v = w; v«1l'a(Xj») = w«1l'a(Xj») for all (Xj) E Am, hence also for Xj = Zj E F. So n is well defined, and therefore homomorphic. T is just the substitution Zj f-+ tj' T(w«zj») = = = = w«tj», homomorphic. We have Tn(Pa) T(1l'a(Zj)j<m) 1l'a(t) Pa, so as P generates S, Tn = i. (c) is proved. Tn Assume (c), i: SeA factoring as across P projective in VA. For any = = point v (va) E Sk we have P (n(va» E pk. By Lemma 1.2, recalled above, there is (CPa(xfJ»a<k in a free algebra on m generators xfJ' with P repre sentable as (CPa(ufJ» for some U = (ufJ) E pm, and cP satisfying all equations that P satisfies. I claim that v has a parametric analysis <1>: Am -+ Ak defined = = <I>«afJ)fJ<m) (cpa(afJ»a<k. <I> takes the value v at t Tm(u), since each CPa(t) = = = = = CPa(T(ufJ)fJ<m) T(CPa(ufJ)fJ<m) T(Pa) Tn(va) Va. As for the equations, cP = (CPa) satisfies all equations that P satisfies; P = nk(v) satisfies all equa tions that v satisfies; and each <I>«ap» = (cpa«ap») satisfies all equations that cP satisfies. Thus (a) holds, and the proof is complete. 0 Theorem 1 gives us two characterizations of k-parametrizable algebras. We call an algebra A k-projective in a variety V if given any surjective morphism of V. t: T -+ Y, and any morphism f: A -+ Y and k-generated subalgebra i: SeA, there exists f*: S -+ T such that commutes: tf* = fi. By a singular subalgebra of A we mean just a morphism B -+ A. It is finitely generated, or projective, or ... , if lJ is so. THEOREM 2 The following properties of an algebra A are equivalent. (a) A is k-parametrizable. (b) Every k-generoted subalgebra of A is: a sBloolgebra ofa V A -projective singwl:ar subalgebra of A. (c) A is k-projective in VA. Proof Evidently A is k-parametrizable if and only if every point of Ak has a parametric analysis. Evidently again, this is equivalent to every k-generated sub algebra being relatively parametrizable. So the equivalence o(a) and (b) fonows from Theorem 1. Given (b), consider smjective t: T -+ Y in V A and f: A -+ Y, and any k-generated subalgebra i: S -+ A. We have a VA-projective P and a factorization of i as vu, u: S -+ P and v: P -+ A. Then by projectiveness, = = fv: P -+ Y lifts to c: P -+ T, tc being fv. Put f* cu; tcu is fvu fi as required. Finally, (c) implies (b) by putting Y = A and taking a surjection t: P -+ A from a V A -projective. 0
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