August 2000 - Categories - categories.org.au

Mac Lane and Abelian Categories
by Colin McLarty 31 Aug '00

31 Aug '00

This is to correct some remarks I made at the History of Categories meeting in Montreal in August, and to substantiate what I have written in the on-line Encyclopedia Britannica entry on Saunders Mac Lane. Mac Lane deserves credit for both the name "Abelian category" and the concept, in his paper "Duality for groups" (BAMS 1950, 485-516). Mac Lane 1950 defines "Abelian categories" in section III. He says an "Abelian category" is a category with a generator and a cogenerator and with zero object and biproducts (which he calls "free-and-direct products). He proves that in such a category, the arrows from any object A to another B form an additive commutative monoid (he actually says "semigroup", p.511); and that any such category is isomorphic to a category of commutative monoids (p.512). Of course this is far from our current definition of Abelian categories. But Mac Lane comes very close to the current definition with his "Abelian bicategories" on p.513 (using terms from 503). The definition makes heavy use of "submaps" and "supermaps". First, let me simplify by taking "submaps" to be monics, and "supermaps" to be epics. In these simplified terms Mac Lane's axioms for an "Abelian bicategory" say it is an "Abelian category" such that every arrow has an epic-monic factorization and: LC-1 the subobjects of any object A form a complete lattice. LC-2 sups of subobjects are preserved by direct images. and furthermore: ABC-1 The arrows from any A to another B form a commutative group. ABC-3 every subobject is a kernel and dually. These imply that every arrow has a kernel (sup of all subobjects killed by the arrow) and dually. Mac Lane's other axioms ABC-2 and ABC-4 and 5, are redundant in this simplified case. So in this simplified form an "Abelian bicategory" for Mac Lane in 1950 is what we would call an Abelian category with generator and cogenerator and with sups of subobjects (preserved by direct image). Mac Lane talks about pre-images of sups also, in terms not exactly ours today, and I have not worked out the connection. Now, to stop simplifying, Mac Lane's "submaps" and "supermaps" are not defined but axiomatized. A category with submaps is a category with a distinguished class of monics satisfying certain axioms (which imply that every split monic is in the class, p.499). "Supermaps" are axiomatized dually. And the axioms imply that every arrow has a "supermap-submap" factorization. Actually the axioms begin with a much narrower class of selected monics called "injections", and corresponding epics called "projections", such that every arrow factors uniquely as a projection followed by an isomorphism followed by an injection. The axioms pose further conditions, apparently all based on the idea that injections should correspond to certain set theoretic inclusions. I have not examined them closely. A submap is defined to be any isomorphism followed by an injection, and dually for supermaps and projections. Clearly the motivation for using "submaps" rather than monics was that Mac Lane hoped to extend this approach to include the category of all groups, with normal monics as the "submaps". See the footnote on p.513 saying "proofs of the first and second isomorphism theorems for all groups can be based on 'categorical' axioms". I think it fair to count this business of submaps and supermaps as an inessential complication. Grothendieck radically simplified the axioms, extended them, and found far more penetrating applications than Mac Lane had in mind in 1950. I am sure Grothendieck did not know of Mac Lane's paper directly, and I suspect that once he heard something like this was possible he worked it out for himself. One key to Grothendieck's improvements is that Grothendieck proceeds in more purely categorical terms. But the name and the concept are from Mac Lane

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Right exact functors
by Michael Barr 31 Aug '00

31 Aug '00

I thought this would be a simple application of the snake lemma, but I cannot do it all. Does anyone know if it is true and, if not, why it is true for tensor product? Suppose X is an abelian category and T:X --> Ab is a right exact functor. Say an object E is T-effaceable if for every exact sequence 0 --> A --> B --> E --> 0, the induced TA --> TB is injective. Now suppose that 0 --> E' --> E --> E'' --> 0 is an exact sequence in which E and E'' are T-effaceable. Does it follow that E' is? Does it help if you suppose that every object has an effaceable cover (and therefore an effaceable resolution)? Then you could define homology, but you want it independent of the resolution and that is where this question actually comes from, trying to prove independence of resolution. Michael

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preprint: Exactness and stability in homotopical algebra
by grandis＠dima.unige.it 30 Aug '00

30 Aug '00

The following preprint is available: M. Grandis, Exactness and stability in homotopical algebra, Dip. Mat. Univ. Genova, Preprint 419 (2000) Abstract. Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general 'categories with homotopies' having homotopy kernels and cokernels, but become more interesting under suitable 'stability' hypotheses, satisfied - in particular - by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of 'homotopical homology'. Available as ps-file at: ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/Exa.Aug00.ps (255 K) or via home page: http://www.dima.unige.it/STAFF/GRANDIS/ With best regards, Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis(a)dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752

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preprint: Relative Stone Locales
by Marta Bunge 30 Aug '00

30 Aug '00

This is to announce the availability of a paper by Marta Bunge, Jonathon Funk, Mamuka Jibladze and Thomas Streicher, "Relative Stone Locales", in the following addresses: http://www.math.mcgill.ca/~bunge/current-papers.html http://www.math.mcgill.ca/~bunge/rsl.pdf(.ps,.dvi) Thanks very much. Regards, Marta _________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com. Share information about yourself, create your own public profile at http://profiles.msn.com.

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Basic Concepts ...
by N Ghani 30 Aug '00

30 Aug '00

Is there some way in which CUP could be convinced to reprint some copies of Kelly's book "Basic Concepts of Enriched Category Theory"? I'm sure there would be a huge demand for it. Neil

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preprints: Homotopies and automorphisms...; Local subgroupoids II
by Ronnie Brown 29 Aug '00

29 Aug '00

The following preprints are available on http://arXiv.org http://arXiv.org/math.CT/0008117 R. Brown, I. Icen Homotopies and automorphisms of crossed modules of groupoids ABSTRACT: We give a detailed description of the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2-dimensional holonomy to be developed elsewhere. 19pp http://arXiv.org/math.DG/0008165 R. Brown, I. Icen, O.Mucuk Local subgroupoids II: Examples and properties ABSTRACT: The notion of local subgroupoid as a generalisation of a local equivalence relation was defined in a previous paper by the first two authors. Here we use the notion of star path connectivity for a Lie groupoid to give an important new class of examples, generalising the local equivalence relation of a foliation, and develop in this new context basic properties of coherence, due earlier to Rosenthal in the special case. These results are required for further applications to holonomy and monodromy. 18pp [The first paper is on http://www.bangor.ac.uk/ma/research/preprints/00/algtop00.html preprint 00.03, and on math.DG/9808112, and is to appear in Topology and Appl.] -- Prof R. Brown, School of Informatics, Mathematics Division, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Centre for the Popularisation of Mathematics http://www.bangor.ac.uk/cpm/

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Ronnie Letter
by Peter Freyd 28 Aug '00

28 Aug '00

Copyright 2000 Newspaper Publishing PLC The Independent (London) August 27, 2000, Sunday SECTION: SPORT; Pg. 27 LENGTH: 145 words HEADLINE: LETTER: THE JOY OF MATHS BYLINE: R. Brown BODY: IT IS UNTRUE to say, as Geoff Poole does (Letters, 13 August), that university maths courses have not changed in 30 years. It depends where you go. Prospective students should look for courses in which a project is a requirement for the final assessment; where the project does allow for a "maths in context" flavour when desired; and in which the evaluation of mathematical method is seen as a part of the whole course. Such considerations are also essential for prospective teachers. Those of us who are working on raising public awareness of mathematics believe it is essential to popularise it. It is an essential component of high-technology because it allows for precision and hence deduction and calculation. This kind of rigour may not be easy, but it makes those who can do it very employable. R BROWN University of Wales Bangor, Gwynedd

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CFP: IJCAR 2001 - International Joint Conference on Automated Reasoning
by IJCAR Publicity Chair 22 Aug '00

22 Aug '00

+------------------------------------------------------------------+ | | | IJCAR 2001 | | | | The International Joint Conference on Automated Reasoning | | | | June 18-23, 2001, Siena, Italy | | | | http://www.dii.uni-si.it/~ijcar/ | | | | CALL FOR PAPERS / TUTORIALS / WORKSHOPS | | | +------------------------------------------------------------------+ CALL FOR PAPERS =============== The International Joint Conference on Automated Reasoning (IJCAR) is the fusion of three major conferences in Automated Reasoning: CADE (The International Conference on Automated Deduction), TABLEAUX (The International Conference on Automated Reasoning with Analytic Tableaux and Related Methods) and FTP (The International Workshop on First-Order Theorem Proving). These three events will join for the first time at the IJCAR conference in Siena in June 2001. IJCAR 2001 invites submissions related to all aspects of automated reasoning, including foundations, implementations, and applications. Original research papers and descriptions of working automated deduction systems are solicited. Topics ------ LOGICS of interest include propositional, first-order, classical, equational, higher-order, non-classical, constructive, modal, temporal, many-valued, substructural, description, and meta-logics, type theory and set theory. TECHNIQUES of interest include model-elimination, tableaux, sequent calculi, resolution, connection method, inverse method, term rewriting, induction, unification, constraint solving, decision procedures, model generation, model checking, semantic guidance, interactive theorem proving, logical frameworks, and AI-related methods for deductive systems such as proof planning and proof presentation. APPLICATIONS of interest include hardware and software development, systems analysis and verification, functional and logic programming, proof carrying code, deductive databases, knowledge representation, computer mathematics, natural language processing, linguistics, planning and other AI areas. Submissions - Research papers and system descriptions ----------------------------------------------------- Submitted research papers and system descriptions must be original and not submitted for publication elsewhere. Research papers can be up to 15 proceedings pages long, and system descriptions can be up to 5 pages long. The proceedings of IJCAR 2001 will be published by Springer-Verlag in the LNAI series. All submissions must be received by January 14, 2001. Submissions that are late or too long or require substantial revision will not be considered. Authors of accepted papers will be requested to sign a form transfering copyright of their contribution to Springer-Verlag. IN THE RESEARCH PAPER CATEGORY, SUBMISSIONS OF THEORETICAL, PRACTICAL AND EXPERIMENTAL NATURE ARE EQUALLY ENCOURAGED. Submissions - Short papers -------------------------- Short papers are intended for quick dissemination of work in progress or results not substantial enough for a full research paper. Their length is limited to 10 pages. Submissions under this category will not be formally refereed, but their content and relevance will be reviewed. Those submissions accepted will be published in a technical report, which will be available at the conference. Authors of accepted papers are expected to present a brief outline of their work at the conference and to prepare a poster for display at the conference venue. The submission deadline is April 2, 2001. Submission details - All categories ----------------------------------- Authors are strongly encouraged to use LATEX2e and the Springer llncs class files. The primary means of submission is electronic. More submission details can be found at the IJCAR 2001 web site. Best Student Paper Award ------------------------ A prize of 500 Euros will be given to the best paper, as judged by the program committee, written solely by one or more students. A submission is eligible if all authors are full-time students at the time of submission. This should be indicated in the submission letter. The program committee may decline to make the award or may split it among several papers. Organization ------------ Conference Chair: Fabio Massacci University of Siena Dipartimento di Ingegneria dell'Informazione via Roma 56 53100 Siena, Italy Phone: +39 0577 234607 FAX: +39 0577 233602 Email: ijcar-cch(a)dii.unisi.it Workshop Chair: D. Hutter (Saarbr"ucken) ijcar-workshop(a)dii.unisi.it Tutorial Chair: T. Walsh (York) ijcar-tutorial(a)dii.unisi.it Program Co-Chairs: Rajeev Gor'e (ARP-ANU, Australia) Alexander Leitsch (TU-Wien, Austria) Tobias Nipkow (TU-M"unchen, Germany) collective Email address: ijcar-pch(a)dii.unisi.it Publicity Chair: P. Baumgartner (Koblenz) Treasurer: E. Giunchiglia (Genova) Invited speakers ---------------- N. Jones (DIKU, DK) L. Paulson (Cambridge, UK) H. Schwichtenberg (M"unchen, D) A. Voronkov (Manchester, UK) D. Zeilberger (Temple Univ., USA) Program committee ----------------- R. Alur (Philadelphia) F. Baader (Aachen) M. Baaz (Wien) B. Beckert (Karlsruhe) R. Caferra (Grenoble) R. Dyckhoff (St. Andrews) U. Furbach (Koblenz) D. Galmiche (Nancy) H. Ganzinger (MPI Saarbr"ucken) J. Goubault-Larrecq (INRIA Rocq.) R. H"ahnle (Chalmers) J. Harrison (Intel, Hillsboro) D. Kapur (New Mexico) H. Kautz (ATT, Florham Park) M. Kohlhase (Saarbr"ucken) Z. Manna (Stanford) P. Patel-Schneider (Bell Labs) F. Pfenning (Pittsburgh) A. Podelski (MPI Saarbr"ucken) W. Reif (Augsburg) G. Salzer (Wien) M. Vardi (Houston) Important dates --------------- (all dates in 2001) January 14 Submission deadline - Research papers and system descriptions March 19 Notification of acceptance - Research papers and system descriptions April 2 Submission deadline - Short papers April 12 Camera-ready copy due - Research papers and system descriptions April 30 Notification of acceptance - Short papers May 14 Camera-ready copy due - Short papers June 18 - June 23 IJCAR 2001 CALL FOR TUTORIALS ================== Scope ----- It is planned to hold a number of tutorials within the technical programme of the confer ence. We invite proposals for these tutorials (as well as suggestions for topics that might be covered). The topics of the tutorials can cover any area related to automated reasoning and any related cross-disciplinary areas that might be of interest (constraints, formal methods, ...). At present, the tutorials are scheduled to take place on Monday 18th and Tuesday 19th June. How to Propose a Tutorial ------------------------- Proposals should be in English and between one and two pages in length. They should contain: * The title of the tutorial. * The names, and affiliations of the person or persons who will present the tutorial. * A brief technical description of the topics covered by the tutorial. * Contact details (email, web page, phone, fax, etc). * A list of tutorials previously given in this or related areas. Proposals should be submitted electronically (in ASCII, Ghostscript compatible Postscript or LaTeX) at the follwing address: Toby Walsh, IJCAR Tutorial Chair Artificial Intelligence Group Department of Computer Science University of York York YO10 5DD, U.K. Email: ijcar-tutorial(a)dii.unisi.it Tel: +44 1904 432745 Fax: +44 1904 432767 Important dates --------------- Tutorial proposal deadline: January 15, 2001 Notification of acceptance: January 29, 2001 Tutorials: June 18+19, 2001 CALL FOR WORKSHOPS ================== Scope ----- Researchers and practitioners are invited to submit proposals for workshops on IJCAR related topics as mentioned in the "Call for Papers". Proposals that promise to bring new topics into IJCAR, of either practical or theoretical importance, or provide a forum for more detailed discussion on central topics of continuing importance are also welcomed. Workshops that close the gap between automated reasoning and related areas, like for instance formal methods or software engineering, are especially encouraged. Recent workshops of participating conferences have included, for instance, automated model building, automation of proofs by induction, empirical studies in logic algorithms, mechanization of partial functions, proof search in type-theoretic languages, strategies in automated deduction, automated theorem proving in software engineering and in mathematics, and integration of symbolic computation and deduction. Submission Details ------------------ Anyone wishing to organize a workshop in conjunction with IJCAR should send in postscript format (e-mail preferred) a proposal no longer than two pages to the workshop chair (ijcar-workshop(a)dii.unisi.it) by January 1, 2001. Proposals should consist of two parts. First, a short scientific justification of the proposed topic, its significance and the particular benefits of the workshop. A second part should include the proposed format and agenda, the procedures for selecting papers and participants, and contact information for the organizers. In particular it should also include estimated dates for paper submissions, acceptance of notification (before May 1, 2001) and camera ready copy. Proposals will be evaluated, and decisions will be communicated by January 15, 2001. Further information about the arrangements for workshops can be obtained from the IJCAR 2001 Web site. Important dates --------------- Workshop proposal deadline: January 1, 2001 Notification of acceptance: January 15, 2001 Workshops: June 18+19, 2001 Workshop chair -------------- Dieter Hutter (Saarbr"ucken, D) ijcar-workshop(a)dii.unisi.it Sponsors -------- UniversitÃ degli Studi di Siena, the University of Siena AI*IA, l'Associazione Italiana per l'Intelligenza Artificiale CADE Inc., The Conference on Automated Deduction. EATCS, The European Association for Theoretical Computer Sciences. ECCAI, The European Coordinating Committee on Artificial Intelligence. ERCIM, The European Research Consortium for Informatics and Mathematics. IJCAI Inc., The International Joint Conferences on Artificial Intelligence. MPS, Monte dei Paschi di Siena -- Peter Baumgartner phone: +49 261 287 2777 mail: peter(a)uni-koblenz.de fax: +49 261 287 2731 WWW: http://www.uni-koblenz.de/~peter/

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CFP Special Issue JSC on Computer Algebra and Mechanized Reasoning
by Manfred Kerber 19 Aug '00

19 Aug '00

Call for Papers Journal of Symbolic Computation Special Issue on Computer Algebra and Mechanized Reasoning Guest Editors: Tom�s Recio, Manfred Kerber _________________________________________________________________ AIM The special issue is related to topics discussed in the context of the ISSAC-2000 symposium and the CALCULEMUS-2000 symposium in August 2000 in St Andrews, Scotland. We invite any work that substantially extends ideas and topics presented in St Andrews. Typical ISSAC-2000 relevant topics are: * Algorithmic mathematics: Algebraic, symbolic, and symbolic-numeric algorithms including: simplification, polynomial and rational function manipulations, algebraic equations, summation and recurrence equations, integration and differential equations, linear algebra, number theory, group computations, and geometric computing. * Computer science: Theoretical and practical problems in symbolic mathematical manipulation including: computer algebra systems, data structures, computational complexity, problem solving environments, programming languages and libraries for symbolic-numeric-geometric computation, user interfaces, visualization, software architectures, parallel or distributed computing, mapping algorithms to architectures, analysis and benchmarking, automatic differentiation and code generation, automatic theorem proving, mathematical data exchange protocols. * Applications: Problem treatments incorporating algebraic, symbolic, symbolic-numeric and geometric computation in an essential or novel way, including engineering, economics and finance, architecture, physical and biological sciences, computer sciences, logic, mathematics, statistics, and uses in education. CALCULEMUS-2000 relevant topics include all aspects related to the combination of deduction systems and computer algebra systems. We also explicitly encourage submissions of results from applications and case studies where such an integration proves particularly important. Typical topics are: * Integration/combination of computer algebra systems/algorithms and deduction systems (either automated theorem provers, or proof-development systems) * Incorporation of deduction techniques in computer algebra * Incorporation of computer algebra techniques in deduction Prospective contributors are warmly invited to contact the guest editors to discuss the suitability of topics and papers. Submission Guidelines ISSAC-related papers must be submitted to Tom�s Recio, CALCULEMUS-related papers to Manfred Kerber. For submissions please follow the instructions provided at http://www.academicpress.com/www/journal/TeX-uk/LaTeXFP.htm. Electronic submissions are strongly encouraged, and may be sent as one e-mail (MIME attachments are allowed). The message should contain (i) the abstract in ASCII and (ii) the whole paper in Postscript. The Postscript form must be interpretable by Ghostscript, and must use standard fonts, or include the necessary fonts. Authors who cannot meet these requirements should submit 5 hard copies by post instead. All submitted papers will be refereed according to the usual JSC refereeing process. To aid planning and organization, we would appreciate an email of intent to submit a paper (including author information, a tentative title and abstract, and an estimated number of pages) as early as possible. Important Dates Submission of papers: 1 November 2000 Notification of acceptance/rejection: 15 February 2001 Submission of revised versions: 15 March 2001 Delivery of camera-ready copies: 1 May 2001 Publication of special issue: planned around July 2001 Guest Editors' Addresses: Tom�s Recio Manfred Kerber Departamento de Matem�ticas School of Computer Science Estad�stica y Computaci�n The University of Birmingham Facultad de Ciencias Edgbaston Universidad de Cantabria Birmingham Avenida de los Castros, s/n B15 2TT 39071 Santander, Espa�a England phone: +34 942 20 14 33 phone: +44 121 414 4787 fax: +34 942 20 14 02 fax: +44 121 414 4281 recio(a)matesco.unican.es M.Kerber(a)cs.bham.ac.uk This information is available as: http://www.cs.bham.ac.uk/~mmk/events/jsc01.html JSC Editor's Web Page: http://www.math.ncsu.edu/~hong/jsc.htm

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errata to "Practical Foundations"
by Paul Taylor 19 Aug '00

19 Aug '00

ERRATA TO "PRACTICAL FOUNDATIONS OF MATHEMATICS" I would like to thank David Benson, Peter Johnstone, Thomas Streicher and Krzysztof Worytkiewicz for their comments in response to my request for corrections. If you know of any other mistakes, please tell me ASAP. First, I apologise to Heinrich Kleisli, Dito Pataraia, Maria Cristina Pedicchio, Dietmar Schumacher and V. Z\"oberlein for my mistakes in their names on pages 179, 403, 533, 540, 563, 566 and 572. Only one actual falsity has come to my attention: exactly a year after I got the first copy of the book, Thomas sent me a simple counterexample to the claim that fibrations preserve pullbacks (Exercise 9.4, page 523). I am very pleased at how few errors have shown up in 17 months, and how trivial they are. This is a tribute to the care that nearly 40 people took in reading individual chapters of the book. Though I am sure that this policy of only sending out single chapters irritated the people in question, it did prove to be a very successful way of ensuring that the entire book got careful and reasonably even attention. In contrast, the most assiduous reader of "Proofs and Types" only got two thirds of the way through the book, starting from the beginning. You may like to read Peter Johnstone's review for the Zentralblatt at http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/Johnstone-review.html (it's on my web site because it's unreadable on the Zentralblatt site). There is an 18 page summary of the contents of the book at http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/summary.dvi http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/html/summary.html The DVI version is intended for you to print, the HTML one for linking into the full text of the narrative that's also on the web. The bibliography is available as a BibTeX file at http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/prafm-ref.bib I understand that 1000 copies were printed in March 1999, of which 825 had been sold when I asked last week, and 400 new ones are to be printed. For over seven years' work, I have had about 3000 pounds in royalties and no permanent job. SIGNIFICANTLY WRONG SYMBOLS AND WORDS p. 37, Lemma 1.6.3, proof box, line 6: should be \exists y.\gamma\land\phi[y] in the left-hand box. p. 144, intro to Section 3.5, and p. 176, Exercise 3.19: the sum of posets or dcpos (not domains). p. 275, final paragraph of Definition 5.5.1: if c then ... (not a). p. 285 Corollary 5.6.12: v(y)=x_1, not u_y=x_1. p. 290, diagram for Lemma 5.7.6(e): (German) f;m and z;n instead of f;n and z;m. p. 362, Exercise 6.23: T\Theta instead of T(\Gamma\times\Theta) at the top right of the diagram. p. 397, introduction to Section 7.5: \mu = U\epsilon F. p. 415, Notation 7.7.3(b) Q_X: H_X \to U `X'. p. 519, Proposition 9.6.13 [c=>a]: The infinitary version of Example 2.1.7, rather than of its converse {Exercise 2.14). The full list of actual changes (including broken sentences and places where inserting a couple of extra words could improve understandability) is at http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/errata.html REMARKS ON OTHER POINTS THAT I FEEL I UNDERSTAND BETTER SINCE PUBLICATION The Substitution Lemma (1.1.5, p. 7) correctly forbids x to be free in a, because the substitution [a/x]^*u is meant to result in a term that doesn't involve x. See Definition 4.3.11(b) for why. Arguably, contexts should be emphasised from the start, especially as, in this book, they turn into the objects of the classifying category. That the proof of the Substitution Lemma tolerates x being free in a entirely misses the point, and merely illustrates the tyranny of notation over concepts In his review, Peter Johnstone teased me for Exercise 1.1 (Bo Peep's theorem). To add fuel to his fire, I now think that the natural proof of this - the very earliest theorem in mathematics - is essentially the argument needed for the Schr\"oder-Bernstein theorem. It would be nice to be able to get an anthropologist to understand that this *is* a significant theorem, and to try to find out at what stage in the development of human cultureS it was learned. p. 358, Remark 6.7.14: It is probably true in the concrete case of ordinals in Pos and Set that a sh-coalgebra is well founded in the sense of Definition 6.3.2 iff < is a well founded relation (Definition 2.5.3). Inability to formulate the abstract result for ordinals in categories between which there is an adjunction F -| U is the reason why I have not finished [Tay96b]. p. 502, Example 9.4.11(d): The closed point of the Sierpi\'nski space does classify closed subsets intuitionistically. Any point of this space can be expressed as the join of a directed diagram taking only \bot and \top as values, whilst (the dual of) the equation in Exercise 9.57 characterises support classifiers [Tay98]. Section 9.4: Beware that, whilst my approach to the Beck-Chevalley condition does ensure that pullbacks of >---|>-maps are >---|>-maps, such pullbacks do not always exist in the category of locally compact spaces. Paul

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