November 1993 - Categories - categories.org.au

(A\otimes B)(C\otimes D)=(AC)\otimes (BD)
by stiller＠blaze.cs.jhu.edu 02 Dec '93

02 Dec '93

Is there a categorical analogue of the following well-known matrix identity: If A, B, C, D are matrices over a field and juxtaposition denotes matrix multiplication and \otimes denotes Kronecker product, then (A\otimes B)(C\otimes D)=(AC)\otimes (BD) whenever the dimensions are consistent? Note: I am a category theory novice, but the proof of this identity uses so few of the field axioms, and is so useful, (and curiously looks like the interchange of horizontal and vertical compositions of natural transformations which I presume is coincidental) that I was just curious if an analogue of this identity is true in more general categories than matrices over rings. -- Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University. stiller(a)cs.jhu.edu. "Tertan I am, but what is Tertan? Of this time, of that place, of some parentage, what does it matter?" ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

5 4

Initial model of higher sketches
by Hiroyuki Miyoshi 29 Nov '93

29 Nov '93

Does anyone know results about the existance of initial model (in Cat or other categories) of 2-sketch or V-sketch with indexed limits, and that of other equivalent (or more general) formulations (e.g. higher-dimensional sketch of Power and Wells)? I think of results analogous to the existance of the (family of) initial model in Set of FL or FLS sketch. Any relevant informations and references are much welcome. Thanks in advance. ---- Hiroyuki Miyoshi Department of Mathematics, Faculty of Science and Technology, Keio Univ. OFFICE: c/o Nobuo Saito, Faculty of Environmental Information, Keio Univ. 5322 Endoh, Fujisawa 252, JAPAN EMAIL: miyoshi(a)slab.sfc.keio.ac.jp ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

3 2

Sydney ftp site: current contents
by walters_b＠maths.su.oz.au 24 Nov '93

24 Nov '93

SYDNEY CATEGORIES COMBINATORICS and COMPUTER SCIENCE CCC Material Available by Anonymous FTP from maths.su.oz.au 129.78.68.2 ======================================================================== This file is README in the sydcat directory of maths.su.oz.au, 129.78.68.2, accessible by anonymous ftp. The sydcat directory is for FTP distribution of recent publications, programs, seminar listings and other material of the Sydney Category Theory, Combinatorics, and Computer Science Group. ======================================================================== This group consists of staff and students at the University of Sydney Macquarie University University of New South Wales University of Technology Sydney Sydney, Australia, including the following: Murray Adelman murray(a)macadam.mpce.mq.edu.au Brian Day Lee Flax flax(a)macadam.mpce.mq.edu.au Robbie Gates gates_r(a)maths.su.oz.au Amitavo Islam islam_a(a)maths.su.oz.au Barry Jay cbj(a)socs.uts.edu.au Mike Johnson mike(a)macadam.mpce.mq.edu.au Giulio Katis katis_p(a)maths.su.oz.au Max Kelly kelly_m(a)maths.su.oz.au Mark Leeming leeming_m(a)maths.su.oz.au Stephen Ma ma_s(a)maths.su.oz.au Wafaa Khalil khalil_w(a)maths.su.oz.au Wesley Phoa wes(a)cs.unsw.oz.au Usha Sridhar sridhar_u(a)maths.su.oz.au Ross Street street(a)macadam.mpce.mq.edu.au Sun Shu-Hao sun_s(a)maths.su.oz.au Dominic Verity domv(a)macadam.mpce.mq.edu.au Henry Weld weld_h(a)maths.su.oz.au Bob Walters walters_b(a)maths.su.oz.au Karl Wehrhahn wehrhahn_k(a)maths.su.oz.au =======================Available material=============================== The following files and directories are available: ADDRESS lists are in sydcat/addresses BIBLIOGRAPHY files are in sydcat/bibliography BOOK descriptions are in sydcat/books PAPERS:dvi files of papers of author are in sydcat/papers/author CONFERENCE details are in sydcat/conference SEMINARS - some details are in sydcat/seminars SOFTWARE - programs are in sydcat/software Here is the current list of contents: addresses/catcurrent The Category Theory address list maintained by Max Kelly and Michael Johnson Updated regularly addresses/structdir Vaughan Pratt's email address list Updated October 1993 Release 3.0 Master copy: Boole.Stanford.EDU:~ftp/pub/struct.dir Maintainer: Vaughan Pratt, pratt(a)cs.stanford.edu bibliography A directory for bibliographies: contains some Bibtex files of sydcat publications bibliography/reports List of Research Reports of the Pure Mathematics Department, University of Sydney, Australia Updated regularly. books/hibi Contents and bibliographic details of Takayuki Hibi "Algebraic combinatorics on convex polytopes" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia books/walters Information about, corrections to, and bibliographic details of R.F.C. Walters "Categories and Computer Science" published by Carslaw Press in Australia and by Cambridge University Press elsewhere Contents and bibliographic details of RFC Walters "Number theory an introduction" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia books/wehrhahn Contents and bibliographic details of KH Wehrhahn "Combinatorics: an introduction" Carslaw Publications PO Box 615 Glebe, NSW 2037 Australia conferences Details of conferences papers/jay A directory containing papers of Barry Jay Title: Compositional Characterization of Observable Program Properties Authors: B. Steffen,C. Barry Jay, M. Mendler Process: Aarhus TR and accepted by Journal Title: Coherence in Category Theory and the Church-Rosser Property Author:CBJ Process: published in Studia Logica Title: Modelling reduction in confluent categories Author: CBJ Process: Published in the Durham Proceedings Title: Fixpoint and Loop Constructions as Colimits Author: CBJ Process: Como Proceedings Title:Long $\beta\eta$ normal forms and confluence (revised) Author: CBJ Process: static cf virtues.tex Title: Extending properties to categories of partial maps Author: CBJ Process : static Title: Partial Functions, Ordered Categories, Limits and Cartesian Closure Author: CBJ Process: proceedings of the HOW Title: Tail recursion through universal invariants Author: CBJ Process: to appear in TCS papers/johnson M.S.Johnson Linear term rewriting systems are higher dimensional string rewriting systems, {\em Proc of the Institute for Mathematics and its Applications\/} (1991), 101--110. M.S.Johnson (with G.P.~Monro and C.N.G.~Dampney) A mathematical foundation for ERA, {\em Proc of the Institute for Mathematics and its Applications\/} (1991), 77--84. M.S.Johnson (with R.F.C.~Walters) Category theoretic modelling of digital circuits and systems, {\em Proc of the Pan-Commonwealth Conference on Mathematical Modelling} {\em in Circuit Design}, Commonwealth Science Council (1992), 199--213. M.S.Johnson (with R.F.C.~Walters) Algebra objects and algebra families for finite limit theories, {\em Journal of Pure and Applied Algebra} {\bf 83} (1992) 283--293. M.S.Johnson (with S.-H. Sun) Remarks on representations of universal algebras by sheaves of quotient algebras, M.S.Johnson {\em Proceedings of the Canadian Mathematical Society\/} {\bf 13} (1992) 299--307. M.S.Johnson (with C.N.G.~Dampney and P.~Deuble) Taming large complex information systems, M.S.Johnson {\em Proceedings of Complex Systems} '92, IOS Press, Amsterdam, 210--222. M.S.Johnson (with R.~Buckland) An application of logic programming in pure mathematics, to appear in {\em Proceedings of ACSC}, 1993. M.S.Johnson (with C.N.G.~Dampney) Category theory and information systems engineering, {\em Proceedings of AMAST93}, Unviersity of Twente, Holland, 95--103. M.S.Johnson On the value of commutative diagrams in information modelling, to appear in {\em Springer Workshops in Computing}, 16 pages. papers/kelly G.M. Kelly, Stephen Lack: Finite-product-preserving functor, Kan extensions, and strongly-finitary 2-monads G.M. Kelly, Stephen Lack, R.F.C. Walters, Coinverters in categories with structure galaus.dvi G.Janelidze and G.M.Kelly, Galois theory and a general notion of central extension. (A4 version) galam.dvi G.Janelidze and G.M.Kelly, Galois theory and a general notion of central extension. (American quarto version) janmark.dvi G.Janelidze and L.Ma'rki, Radicals of ringa and pullbacks. papers/phoa Wesley Phoa bohm.ps "From term models to domains" --describes a category of `synthetic domains' for the closed term model in which terms with the same Bohm tree are identified Wesley Phoa graph.ps "Building domains from graph models" --describes a category of `synthetic domains' in the realizability topos arising from the r.e. graph model of the lambda-calculus Wesley Phoa pcf.ps "A note on PCF and the untyped lambda-calculus" --proof of computational adequacy of an untyped translation of call-by-name PCF Wesley Phoa poly.ps "A simple categorical semantics for first-order polymorphism" --describes how any cartesian closed category can be used to model ML polymorphism, using the notion of `polynomial category' M.P. Fourman and Wesley Phoa sml.ps "A proposed categorical semantics for Pure ML" (with M. P. Fourman, LFCS) --sketch of a semantics for SML using synthetic domain theory; focuses on the Modules system Wesley Phoa subtypes.ps "Using fibrations to understand subtypes" --informal account of categorical models for subtyping and bounded quantification Wesley Phoa synth.ps "Effective domains and intrinsic structure" --describes a category of `synthetic domains' in the effective topos Wesley Phoa tech.ps Replacing fibs.ps, topoi.ps, eff.ps, these notes provide an introduction to (some aspects of) a) fibrations and polymorphic lambda calculus b) constructive logic, categorical logic and topos theory c) Kleene realizability; PERs and omega-sets d) the effective topos; modest sets and how they model polymorphism They assume some basic knowledge of category theory, logic and typed lambda calculus. No familiarity with indexed categories or with categorical logic or topos theory is required. The notes do not attempt to be comprehensive, but simply try to give a reasonably relaxed account of the material. They are about 150pp including the index and appendixes. There are plenty of exercises. papers/sun Shu-Hao Sun, RFC Walters, Representations of modules and cauchy completeness Shu-Hao Sun, Adjunction of the associated sheaf functor of non-commutative rings Shu-Hao Sun, Non-commutative Deligne formula Shu-Hao Sun, Biregular rings and their duality Shu-Hao Sun, Non-commutative quasi-coherent sheaves Shu-Hao Sun, Equivalence of algebraic and geometric local cohmology Shu-Hao Sun, Structure sheaves for non-commutative rings Shu-Hao Sun, Duality on compact prime ringed spaces Shu-Hao, Generalized Grothendieck topologies papers/walters Aurelio Carboni, Stephen Lack, R. F. C. Walters, An introduction to extensive and distributive categories, Jun92. S. Carmody, R.F.C. Walters, The Todd-Coxeter Procedure and Left Kan Extension, Mar91 S. Carmody, R.F.C. Walters, Computing quotients of actions of a free category, Mar91. M. S. Johnson, R.F.C. Walters, Algebra Families, Apr92 P. Katis, N. Sabadini, RFC Walters, On discrete dynamical systems and concurrency, Dec93 G.M. Kelly, Stephen Lack, R.F.C. Walters, Coinverters in categories with structure, Jun92 Wafaa Khalil, R.F.C. Walters, An imperative language based on distributive categories II Apr92 Wafaa Khalil, R.F.C. Walters, Functional processors and operations on them in extensive categories, May93 Wafaa Khalil, Eric Wagner, R.F.C. Walters, Fix-point semantics for programs in distributive categories, May93 Mark Leeming, R.F.C. Walters, Computing left kan extensions using the Todd-Coxeter procedure N. Sabadini, S. Vigna, RFC Walters An automata-theoretic approach to concurrency through distributive categories: on morphisms, Jan93 N. Sabadini, R.F.C. Walters On functions and processors: an automata-theoretic approach to concurrency through distributive categories, Nov92 N. Sabadini, H. Weld, R.F.C. Walters, Distributive automata and asynchronous circuits, May93 N. Sabadini, S. Vigna, RFC Walters A notion of refinement for automata, Jan93 Shu-Hao Sun, RFC Walters, Representations of modules and cauchy completeness, Mar93 R.F.C. Walters, An imperative language based on distributive categories, 91 seminars/sydcat sydcat.tex is a listing of seminars given at the Sydney Category Seminar Not being currently maintained. seminars/cics cics.tex is a listing of seminars given at the Sydney Categories in Computer Science Seminar. software/kan_1.0 kan (vers 1.0) (Sean Carmody, Craig Reilly, Bob Walters) An implementation of the algorithm developed in 1990 by Carmody & Walters to compute (finite) left Kan extensions is now operational (programmed by Reilly and Carmody). There are also some sample input files as well as a file called KAN.info which gives further details of the program and one called README which describes the sample input files. If you experiment with the program, we would be very interested to hear any comments or suggestions (especially with regards any bugs which you -- hopefully won't -- find). Future versions of the program will include a more standardised i/o format, and will renumber the elements of the output sets will be of the form {1,2,..,n} (a set which may currently be given as {1,3,7} would become {1,2,3}). Sean Carmody. email: carmody_s(a)maths.su.oz.au 21 May 1991 software/kan_2.0 A later version of kan, but Sean left Sydney for Cambridge before it was properly documented. software/buckland Representing pasting schemes in Prolog A program written by Richard Buckland, with Michael Johnson. For details see programs/buckland/readme Further details email t-richbu(a)microsoft.com mike(a)macadam.mpce.mq.edu.au -- Bob Walters Department of Pure Mathematics, University of Sydney, NSW 2006, Australia Internet: walters_b(a)maths.su.oz.au Phone: +61 2 692 2966 FAX: +61 2 692 4534 ==========================Instructions====================================== FTP LOGIN. Give the following commands. ftp maths.su.oz.au Login: anonymous (if you don't have an account on maths) Paswd: yoursurname (though any string will work) bin (if you are retrieving a .dvi file) prompt off (if you want no ? prompts from mget) cd sydcat (change directory to _public/sydcat ls -lt (see what's there, most recent first) mget filename-1 ... filename-n (e.g. mget catcurrent.Z) quit (exit from FTP) DVI. If you wish to print paper, calg say, retrieve calg.dvi and associated .eps and .sty files from the subdirectory calg (cd calg first). You must first give the bin command to ftp since .dvi files are not text files. You will then need a dvi to postscript converter which will include the .eps files. Print the resulting postscript file on your host. PROBLEMS. If you have problems in either retrieving or compiling papers, please contact Bob Walters. NOTE. Please note that the IP satellite link between Australia and the rest of the world is saturated most of the time. Large file transfers to non-Australian sites should be spaced out, and should preferably take place between the hours 2300 and 0800 local Eastern Australian time (the local time appears in the ftpd banner at connection). ============================================================================ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1 0

Arrows as Axioms
by momathie＠mathp7.jussieu.fr 22 Nov '93

22 Nov '93

Ce dont parle F.Dacar est largement etudie dans : 1. H. Andreka et I. Nemeti , "Generalization of the Concept of Variety and Quasi-Variety to Partial Algebras through Category Theory" Diss. Math. CCIV , Warszawa , 1983. On peut en trouver une "version modernisee" dans : 2. F. Cury ,"Completion et Completude selon H.Andreka. et I.Nemeti. Diagrammes 29 , Paris 1992. Le cas particulier de "il existe un unique" se touve dans : 3. L. Coppey, Theories Algebriques et Extensions de Pre-faisceaux" Cahier de Top. et de Geometrie Diff. , Vol. XIII - 1 et XIII - 4 , Paris 1972 , 4. L. Coppey et C. Lair , "AMEN, Algebricite - Monadicite - Esquissabilite et Non-algebricite , Diagrammes 13 , Paris 1985. Anterieurement, H. Andreka, I. Nemeti et I. Sain avaient meme etendu la satisfaction (par un objet) d'un ensemble A de fleches en la satisfaction d'un ensemble de familles projectives de fleches. Voir, par exemple : 5. A. Andreka et I. Nemeti , "A General Axiomatizability Theorem formulated in terms of Cone Injective Subcategories" , Coll. Math. Soc. , J. Bolyai 29 , Univ. Alg. Esztergon (Hungary) 1977. 6. I. Nemeti et I. Sain , "Cone Implicational Subcategories and some Birkhoff-Type Theorems" Coll. Math. Soc. J. Bolyai 29 , Univ. Alg. Esztergon (Hungary) 1977. 7. H. Andreka et I. Nemeti , "Los Lemma holds in every Category" Studia Scientiarum Mathematicarum Hungarica 13 , 1978. 8. H. Andreka et I. Nemeti , "Injectivity in Categories to represent all First Order Formulas" , Demonstratio Math. Vol XII - 3 , 1979. L'aspect completement categorique des choses consiste a satisfaire des ensembles de cones projectifs (d'indexations non necessairement discretes). Ceci a ete traite originellement (a la suite des travaux de H. Andreka, I. Nemeti et I. Sain) tout d'abord par : 9. R. Guitart et C. Lair , "Calcul Syntaxique des Modeles et Calcul des Formules Internes" , Diagrammes 4 , Paris 1980. Le lien systematique avec les categories de modeles d'esquisses (i.e. de theories du premier ordre) est largement explicite par : 10. C. Lair , "Categories Qualifiables et Categories Esquissables" Diagrammes 17 , Paris 1987. Ces derniers temps, certains reprennent les idees qui se trouvaient dans 1. pour, vraisemblablement, aboutir dans quelques annees a celles qui se trouvent dans 10?? 11. J. Adamek et J. Rosicky , "Locally Presentable and Accessible Categories" Livre a paraitre. 12. H. Makkai , "Generalized Sketches as a Framework for Completeness Theorem" Pre-print , 1993. Monique Mathieu, Universite Paris 7, Denis Diderot. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

2 1

normed categories
by koslowj＠math.ksu.edu 22 Nov '93

22 Nov '93

In "METRIC SPACES, GENERALIZED LOGIC, AND CLOSED CATEGORIES" (Rend. del Sem. Mat. e Fis. di Milano 43 (1973)) Bill Lawvere suggests the notion of a "normed category" as a category enriched in a suitable (symmetric monoidal) closed category S(R). (Here R denotes the interval [0,\infty], ordered by >= and with + as tensor and truncated - as internal Hom.) Has anybody worked out the details? Presumably, the objects of S(R) are to be subsets of R. This is suggested by the claim that there ought to be a closed functor inf from S(R) to R that induces the passage from normed categories to metric spaces (and by the use of the same symbol "S" that is used to denote the category of sets). As with any commutative monoid, the power-set of R under inclusion carries a symmetric monoidal closed structure: A + B = { a+b | a\in A and b\in B } (point-wise) defines the tensor and C - A = { x\in R | A + {x} contained in C } (not point-wise!) defines the internal Hom. {0} is the unit object. Clearly, inf turns into a strong functor as required. But how am I to interpret "the fundamental property of a normed category" (top of p. 140), namely (*) |f| + |g| >= |fg|? This inequality seems to apply to ordinary (= Set-enriched) categories X that carry an extra structure, namely a function |-| from Mor(X) to R. Presumably, this function also ought to satisfy (**) 0 >= |id_Y| for any X-object Y. E.g., one could define |f| to be 0 for every isomorphism of X, and 1 for every other morphism. In general this does NOT yield an S(R)-enriched category in the sense defined above. In fact, the definition above only seems to support the equality |f| + |g| = |fg|. Can further morphisms be added to S(R) without destroying the monoidal closed structure? Yes, if A +{q} is contained in B one can interpret q as a morphism from A to B. (The original inclusions arise for q = 0.) All these new morphisms f satisfy x <= f(x). But in view of the preceding paragraph one would prefer to have morphisms g that satisfy x >= g(x). Since this latter condition does not allow any new morphisms with domain {0}, the monoidal closed structure cannot be maintained. I must be missing something obvious. E.g., the definition of S(R) above doesn't use the symmetric monoidal closedness of R itself at all. Could someone please point me in the right direction? Thank you! J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj(a)math.ksu.edu | Jimi Hendrix (Voodoo Chile) aberne(a)dhvrrzn1.uni-hannover.d400.de ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

2 1

None
by cfw2＠po.CWRU.Edu 19 Nov '93

19 Nov '93

The work described in Point 3 of France Dacar's recent message looks very much like a special case of work of Andreka and Nemeti and Guitart and Lair. The basic references are given below in BibTeX input form, which is (I fondly believe) self-explanatory: @ARTICLE{ author = "Andreka, H. and I. Nemeti", title = "Formulas and ultraproducts in categories", journal = "Beit. zur Alg. und Geom.", volume = 8, year = 1979 } @ARTICLE{ author = "Guitart, Ren\'e and Christian Lair", title = "Calcul Syntaxique des Mod\`eles et Calcul des Formules Internes", journal = diagrammes, year = "1980", volume = "4", } I do not have the Andreka and Nemeti article. These articles are also relevant: @ARTICLE{ author = "Guitart, Ren\'e and Christian Lair", title = "Limites et Co-limites pour Repre\-senter les For\-mu\-les", journal = diagrammes, year = "1982", volume = "7", } @ARTICLE{ author = "Guitart, Ren\'e", title = "On the Geometry of Computations (I)", journal = cahiers, year = "1986", volume = "27", pages = "107--136", } -- 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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1 0

None
by strecker＠math.ksu.edu 19 Nov '93

19 Nov '93

Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. I believe that this was first done by Banaschewski and Herrlich Houston Math. J. (1977) 149-171. See also Section 22 of the book "Abstract & Concrete Categories" [Wiley Interscience]. George Strecker ++++++++++++++++++++++

1 0

None
by koslowj＠math.ksu.edu 19 Nov '93

19 Nov '93

Concerning question 1 of France Dacar: > Q1: In "Algebraic Theories", there are several propositions > (say 4.22, 4.23, 4.24) which include the following assumption: > "Let (K,E,M) be a regular category _such that every equalizer is in > in M_." This mystifies me, because an equalizer is _always_ an > M-monic, for any image-factorization system (E,M). In any > category, if an equalizer f factors as f = ge (composition to the > left...) with e epi, then e is an isomorphism. This is quite > trivial; since I cannot imagine that this could have been > overlooked for as long as image-factorization systems were around > up to the Manes' book, I am starting to doubt my ability to spot > a mistake in my reasoning... > I don't have my copy of Manes' AT handy, but it seems to me that the assumption that E consists of epis is not warranted. After all, E = all morphisms and M = all isos constitute an image-factorization system, and not all equalizers are isos. Regards, J"urgen -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj(a)math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++

1 0

Three questions: on equalizers, e-mail to Manes, arrows as axioms
by France.Dacar＠IJS.si 18 Nov '93

18 Nov '93

Greetings. These days I'm going through a little categories-refresher course. Among my reading matter are the "Categories, Functors..." by Arbib and Manes, and Manes' "Algebraic Theories". My three questions are related to these. Q1: In "Algebraic Theories", there are several propositions (say 4.22, 4.23, 4.24) which include the following assumption: "Let (K,E,M) be a regular category _such that every equalizer is in in M_." This mystifies me, because an equalizer is _always_ an M-monic, for any image-factorization system (E,M). In any category, if an equalizer f factors as f = ge (composition to the left...) with e epi, then e is an isomorphism. This is quite trivial; since I cannot imagine that this could have been overlooked for as long as image-factorization systems were around up to the Manes' book, I am starting to doubt my ability to spot a mistake in my reasoning... Q2: How can I reach Manes or Arbib? They're not on the structures mailing list. I have collected quite an errata on their book which they still might find useful. Q3: This last question is about Universal Algebra vs. Monads and friends. My knowledge of the relevant literature is rather narrow, so this question has been probably adressed somewhere; I'd like the relevant pointers. As a kind of self-imposed exercise I have tried to fill in the gap I felt there exists between the old good techniquies of UA and that of high-flying categorial treatment involving monads and such. I developed a little "elementary theory of UA in categories" which uses arrows as axioms, in the following sense. Let C be any category, and A a set of arrows in C. Say that an object c of C _satisfies_ A if for every arrow p: a ---> a' in A, any arrow f: a ---> c factors through p as f = f'p. Denote by C:A the full subcategory on all the objects that satisfy "axioms" A. Then, playing around with conditions imposed on the category C and on the axioms A, I can, step by step, reconstruct most of the "standard" results of UA, up to and beyond the Birkhoff's theorem. All along the reasoning stays close to that in AU, just translated into the language of arrows. At certain point a monad emerges, and then I can make the jump upwards to more rarefied regions. For me this treatment by arrows as axioms provided the "missing link". On the road up to the final ascent to monads there are plenty of points where one can go off in some other direction and still get some benefit by transplanting part of UA techniques "in abstract" to an un-algebraic matter. One can also generalize: there are other conditions on an object expressible by a single arrow, one can allow restricted boolean combinations of one-arrow conditions as basic statements (say Horn implications), and so on. If this was already done, where can I look it up? If not, is it worth publishing, and where? ------------------------------------------------------------------------ France Dacar Email: france.dacar(a)ijs.si Computer Science Department Phone: +386 61 1-259-199 / 768 Jozef Stefan Institute Fax: +386 61 1-258-058 Jamova 39, 61000 Ljubljana, Slovenia ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

2 1

higher dimensional groupoids
by mas010＠bangor.ac.uk 18 Nov '93

18 Nov '93

Higher dimensional groupoids One reason to see how higher dimensional categories exist and are useful is to look at the groupoid case. Higher dimensional groupoids form a useful higher dimensional version of groups and the fundamental group. The old result that "a set with two compatible group structures is just an abelian group" (basically, 1932, with Cech's description of higher homotopy groups) was for long regarded as an obstruction to such a programme. In fact, sets with 2 compatible groupoid structures model 2-dimensional homotopy theory, and so on for n compatible groupoid structures. This makes them of course very complicated, even in dimension 2. This fact is interesting in itself. A clear problem was to define a higher homotopy groupoid. One solution was by the writer and PJ Higgins, as may be found in joint work in (Proc London Math Soc 1978, for dimension 2, JPAA 1981 for all higher dimensions). This gives what we called the fundamental omega-groupoid of a filtered space. Later, Loday found a more complex and more general model, the fundamental cat^n-group of an n-cube of spaces. The existence of these functors is non-trivial. My purpose in considering higher dimensional groupoids since about 1966 was the possibility of higher dimensional van Kampen Theorems, modelled on the groupoid version. The references are the above papers with Higgins, and also with J-L Loday (Topology, 1987, Proc London Math Soc 1987), with his more complicated algebraic model. These theorems and models now allow for specific computations of some homotopy invariants and even some homotopy types (my paper in the Adams Memorial Colloquium). The intuition is that one needs algebraic structures which can model the geometric notion of subdivision, and which have structure in a range of dimensions, to carry the information about how bits of a space fit together. Multiple groupoids (or categories) seem well placed for that.The intuition is related to old ideas in topology of "What is a cycle?". A cycle should be some kind of composition of the little pieces. How should one accomplish this, algebraically? One has to move away from a linear notation and an always defined composition. Taking the free abelian group on the little bits seems like a cop-out (but of course, it has its uses!). One aspect of the theory is the equivalence of various views of a given structure. This is referred to by Ross Street in his communication, with regard to infinity-categories. So we have a set of equivalences between crossed complexes, omega-groupoids (RB-PJH, JPAA 1981), infinity-categories (CTGDC 1981), cubical T-complexes (CTGDC 1981), simplicial T-complexes (Ashley's thesis of 1978, published in Diss Math 155, 1988), polyhedral T-complexes (Jones, 1984, published as Diss Math 156, 1988). Here T-stands for "thin". These canonical structures are a generalisation of identities which appear only from dimension 2. The idea appears in old axioms for groups; with Keith Dakin in Bangor in 1975; and independently with Roberts in Australia at the same time, with the name "hollow". Thin elements play a crucial role in computations in and manipulations with these objects. Intuitively, thin elements have commutative boundary, and commutative boundaries have thin fillers. Thin elements also define compositions: all faces but one of a thin element "compose" to give the remaining face. For more on the philosophy of this, see Jones' thesis, which deals with compositions of general faces. Even more complex is the equivalence between cat^n-groups and crossed n-cubes of groups (Ellis-Steiner, JPAA, 1987). These equivalences allow a translation from a linear notation to a "higher dimensional" notation. Verity's equivalence in the category case mentioned by Ross Street, is in the same spirit as, and generalises, the equivalence between simplicial T-complexes and infinity-groupoids (go through crossed complexes, combining RB-PJH with Ashley). These things work. I just realised that one can do many sums with results of the first paper of RB-PJH, which for example allows the computation not only of \pi_2(BH \cup CBG) when G is a subgroup of H, for specific G,H, but also of the first k-invariant, i.e. the 2-type, of this space. These ideas are all modelled on groupoids, as is natural for the homotopy applications and the Generalised Van Kampen Theorems. A categorical version has not been done in full generality (polyhedrally). One does not expect the same type of application. There are relations with cohomology and knot theory which need more investigation. A lot of these ideas were discused in the visit of Ross and Dominic Verity to Bangor in May/June, 1993. Ronnie Brown ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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