January 1990 - Categories - categories.org.au

another correction
by RROSEBRUGH 01 Jan '90

01 Jan '90

Cristina Pedicchio has asked me to correct the e-mail address transmitted for her. It should be ti2tsg24(a)icineca2.bitnet and not ti2tsg2h@.... Bob Rosebrugh

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None
by mas010＠vaxa.bangor.ac.uk 01 Jan '90

01 Jan '90

In the groupoid case, there are various possibilities for `higher dimensional objects', namely crossed complexes, infinity-groupoids, omega-groupoids, simplicial T-complexes, poly-T-complexes (see the references in my article on `Some problems in non-abelian homological and homotopical algebra' SLNM 1418, ed M Mimura). Crossed complexes are useful because of their close relations to classical tools (chain complexes, universal covering spaces, relative homotopy groups). Omega-groupoids (=cubical infinity-groupoids with connections, and so thin structures) are useful because of the clear compositions, monoidal closed structures, and context for proving the generalised Van Kampen Theorem. Simplicial T-complexes are useful because of the relation with classical simplicial techniques, and because of a lingering suspicion in some quarters that it is old fashioned to use cubical sets. The aim of the poly-theory was to to give a context for group theory methods such as Van Kampen diagrams (see D.Johnson, Presentations of groups, LMS Lecture notes). In any case, what is wrong with pentagons, or rhombic dodecahedra? How do Stasheff polyhedra fit into a general theory similar to simplicial sets? What algebraic structures do Stasheff T-complexes correspond to? What is not so clear is what are the valuable features of infinity-groupoids. It is somehow interesting to know they are there, but what does one do with them when one has got them? I wish I knew. For example, translating the monoidal closed structure on omega-groupoids to the infinity-groupoid case, with explicit formulae, seems a formidable task. (It is not so good for the simplicial case, either!) There are also problems. For example, what is the crossed complex corresponding to the free infinity-groupoid on one generator of dimension n? It should be the fundamental crossed complex of the n-cell with its globular (hemi-spherical ) subdivision. But how does one prove this? The corresponding results for the cubical and simplicial case use basic facts on the homotopy theory of geometric realisations for the (current) proofs. A paper of Spencer and Wong (CTGD 24 (1983) 164-192) shows the advantage of working with double categories with thin structure rather than with 2-categories. Part of the aim of the thesis of F. Al-Agl was to get similar methods in the n-dimensional case. Intuitively, one hopes to replace pasting arguments by calculations with thin elements, since these obey the transparent rule: any composition of thin elements is thin. One can also make inductive arguments about how degenerate is a thin element (see Proposition 4.4 of Brown-Higgins JPAA 22 (1981) 11-41: the whole theory of omega-groupoids is designed to make this Proposition work, since the immediately following lemma gives the crucial result that certain choices previously made do not affect the final composition, and this is proved by showing that an (n+1)- dimensional cube is degenerate in direction n+1, so that its two opposite faces in this direction are the same). Trying to do this last proof of the Van Kampen Theorem in infinity-groupoids is a nightmare of intricately coiling tubes (see Whitney's tube systems?), so that the work with Loday on cat-n-groups adopted an entirely different approach, using sophisticated simplicial techniques, which in some mysterious way accommodate all these local problems. Clearly n-categories arise in nature. But it would seem useful to assess the advantages and disadvantages of the various possible categories in which to work before committing oneself to setting up a homotopy theory in one setting rather than another. The advantage of having these (highly non-trivial) equivalences of categories is that one can swap round at will, not noticing the technical nature of the machine which allows one to do this. But as Vaughan Pratt emphasised, there is still the problem of proving that infinity-categories are equivalent to omega-categories (=cubical infinity-categories with connections), and this is not yet solved by the work of Al-Agl/Steiner, although that does give a beautiful cubical setting equivalent to infinity-categories. I was very interested to see the high priority given by Vaughan to this question. Ross Street in an extra informal lecture at Montreal explained the background in constructing `cohomology with coefficients in an n-category' for his work on nerves of infinity-categories (following up suggestions of John Roberts). I have tended to concentrate on the corresponding possibilities for the groupoid case, but work of Pachkoria explained at Montreal shows the geometric possibilities of cohomology with coefficients in a commutative monoid. Recent work of Larry Breen gives Schreier systems with coefficients in a crossed square, while Bullejos and Cegarra have considered coefficients in a reduced `crossed module of length 2' (a la Conduche). Brown-Higgins deal in a recent preprint (`The classifying space of a crossed complex', MPCamb Phil Soc, in press) with coefficients in a crossed complex. It all looks very promising. Ronnie Brown ===================================

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Free categories
by P. B. Johnson 01 Jan '90

01 Jan '90

The underlying functor : cat ----> graph, from small categories to small graphs, is tripleable. I would like to consider the same question over an arbitrary topos, and would appreciate any hints from topos theorists. It seems to me that, given a topos E, the underlying functor: cat(E) ----> graph(E) is "equationally defineable", that is to say, each internal category in E can be described by (in fact finitary) operations on some object of graph(E), subject to some equations. Then cat(E) ----> graph(E) is tripleable if and only if it has a left adjoint. It seems that constructing free categories over graphs requires some kind of countable colimits, which I don't imagine are always available. Perhaps the existence of a natural numbers object (which seems to be in some sense the most primordial of infinite colimits that a topos might possess) ensures the existence of free categories. Can such toposes with free categories be characterized? Thank you, Paul. =================================

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by unknown＠categories.mta.ca 01 Jan '90

01 Jan '90

Appended to this is a file listing sites containing material which has been announced on the categories list as available by anonymous ftp. The file is available by ftp in [anonymous.categories] at macc2.mta.ca (138.73.1.2) and it would be happy to be expanded. It will be resent to the categories list occasionally. I would be glad for any suggestions re format and appropriate data for this file, but it is not intended to include a lengthy, article-by-article description. Bob Rosebrugh +++++++++++++++++++++++++++ This is FTP.SITES in [anonymous.categories] at macc2.mta.ca (138.73.1.2) The following sites have anonymous ftp directories with documents of interest to subscribers to the categories list: Machine Directory(ies) Comment ------- ------------- ------- boole.stanford.edu /pub papers by Casley, Crew (36.8.0.65) Pratt et al. maths.su.oz.au /sydcat papers by the Sydney (129.78.68.2) categories group; computational cat theory software, seminar notices etc. theory.doc.ic.ac.uk /theory papers by Abramsky, Taylor, (146.169.2.37) Vickers et al.; diagram macro package by Taylor triples.math.mcgill.ca /pub papers by Barr, Reyes, (132.206.150.30) Seely et al.; diagram macro package by Barr macc2.mta.ca [.mathcs.rosebrugh] papers by Rosebrugh and Wood (138.73.1.2) !note Vax syntax for cd as e.g. cd [.mathcs.rosebrugh] File updated 2 January 1992 ======================================

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Re: groups in GROUPOIDS
by MAS013＠vaxa.bangor.ac.uk 01 Jan '90

01 Jan '90

Here is a reply to Dave Yetter's query. The Magic words are "crossed modules". The basic reference is : R.Brown and C.B.Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc.Konink. Ned. Akad. Weten. 79, 1976, 296-302. The results seem to have been known earlier. In fact if one thinks of a groupoid as being a generalisation of an equivalence relation, the proof of the equivalence of crossed modules and groups in groupoids is but a slight generalisation of the undergraduate proof that normal subgroups and congruences are equivalent ways of presenting the same information. Does this give the sort of structure theory required? Tim Porter. ++++++++++++++++++++++

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Re: categories list
by Bob Rosebrugh 01 Jan '90

01 Jan '90

For the next two weeks I will be absent and this list will be moderated by my colleague Daniel Perron. That would not normally be remarked here, but at the moment faculty at Mount Allison University (including me) have been on strike for two weeks and there is a possibility that access to computing services may be suspended. If that were to happen during my absence, the contingency plan to move the list to another machine could not be implemented until May 4, with a resulting interruption of service. Best regards to all, Bob ===================================================================

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Complete Distributivity paper - abstract/ftp
by rrosebrugh＠mta.ca 01 Jan '90

01 Jan '90

The preprint whose abstract follows is available by anonymous ftp from the machine sun1.mta.ca in the directory pub/papers/rosebrugh as ccd4.{tex,dvi}. The source file is LaTeX and also requires the files catmac.sty (for diagram macros) and cat.bib which are found in the same directory. Paper copies are available from Rosebrugh - be sure to include a postal address if requesting one. Constructive complete distributivity IV Robert Rosebrugh and R. J. Wood Abstract A complete lattice L is constructively completely distributive, ccd, when the sup arrow from down-closed subobjects of L to L has a left adjoint. The Karoubian envelope of the (bi-)category of relations is equivalent to the (bi-)category of ccd lattices and sup-preserving arrows. The equivalence restricts to an equivalence between ideals and ``totally algebraic'' lattices. Both equivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to an equivalence with ccd lattices in a topos. ==============================================================================

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by Bob Rosebrugh 01 Jan '90

01 Jan '90

This is the routine distribution for the categories mailing list. It is the file routine.dist in [.categories] and was last updated on January 4, 1993. Subscribers should note that the From: field of a categories posting is categories(a)mta.ca (which is different from some other mailing lists). Thus, an automatic reply is redirected to the entire list (unless another intention is clearly detected by the moderator). Administrative items (address changes etc.) can be sent to categories-request(a)mta.ca, or directly to the moderator. Usually, items of this sort sent to categories(a)mta.ca will not be posted. The archives of postings on categories are held at the ftp site macc2.mta.ca (138.73.1.2) in the directory [.categories]. Note that this is a Vax VMS system and so the cd command requires appropriate syntax i.e. cd [.new] moves one node down the directory tree to new cd [-] moves one node up the tree. The postings are filed in yearly subdirectories called [.90], [.91] etc. Within those subdirectories there are monthly files, and an annual list of dates and subjects of postings. In the [.categories] directory there is also a file called ftp.sites with information about ftp sites holding files of interest to subscribers. Several TeX diagram macro packages are in the subdirectory [.macro]. Information about ftp sites with holdings of interest is welcome. This ftp site will also archive files which represent preprints in category theory for persons without access to an ftp archive. It is preferred that these be in TeX, but other submissions will be considered. Submissions should be sent to the moderator by e-mail only, and should include a short abstract suitable for posting to the categories list. Submissions must be ascii files only (so TeX source code in any flavour is fine, but dvi files are not acceptable currently.) If you need detailed instructions on how to use ftp ask anyone knowledgable about the Internet at your site, or write to me. Bob Rosebrugh Phone: +1-506-364-2538 Department of Mathematics and Computer Science Fax: -364-2210 Mount Allison University Sackville, N. B. E0A 3C0 Email: rrosebrugh(a)mta.ca Canada +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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ftp access: Adjoint Char'n of Cat of Sets
by Bob Rosebrugh 01 Jan '90

01 Jan '90

The paper whose abstract follows is available by anonymous ftp in the directory pub/papers/rosebrugh at sun1.mta.ca (138.73.1.12) as the files accs.{dvi,tex} If you download the latter you need to know that it is in LaTeX and uses the bibliography file accs.bib and Mike Barr's diagram macros, diagram.tex, both found in the same directory. Don't hesitate to contact me if you experience any difficulties or need ftp instructions. I will gladly forward a paper copy. If you request one be sure to include a full postal address. Bob Rosebrugh +++++++++++++++++++++++++++++++++++ An adjoint characterization of the category of sets Robert Rosebrugh and R. J. Wood Abstract If a category B with Yoneda embedding Y : B ---> CAT(B^op,set) has an adjoint string, U -| V -| W -| X -| Y, then B is equivalent to set. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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Re: Cahiers de topologie et geom. diff. cat.
by unknown＠categories.mta.ca 01 Jan '90

01 Jan '90

Dear Rosebrugh: Thank you for your letter.... For some time, I have thought of some electronic version of the ``Cahiers'', and it would be good if Categoricians gave their advice on it. The idea is the following one (but it needs some more elaboration both on the material and conceptual side): articles _could_ be transmitted via e-mail, then sent to a referee by e-mail (it would be quicker than the present process); naturally authors lacking e-mail could send either paper manuscripts or disks. Then: either the paper is accepted, or it is rejected, or (and this should be the more usual case) it is asked that the author divides the paper in two parts: one with the main results, ideas and methods, to be published in the usual form in the ``Cahiers''; another, longer with more technical parts, which would be sent to the subscibers by e-mail as soon as the paper is accepted. Moreover, after acceptance, the title and abstract of the paper could be included in a Bulletin Board so that any interested person could obtain a copy. Anyway, preprints already circulate, and a real `papered' article keeps its value longer than electronic medias, so that it should not be too bad for the ``Cahiers'', the subscibers of which are essentially University libraries for which they'll retain their documentary value. What do you (and others) think of it? The problem would be the `standard' software. Sincerely, A.C. Ehresmann <EHRES(a)FRMAP711.BITNET> (The above was received by ordinary mail and transcribed by B. Rosebrugh.)

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