October 1991 - Categories - categories.org.au

omega-cats
by James Stasheff 30 Oct '91

30 Oct '91

Since the associahedra (my K_n complexes) exist for all n, can they be regarded as (generating) an \omega-category?? jim =============================

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homotopy theory, but really ...
by James Stasheff 23 Oct '91

23 Oct '91

Doug Ravenel (URochester) asks: %Date: Fri, 18 Oct 91 18:01:01 EDT %From: Doug Ravenel <drav(a)uhura.cc.rochester.edu> %Subject: Query on homotopy commutative diagrams %Dear experts, % I recently reproved the following lemma about homotopy commutative % diagrams after Peter May found an error in the proof I published 10 years ago. %(The following is written in LateX.) \documentstyle{article} \begin{document} Suppose are given a collection of pointed CW--complexes (or CW--spectra) $X_{i, j}$ for $i, j \in \ints$, maps $f:X_{i, j} \to X_{i + 1, j}$ and $g:X_{i, j} \to X_{i, j + 1}$ (we can safely omit the subscripts on $f$ and $g$) so that the diagram $$ \begin{array}{ccccc} X_{i, j} &\stackrel{f}{\longrightarrow} &X_{i + 1, j} &\stackrel{f}{\longrightarrow} & \cdots\\ {}\sp{g}\downarrow & &\downarrow {}\sp{g}\\ X_{i, j + 1} &\stackrel{f}{\longrightarrow} &X_{i + 1, j + 1} &\stackrel{f}{\longrightarrow} & \cdots\\ {}\sp{g}\downarrow & &\downarrow {}\sp{g}\\ \vdots & &\vdots \end{array} $$ commutes up to homotopy, and for each $(i, j)$ we are given a homotopy $$ X_{i, j} \times I \stackrel{h}{\longrightarrow} X_{i + 1, j + 1} $$ with $h(x, 0) = fg(x)$ and $h(x, 1) = gf(x)$. Then this diagram is homotopy equivalent to a {\em strictly} commutative diagram with spaces $X'_{i, j}$ and maps $f'$ and $g'$, each of which is a cofibration (i.e., an inclusion). The construction of this new diagram depends on the given homotopies $h$. For each $(i, j)$ there are homotopy equivalences $$ \alpha:X_{i, j} \to X'_{i, j} \qquad\mbox{and}\qquad \beta:X'_{i, j} \to X_{i, j} $$ with $f'\alpha$ homotopic to $\alpha f$, $g'\alpha$ homotopic to $\alpha g$, $\beta f'$ homotopic to $f\beta$, and $\beta g'$ homotopic to $g\beta$. \end{document} My first question is, has anyone heard of this before? The proof is elementary and should have been done thirty years ago. I would like to generalize this result to $n$--dimensional diagrams equipped with suitable higher homotopies. The proof I have in mind is still elementary in spirit but could involve some tricky combinatorics that I would like to avoid. In particular it reqires a tessellation of $n$--space with certain properties. I suspect that all of the relevant higher homotopies have been written down somewhere but I do not know where. I also suspect that someone has developed some machinery that will enable one to prove this quickly. Any suggestions? Thanks for your help. Doug Ravenel \end{document} Seems to me it is a case of rectification but I defer to the experts for the precise details/reference, especially with regard to the topology involved. jim ======================

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Monograph: Category Theory for the Working Computer Scientist
by longo%FRULM63.BITNET＠mitvma.mit.edu 21 Oct '91

21 Oct '91

Categories, Types, and Structures An Introduction to Category Theory for the Working Computer Scientist by Andrea Asperti and Giuseppe Longo Categories, Types, and Structures provides a self-contained introduction to general category theory and explains the mathematical structures that have been the foundation of programming language design for the past two decades. The authors observe that the language of categories could provide a powerful means for standardizing methods and language, and they offer examples including recursion theory, the early dialect of LISP, Edingurg ML, and current work in polymorphism and modularity. In Part 1, the book familiarizes readers with categorical concepts through examples based on elementary mathematical notions such as monoids, groups, and topological spaces, as well as elementary notions from programming language semantics such as partial orders and categories of domains in denotational semantics. Part 2 pursues the more complex mathematical semantics of data types and programs as objects and morphisms of categories. In particular, the semantics of type-free lambda calculus and of its more recent higher-order versions is closely discussed. The basis for a theory of internal category theory is also developed as a tool for the categorical understanding of polymorphism in functional programming. Andrea Asperti is Charg de Recherche at INRIA-Rocquencourt. Giuseppe Longo is Directeur de Recherches CNRS at the Departement de Mathematiques et Informatique, Ecole Normale Superieure, Paris, and is former Professor Computer Science at the University of Pisa. The first draft of the book was written while Longo was teaching a course on the topic at Carnegie Mellon University, in 1987/88, and Asperti was visiting there as a graduate student. September 1991 307 pages ISBN 0-262-01125 #29.25/$43.95 MIT PRESS 55 Hayward ST. Cambridge Mass 02142 USA EUROPE: 14 Bloomsbury Square London WC1A 2LP U.K. Facsimile: 071-404-0601 ====================================

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query
by James Stasheff 14 Oct '91

14 Oct '91

What is known about the homology of the classifying space (realization of the nerve) of a category of modules over an algebra A? Anything special if A is the group algebra of a group G?? jim stasheff jds(a)math.unc.edu I am assembling a list of people interested in 2-cats or rather in applied 2-cat theory - e.g. classifying spaces thereof If interested in being added to the list... ===================================

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The Category of Hilbert Spaces
by dyetter＠math.ksu.edu 02 Oct '91

02 Oct '91

Can anyone point me to a reference on the categorical properties of the category of Hilbert spaces and bounded linear operators, or on the functorial properties of the construction of L^2 spaces from measure spaces? In particular I'd like to know a characterization of those maps with respect to which L^2 is functorial. It would also be helpful to know the completeness properties of the category of Hilbert spaces. --David YETTER ===================================

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Parity Complexes
by street＠macadam.mpce.mq.edu.au 02 Oct '91

02 Oct '91

I have just received back from the Macquarie Printer copies of a corrected and modified version of Macquarie Math Report 88-0015 (Jan 1988) entitled "Parity complexes". The new version is Report 91-072 (Sept 1991). If anyone would like a copy of this, please let me know at my email address <street(a)macadam.mpce.mq.edu.au> and I'll send a copy by airmail. Another Macquarie Report which is available is "Modular categories" by Michel Thiebaud. Let me know if you would like this sent. Regards, Ross ==================================

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Bi-Heyting algebras, toposes and modalities
by magnan＠mathcn.umontreal.ca 01 Oct '91

01 Oct '91

"Bi-heyting algebras, toposes and modalities" from G.E.Reyes and H.Zolfaghari has also been installed in the directory pub/reyes. Everyone can now get a copy of this document by doing: ftp triples.math.mcgill.ca anonymous "your email address" cd pub/reyes get biheyting.ps quit Anyone needing help should ask his questions to: magnan(a)mathcn.umontreal.ca ---- Francois Magnan ************************************************** ** Francois Magnan ** ** Departement de mathematiques et statistiques ** ** Universite de Montreal ** ** EMAIL: magnan(a)mathcn.umontreal.ca ** ************************************************** ==================================

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Paper by Cockett and Seely
by Robert Seely 01 Oct '91

01 Oct '91

The article Weakly distributive categories by J.R.B. Cockett and R.A.G. Seely (as submitted to the Durham Symposium Proceedings) is available by anonymous ftp from triples.math.mcgill.ca. The files needed for this paper are in the ftp directory pub/rags/wk_dist_cat/ Full instructions are in the file pub/rags/wk_dist_cat.README : a brief summary is given now: When in the directory pub/rags/wk_dist_cat/, you have a choice: Either take the .dvi (or .ps) file wdcat.dvi (or wdcat.ps) or else take all the .tex and .sty files in this directory (by mget *.tex, get sprite.sty) and tex it yourself. You'll need a file for each section (sorry - but for dual authorship that seemed easier, and I've not got 'round to joining them up!), one for the overall structure, a macro file, two files to produce a par symbol, (one a .sty file), and F. Borceux' diagram macro file. DON'T FORGET to set the ftp to binary (and also faster if you don't ask for a prompt at each file) for transferring .dvi or .ps files. %ftp triples.math.mcgill.ca login as user anonymous with your email address as password ftp>cd pub/rags ftp>dir Then do one of these: ftp>prompt ftp>mget *.tex ftp>get sprite.sty or: ftp>bin ftp>get wdcat.dvi (or ftp>get wdcat.ps ) ftp>quit That should do. Any problem reports or further enquiries should be directed to rags(a)triples.math.mcgill.ca Paper copies of the articles are available from either author. - Robert Seely PS - You will also find a copy of the paper A Logical Calculus for Polynomial-time Realizability by J.N. Crossley, G.L. Mathai, and R.A.G. Seely as the .tex file ptime.tex (in pub/rags/). Help yourself. ===================================

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ftp on macc2.mta.ca
by Bob Rosebrugh 01 Oct '91

01 Oct '91

The articles: Constructive Complete Distributivity II by R. Rosebrugh and R.J. Wood and Constructive Complete Distributivity III by R. Rosebrugh and R.J. Wood are available by anonymous ftp as the LaTeX source files ccd2.tex and ccd3.tex from the directory [anonymous.mathcs.rosebrugh] at macc2.mta.ca They require the files catmac.sty and cat.bib which are in the same directory. To obtain the files: ftp macc2.mta.ca login as user anonymous with your email address as password cd [.mathcs.rosebrugh] (note the VMS directory syntax!!) dir (make sure you are in the right directory) get ccd2.tex (then get cat.bib, etc. as needed) bye Any problem reports or further enquiries should be directed to rrosebrugh(a)mta.ca Paper copies of the articles are available from the first-named author. Bob Rosebrugh p.s. this seems like a good opportunity to remind subscribers that the categories mailing list is no longer reliably reachable on bitnet. Material for posting should go to categories(a)mta.ca and administrative details to me (or to categories-request(a)mta.ca, which is now available.) ===============================

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