October 2003 - Categories - categories.org.au

Re: Categories of elements
by Jpdonaly＠aol.com 03 Oct '03

03 Oct '03

Dear Professor Lawvere, Thanks for your clarifications and views in response to my latest note. Coming from an applications-oriented environment, I do assume a set of Zermelo-Fraenkel axioms with a universe of small sets (as prescribed in CWM) in order to ensure access to a fully viable arithmetic of natural transformations. This seems to allow for more than enough categories for my purposes, but it certainly does give the category of small functions a prominence which can feel artificially restrictive at times. Thus I would be especially attentive to any comments which you might make specifically on the functorial isomorphism (I presume to call it a "Lawvere isomorphism" ) which, in converting the Yoneda picture (function-valued natural transformations) of categorical duality into the Lawvere picture (cocompatible functors), represses the category of small functions and, as I do realize, moves things into the context of the general existence theory of adjunctions and Kan extensions, possibly providing a functorial interpretation of your explanation of the origin of comma categories. By now this isomorphism seems to me to be more of a perspicuous relabelling than a redefiner of concepts, so that I have to plead innocent to your apparent conviction that I agonize over the definition of elements. I am in full accord with the doctrine of elements as you have described it, and the Lawvere isomorphism actually relieves some conceptual agony in this regard by smoothly ensuring that, to within a label, the elements of a function-valued functor constitute a (limit) object which is in the functor's codomain category. But I have to restate my belief that the otherwise perfectly redeemable sentence, "An element of a functor is an attaching functor into the category of elements of the functor," is unacceptably confusing due to the fact that the category of elements of a functor does not in any sense consist of the elements of the functor (as you would describe them). So I would rename it. Pat Donaly

2 1

query
by jim stasheff 02 Oct '03

02 Oct '03

Apparently Graeme's approach to infinite sloop spaces or rather to one fold loop spaces has be further abstracted to produce what is known as a `Segal category' at a quick glance, it seems to me these are related to Fukaya's A_\infty cats as my approach to \Omega X is related to Graeme's anyone seen this worked out or even commented on? jim

1 0

Euler characteristic versus homotopy cardinality
by John Baez 02 Oct '03

02 Oct '03

Dear Categorists - Some of you might be interested in this talk, since it's secretly about attempts to categorify the rational numbers. http://www.math.ucr.edu/home/baez/cardinality/ Euler Characteristic versus Homotopy Cardinality Abstract: Just as the Euler characteristic of a space is the alternating sum of the dimensions of its rational cohomology groups, the homotopy cardinality of a space is the alternating product of the cardinalities of its homotopy groups. The two quantities have many of the same properties, but it's hard to tell if they're the same, since like Jekyll and Hyde, they're almost never seen together: there are very few spaces for which the Euler characteristic and homotopy cardinality are both well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series - and the two then agree! We give examples of this phenomenon and beg the audience to find some unifying concept which has both Euler characteristic and homotopy cardinality as special cases.

1 0

new job & address for me too
by Paul Taylor 02 Oct '03

02 Oct '03

As Tom Leinster has told you about his new job, maybe I should do the same... A project to study "ABSTRACT STONE DUALITY" has been funded by the EPSRC (the UK funding agency for the exact sciences), as a result of which I have a job for three years in the Mathematical Foundations Group, Department of Computer Science, University of MANCHESTER and so have new email and web addresses pt(a)cs.man.ac.uk www.cs.man.ac.uk/~pt I am very much looking forward to working with my new colleagues there, including Peter Aczel, David Rydeheard, Andrea Schalk and Harold Simmons. Please note that, whilst I shall be making regular visits to Manchester, for the time being I shall continue to live and work in LONDON (four hours away by train). If, therefore, you want to send me anything by post, please see my web page or ask me by email for my home address. My email is forwarded from pt(a)dcs.qmw.ac.uk, pt(a)dcs.qmul.ac.uk and pt(a)di.unito.it, so please do not send it to multiple addresses. -------------------------------- Although funding for this project was in fact confirmed two months ago, I delayed making this announcement because I wanted to say at the same time that all of my WEB PAGES have been thoroughly revised, (though, needless to say, they haven't yet). Nevertheless, amongst my PAPERS linked from www.cs.man.ac.uk/~pt are - the ABSTRACT STONE DUALITY papers and proposal, - the HTML version of my book, "PRACTICAL FOUNDATIONS", - my older papers on CONTINUOUS, STABLE & SYNTHETIC DOMAIN THEORY - also on INTUITIONISTIC SETS AND ORDINALS and CATEGORICAL RECURSION PLUS *** NEW!!! *** - my teaching materials for the first year computer science course INTRODUCTION TO ALGORITHMS that I taught at QMW. - the complete text of Jean-Yves Girard's book PROOFS AND TYPES, as this is now out of print. - a translation of Gauss's second proof (1815) of the "fundamental theorem of algebra" (every polynomial has a complex root) Then of course there are (La)TeX MACROS (still in need of new web pages) - my famous DIAGRAMS package (which now supports PDFTEX) - PROOF TREES and BOXES - QED macros (with new documentation) - supermarket bar codes (OK, nothing to do with category theory) The papers are available in the usual variety of formats, whilst their web pages have been designed to allow navigation entirely in DVI or PDF format, using XDVI, XPDF or ACROREAD. This of course is prone to bugs, so please tell me if any of the links go astray. If there are any other (mathematical or programming) materials of mine that you still consider useful, but which haven't been included or updated in the new web pages, please ask. -------------------------------- Amongst 48 proposals that they considered in that quarter, the EPSRC Computer Science panel ranked this one equally amongst the top three, on the basis of four outstanding referees' reports. My two nominated referees were GRAHAM WHITE and RICHARD WOOD, to whom I would like to express my appreciation for their support. Graham had in fact also helped me to write the proposal in the first place. EPSRC rules did not allow me to submit this proposal in my own name, as I did not have a teaching (or indeed any) job. (In fact, they don't even allow people to buy themselves out of their teaching, just to employ other people to do the research for them, which makes some sense in experimental sciences, but not really in mathematics.) The success of this proposal follows a period of 20 months during which I continued to work on this research programme and attend conferences funded from my own savings --- and, I would like to make clear, NOT from social security. My attendance at SOME of the meetings was paid for by Birmingham, Dalhousie, St. Mary's and Turin Universities and the EU APPSEM project. I would not, however, have had the emotional energy to do this without the constant support of my partner, RICHARD SYMES, who let me pursue Plan A, even though (being a modern languages graduate) he had no clue what it was about, or - until he saw the referees' reports - whether it had any prospects of success. I would also like to thank MARIANGIOLA DEZANI of the Universita` degli Studi di Torino for her offer of funding, even though it turned out that certain Italian bureaucracy didn't allow it. Her support, and her personal hospitality when I visited Turin, meant a lot to me. -------------------------------- Finally, I feel that having paid for myself to attend conferences gives me the prerogative to make a comment on the way that they are organised: I deplore the practice of charging those like me (and numerous others, some of them important members of our community) who have had to pay for their own research, to subsidise "senior" people as guest speakers, with longer time-slots at conferences. These people have research grants and professorial salaries from rich universities. I have noticed, however, that it is the guest speakers who most commonly over-run their allotted time, and sometimes present material that would not have been accepted from others by the referees. Paul Taylor

1 0