From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4527 Path: news.gmane.org!not-for-mail From: "R Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: KT Chen's smooth CCC, a correction Date: Wed, 27 Aug 2008 11:51:31 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020006 13710 80.91.229.2 (29 Apr 2009 15:46:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:46 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Aug 27 16:43:20 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 27 Aug 2008 16:43:20 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYQvB-0002FL-40 for categories-list@mta.ca; Wed, 27 Aug 2008 16:43:13 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 63 Original-Lines: 94 Xref: news.gmane.org gmane.science.mathematics.categories:4527 Archived-At: Dear Bill and Colleagues,=20 I would like to explain my own interest in function spaces and function = objects since it has a different origin to what Bill explains and a = different direction which could be of interest for comment and = investigation.=20 Michael Barratt suggested to me in 1960 the problem of calculating the = homotopy type of the space X^Y by induction on the Postnikov system of = X, in contrast to Michael's own work on Track Groups, where he used a = homology decomposition of Y, and using Whitney's tube systems gave = explicit description of some group extensions in examples of the = Barratt(-Puppe) exact sequence. (amazing!!?)=20 Now the first Postnikov invariant in its simplest form is a Sq^2 but the = extension is described by a Sq^1. How did the one transform into the = other? Clue: the Cartan formula for Sq^2 on a product. How did a product = get into the act? Answer: the evaluation map!=20 Trying to write all this down led to using a number of `exponential = laws' in spaces, spaces with base point, simplicial sets, pointed = simplicial sets, chain complexes, simplicial abelian groups, etc. So it = was dinned into me that an exponential law depended on the product as = well as the function object. So why not try the known weak product for = topological spaces? Surprise, surprise, it all worked, and was part 1 of = my thesis, submitted 1961, with a sketchy account of what we now call = monoidal closed categories, exemplified, but not developed in general = terms.=20 Subsequent work with Philip Higgins has continued to use monoidal closed = categories in algebraic topology. Indeed the category of crossed = complexes is cartesian closed, but the homotopy theory one wants is = given by the (different) monoidal closed structure. So the category of = filtered spaces is usefully enriched over this monoidal closed = category.=20 My question is then: what is the potential influence of this need for = monoidal closed? It clearly does not lead to topos theory as such. It = does lead to the possibility of some not previously available = calculations, even of nonabelian homotopical invariants, and is relevant = to the study of local-to-global problems. (I first heard these words = from Dick Swan in connection with sheaf theory.) But in this work = cubical sets became essential, for ease of discussing subdivision, = multiple compositions, and homotopies, and here the monoidal closed = structure is crucial. Kan's initial cubical work was neglected in favour = of the (convenient in many ways) cartesian closed category of simplicial = sets.=20 One specific problem for me was a general notion of symmetry (naively, = and using buzz words (!), higher order groupoids should yield higher = order notions of symmetry!). In a cartesian closed category C we have = for a specific object x not only Aut(x), the isomorphisms of x, but also = AUT(x), the internal group object of automorphisms of x. This has been = developed for the topos of directed graphs, in John Shrimpton's thesis, = but actually the unaccomplished aim was to understand Grothendieck's = Teichmuller Groupoid, and his envisaged computations of this by gluing = or clutching procedures, but which needed topos theory, he claimed! When = I asked for any notes on this he just said nothing was written down, it = was all in his mind. Baffling! =20 In the monoidal case we can get only that END(x) is an internal monoid = wrt tensor. But in some cases we have a candidate for AUT(x), even if = `internal group' wrt tensor makes no sense. One example was worked out = with Nick Gilbert (published 1989). In this case there is a forgetful = functor U to a cartesian closed category, in this case Set, and you can = make up the rest. It worked in this dimension, relevant to homotopy = 3-types, but still did not lead by induction to even higher order = notions of symmetry. Pity!=20 My question is now: given this background, how should we match the = beautiful ideas and insights of Bill with what seem to be some monoidal = closed realities? Could this be important for geometry, and, better = still, even for analysis, and dynamics?=20 Ronnie From: Bill Lawvere To: categories@mta.ca Sent: Tuesday, 26 August, 2008 9:07:58 PM Subject: categories: Re: KT Chen's smooth CCC, a correction Dear Jim and colleagues, By urging the study of the good geometrical ideas and constructions of Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, = Spanier, Steenrod, I am of course not advocating the preferential resurrection of the particular categories they tentatively devised to contain the constructions. ...