From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4529 Path: news.gmane.org!not-for-mail From: "R Brown" Newsgroups: gmane.science.mathematics.categories Subject: Smooth categories etc Date: Thu, 28 Aug 2008 15:03:26 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020007 13720 80.91.229.2 (29 Apr 2009 15:46:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:47 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Thu Aug 28 15:35:42 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 28 Aug 2008 15:35:42 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KYmIm-0005JX-2t for categories-list@mta.ca; Thu, 28 Aug 2008 15:33:00 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 65 Original-Lines: 165 Xref: news.gmane.org gmane.science.mathematics.categories:4529 Archived-At: Dear Bill and Colleagues,=20 In reply here only to Bill (4), but great interest in the other = comments, I entirely agree with the many pointed approach, which was = published in my 1967 Proc LMS article. This is relevant also to higher = homotopy theory. If you define \pi_n(X,P) where P is now a *set* of base = points, then you see this has to have the structure of module over = \pi_1(X,P). As an example of its use, consider the map=20 S^n \vee [0,1] \to S^n \vee S^1 which identifies 0 and 1, where the S^n is stuck to [0,1] at 0. Clearly = \pi_n of the first space on the set of base points P consisting of 0 and = 1 is the free module on one generator over the indiscrete groupoid = \I=3D\pi_1([0,1],P). The main theorem of Higgins and RB implies that = \pi_n of the second space at 0 is the free module on one generator over = the infinite cyclic group, which itself is obtained from \I by = identifying 0 and 1 in the category of groupoids.=20 I know this result can also be obtained using covering space arguments = and homology, but I got into groupoids by trying to avoid this detour to = covering spaces to describe the fundamental group of S^1. It is a = question of finding algebra which models geometry.=20 One of my joint papers, which calculated (using vKT for crossed modules) = some \pi_2(X,x) as a module, was rejected by one journal on the = grounds that the calculations were too elaborate when `the interest is = in the group and not the module'. So much for homological algebra!=20 There is a culture in algebraic topology which neglects the operations = of \pi_1; perhaps it seems an encumbrance when there is only one base = point, but Henry Whitehead commented in 1957 that the operations = fascinated the early workers in homotopy theory.=20 It is amusing to speculate what might be infinite loop space theory, or = little cube operads, if you allow many base points! Could it clear up = the subjects??!! Answers on a postcard please (joke).=20 A standard lesson in mathematics is that you should forget structure at = the latest possible moment. In homotopy theory low dimensional = identifications can and usually do affect higher dimensional homotopy = invariants. To try and cope with this, we need homotopical functors = which carry algebraic information in a range of dimensions, to model how = spaces are glued together. This has been the aim of the various Higher = Homotopy van Kampen theorems. To handle these algebraic structures one = needs category theory ; as one example, I am currently working on a = joint paper using fibred and cofibred categories to relate high and low = dimensional information on colimits and induced structures. =20 The hope also is that because of the wide interest in deformation, i.e. = homotopy, as a means of classification, these tools and methods will = have wider implications.=20 Ronnie Dear Ronnie and Colleagues, Your comments are extremely interesting. Thank you very much for = raising=20 in so striking a manner the question of the relation between general=20 monoidal structures and cartesian closed structures. Below are some observations which show, I think, that everybody should be interested in this relation because it is=20 manyfold and fruitful. (1) While cartesian closed structures have the virtue of being = unique,=20 general monoidal closed structures have the virtue of not being unique.=20 Thus, for example, the cartesian closed presheaf toposes (with their=20 exactness properties and combinatorial truth object) often have a = further=20 monoidal closed structure given by Brian Day's convolution with respect = to=20 a pro-co-monoidal structure on the site. Cubical as well as simplicial=20 sets have both cartesian and non-cartesian closed structures, and that = is=20 'true', not merely 'convenient'. (2) Another category having both cartesian and non-cartesian monoidal = structures is the real interval from zero to infinity with 'x dominates = y'=20 as the morphism from x to y. (Actually, this category is derived by=20 collapsing a natural topos of dynamical systems in 'Taking categories=20 seriously' TAC Reprints.) Categories enriched with respect to the=20 non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise=20 every day in analysis and the rich insights of enrichment theory = (Functor=20 categories, bi-module composition, free categories, etcetera) should be=20 systematically applied to the advance of analysis and geometry, while on = the other hand metric examples inspire further developments of = enrichment=20 theory. Cauchy (who never worked on idempotent splitting in ordinary=20 categories and additive categories in the way that Freyd and Karoubi = did)=20 does not deserve to have his name brandished as a joke to scare one's=20 uncomprehending colleagues in analysis. The kind of completeness that is = inspired by two-sided intervals (unlike the one-sided intervals=20 inaccurately alluded to in common discussions of 'density') indeed = reduces=20 to the one attributed to Cauchy in the particular example of Metric=20 Spaces. The author hoped that observation would contribute to the = advance=20 of analysis and the development of enrichment theory, not to the supply = of=20 buzzwords. In fact, there is an insufficiently known branch of analysis called=20 'Idempotent Analysis', which deals largely with composition of = bi-modules,=20 or more precisely, with the relation between the two closed structures = on=20 the infinite interval. Of course, that monoidal category is isomorphic = to=20 the unit interval under multiplication (still cartesian closed too) = which=20 induces many of the relations between probablility and entropy. (3) Perhaps the most common relation between non-cartesian monoidal=20 categories and cartesian categories arises when a structure such as = vector=20 space is interpreted in a cohesive background. I am sticking to my story = that cohesive backgrounds are basically cartesian closed, due to the=20 ubiquitous role of diagonal maps and also due to the fact that, for=20 example, bornological vector spaces have an obvious monoidal closed=20 structure, whereas topological vector spaces have none. The rumor that=20 topological vector spaces might have a tensor with an adjoint hom is = part=20 of the disinformation that makes functional analysis look more difficult = than it is. A more accurate account of the relation between non-Mackey=20 convergence and closed structure can be found in C. Houzel's paper on=20 Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54: essentially, the topological categories are merely enriched in the=20 genuinely monoidal closed bornological ones. Similarly, the idea that = not=20 all dual spaces are complete seems to be based on a misguided generality = in the notion of Cauchy nets (they should be bounded). (4) Although pointed spaces are somewhat entrenched in algebraic=20 topology, there is an improvement suggested by your own work, Ronnie.=20 Consider the category whose objects are arrows S ---> E where E is a = space=20 (object of a cartesian closed cohesive background category) and S is a=20 discrete space. This category is even a topos if the category of E's = was,=20 as is the larger category of arrows between general pairs of spaces. The = first category is actually an adjoint retract of the second, correcting=20 the discontinuity that arises from the traditional limitation S =3D 1.=20 Intuitively, in the case where the pair of spaces is a subspace = inclusion,=20 the adjoint collapses the subspace to a point if the subspace is=20 connected, but if it is not connected, does not artificially merge its=20 components. There are many applications of this corrected construction = of=20 the space which results from 'neglecting' a subspace, both in algebraic=20 topology and in functional analysis, too numerous to discuss here. Bill