From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7934 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Re: preprint Date: Sat, 23 Nov 2013 12:03:56 +0100 Message-ID: Reply-To: Marco Grandis NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1385237665 27308 80.91.229.3 (23 Nov 2013 20:14:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 23 Nov 2013 20:14:25 +0000 (UTC) To: "F. William Lawvere" , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Sat Nov 23 21:14:31 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1VkJb5-0006p5-Bp for gsmc-categories@m.gmane.org; Sat, 23 Nov 2013 21:14:31 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46643) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VkJaG-0007FI-Td; Sat, 23 Nov 2013 16:13:40 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VkJaG-00072R-Lh for categories-list@mlist.mta.ca; Sat, 23 Nov 2013 16:13:40 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7934 Archived-At: [Please do not use this e-address=2C but only the usual one at dima.unige.i= t] Dear Bill=2C Thank you very much for all these interesting comments and suggestions=2C w= e'll think of them. We also received others from Ronnie Brown=2C Marta Bunge and Bob Pare=3B we= are preparing a revised version. I am quite convinced of the importance of presheaves on nonempty finite set= s. Just for the pleasure of friendly=20 discussing=2C let me add that I like calling them 'symmetric simplicial set= s' :-) - a term against which you protested=20 sometime ago=2C in this list if I remember well=3B at least in a context wh= ere their links with simplicial sets and algebraic topology are relevant. On the other hand=2C I think that the classical term of 'simplicial complex= ' is misleading. In fact=2C they sit inside symmetric simplicial sets (not inside simplicial= sets) as the objects that=20 'live on their points'=2C according to your terminology. I used for them th= e term 'combinatorial space'=2C while I used 'directed combinatorial space' for the (non classical) directe= d analogue (sitting inside=20 simplicial sets)=2C whose objects are defined by privileged finite sequence= s instead of privileged finite subsets.I think that the opposition between = non-directed and directed notions should be kept clear. Thanks again and best wishes Marco On 23/nov/2013=2C at 02.41=2C F. William Lawvere wrote: Dear Ettore and Marco Much of your very interesting paper is actually taking place=20 in an unjustly neglected topos. The classifying topos for nontrivial Boolean algebras is concretely just=20 presheaves on nonempty finite sets. It was one of two examples in my=20 1988 Macquarie talk "Toposes generated by codiscrete objects in=20 algebraic topology and functional analysis" * Like many non-localic toposes=2C this one unites pair of identical=20 but adjointly opposite copies of the base topos of abstract sets=2C namely a subtopos of codiscretes and its negation=2C the subcategory of discretes or constant presheaves. Because in this very special case the Yoneda inclusion is a restriction of the codiscrete inclusion=2C=20 it is well justified to consider this topos as "codiscretely generated "=20 (wrt colimits). This topos contains the category of groupoids as a full reflective subcateg= ory. In fact the reflector preserves finite products=2C and is a refinement of=20 the syntax for presenting groups. It should probably be=20 considered as the fundamental combinatorial Poincare functor . That point of view requires viewing the objects of the topos as=20 combinatorial spaces=2C which is out of fashion even though a full coreflective subcategory is that of classical simplicial complexes=20 ( or "simplicial schemes"according to Godement). The fear is that geometric realization will not be exact. But as Joyal and Wraith=20 pointed out 30 years ago=2C one need only replace the usual interval with = the weak infinite-dimensional sphere as basic parameterizer (of Zitterbewegungen=20 instead of ordinary paths ?) in order to obtain a geometric mo=10rphism of the singular/realization kind from any reasonable topological topos. Actually the groupoids are already at a low level in the sequence of essential subtoposes: if the trivial topos is taken as level minus infinity= =2C the discrete as level zero=2C and the lowest level whose skeleton functor=20 detects connectivity as level one=2C then the lowest level whose UIAO is compatible seems to be three. Explicit consideration of the role of the groupoids may a help in calculating the precise Aufhebung function. Best wishes Bill *the other example was the Gaeta topos on countably infinite sets=2C=20 closely related to the Kolmogoroff-Mackey bornological generalizations of Banach spaces=2C which are vector space objects in that topos. = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]