From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7933 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: RE: preprint Date: Fri, 22 Nov 2013 20:41:07 -0500 Message-ID: References: Reply-To: "F. William Lawvere" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1385237611 26709 80.91.229.3 (23 Nov 2013 20:13:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 23 Nov 2013 20:13:31 +0000 (UTC) To: marco grandis , categories Original-X-From: majordomo@mlist.mta.ca Sat Nov 23 21:13:42 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 1VkJaH-0006Wb-Lx for gsmc-categories@m.gmane.org; Sat, 23 Nov 2013 21:13:41 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46637) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VkJZL-00076h-1q; Sat, 23 Nov 2013 16:12:43 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VkJZK-00071j-TJ for categories-list@mlist.mta.ca; Sat, 23 Nov 2013 16:12:42 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7933 Archived-At: Dear Ettore and Marco Much of your very interesting paper is actually taking place in an unjustly= neglected topos. The classifying topos for nontrivial Boolean algebras is concretely just pr= esheaves on nonempty finite sets. It was one of two examples in my 1988 Mac= quarie 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 but adj= ointly opposite copies of the base topos of abstract sets=2Cnamely a subtop= os of codiscretes and its negation=2C the subcategory ofdiscretes or consta= nt presheaves. Because in this very special casethe Yoneda inclusion is a r= estriction of the codiscrete inclusion=2C it is well justified to consider= this topos as "codiscretely generated " (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 the syntax for presenting groups. It should probably be considered as th= e fundamental combinatorial Poincare functor . That point of view requires viewing the objects of the topos as combinatori= al spaces=2C which is out of fashion even though afull coreflective subcat= egory is that of classical simplicial complexes ( or "simplicial schemes"ac= cording to Godement). The fear is that geometric realization will not be e= xact. But as Joyal and Wraith pointed out 30 years ago=2C one need only rep= lace the usual interval with the weakinfinite-dimensional sphere as basic = parameterizer (of Zitterbewegungen 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 ofessent= ial subtoposes: if the trivial topos is taken as level minus infinity=2Cthe= discrete as level zero=2C and the lowest level whose skeleton functor dete= cts connectivity as level one=2C then the lowest level whose UIAO is compat= ible seems to be three. Explicit consideration of the role of the groupoid= s may a help in calculating the precise Aufhebung function. Best wishesBill *the other example was the Gaeta topos on countably infinite sets=2C close= ly related to the Kolmogoroff-Mackey bornological generalizations ofBanach = spaces=2C which are vector space objects in that topos. > From: grandis@dima.unige.it > Subject: categories: preprint > Date: Mon=2C 18 Nov 2013 11:51:29 +0100 > To: categories@mta.ca >=20 > The following preprint is available: >=20 > ------- > E. Carletti - M. Grandis > Fundamental groupoids as generalised pushouts of codiscrete groupoids > Dip. Mat. Univ. Genova=2C Preprint 603 (2013). > http://www.dima.unige.it/~grandis/GpdClm.pdf >=20 > Abstract. Every differentiable manifold X has a =91good cover=92=2C where= =20 > all open sets and their finite intersections are contractible. Using =20 > a generalised van Kampen theorem for open covers we deduce that the =20 > fundamental groupoid of X is a =91generalised pushout=92 of codiscrete =20 > groupoids and inclusions. > This fact motivates the present brief study of generalised pushouts. =20 > In particular=2C we show that every groupoid is up to equivalence a =20 > generalised pushout of codiscrete subgroupoids=2C and that (in any =20 > category) finite generalised pushouts amount to ordinary pushouts and =20 > coequalisers. > ------- >=20 > Before submitting it=2C I would like to know if the =91generalised =20 > pushouts=92 we are using (or similar colimits) have been considered =20 > elsewhere. > (They are not simply connected colimits=2C in the sense of Bob Pare=2C =20 > and indeed they cannot be constructed with pushouts.) >=20 > With best regards to all colleagues and friends. In particular to =20 > Ronnie Brown and Bob Pare=2C whose results are used in this preprint. >=20 > Marco >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]