From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7935 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: preprint Date: Sat, 23 Nov 2013 19:38:02 -0500 Message-ID: References: Reply-To: "F. William Lawvere" 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 1385338819 11051 80.91.229.3 (25 Nov 2013 00:20:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 25 Nov 2013 00:20:19 +0000 (UTC) To: Marco Grandis , categories Original-X-From: majordomo@mlist.mta.ca Mon Nov 25 01:20:23 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 1VkjuZ-0004VD-4Y for gsmc-categories@m.gmane.org; Mon, 25 Nov 2013 01:20:23 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:48815) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Vkjt7-00034D-1w; Sun, 24 Nov 2013 20:18:53 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Vkjt7-0006gw-HE for categories-list@mlist.mta.ca; Sun, 24 Nov 2013 20:18:53 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7935 Archived-At: Dear Marco If I understand the history=2C it is the term "simplicial sets" that was so= mewhat misleading : Classically the simplicial complexes (based =2Cfrom a c= ategorical view=2C actually on finite families not just finite subsets) were an important combinatorial structure and indeed still are. Eilenberg a= nd others recognized the important role in topological calculations for "o= rdered simplices " and the corresponding toposwas said to consist of "semis= implicial sets". But then the prefixes were dropped ! It does not seem accurate to consider that there is an "opposition "betwee= n the ordered and unordered cases. They are just two subtoposesof the distr= ibutive lattice classifier=2C which consists of presheaves onfinite posets.= The classifying topos point of view is helpful=2C because it is the choice= of an algebraic structure=2C of the kind classified=2C in a topological to= pos that gives rise to an exact singular/realization pair. For example any = topological distributive lattice gives rise to such =3Bif it happens to be = totally ordered=2C these adjoint functors factor throughthe simplicial subt= opos. But a boolean algebra whose operations are continuous represents a re= alization that depends only on the subtoposthat depends only on the finite = trivially ordered sets. The symmetry is concretely the action of boolean ne= gation. The fact that any distributive lattice space embeds in a boolean a= lgebra space helps to relate these . There are other subtoposes (i.e. strengthenings of the theory of distributi= ve lattices)=2C so that precise calculation of the Aufhebung andof coHeytin= g boundary should in particular be relevant to combinatorial topology. There may be a connection with your earlier work that employed distributive= lattices in the analysis of diagrams. When one represents ageometric categ= ory in presheaves on a subcategory P=2C it is helpful to consider that P pa= rameterizes the basic figure shapes that determine the structureof a genera= l object. In the category of categories=2C the basic figures arefinite comm= utative diagrams=2C are they not ? Thus=2C if we do not insist on a minimal= istic notion of nerve=2C distributive lattices emerge. Best wishesBill > From: grandis43@hotmail.com > To: wlawvere@hotmail.com=3B categories@mta.ca > Subject: categories: Re: preprint > Date: Sat=2C 23 Nov 2013 12:03:56 +0100 >=20 > [Please do not use this e-address=2C but only the usual one at dima.unige= .it] > Dear Bill=2C >=20 > Thank you very much for all these interesting comments and suggestions=2C= we'll think of them. > We also received others from Ronnie Brown=2C Marta Bunge and Bob Pare=3B = we are preparing a revised version. >=20 > I am quite convinced of the importance of presheaves on nonempty finite s= ets. Just for the pleasure of friendly=20 > discussing=2C let me add that I like calling them 'symmetric simplicial s= ets' :-) - a term against which you protested=20 > sometime ago=2C in this list if I remember well=3B at least in a context = where their links with simplicial sets and > algebraic topology are relevant. >=20 > On the other hand=2C I think that the classical term of 'simplicial compl= ex' is misleading. > In fact=2C they sit inside symmetric simplicial sets (not inside simplici= al sets) as the objects that=20 > 'live on their points'=2C according to your terminology. I used for them = the term 'combinatorial space'=2C > while I used 'directed combinatorial space' for the (non classical) direc= ted analogue (sitting inside=20 > simplicial sets)=2C whose objects are defined by privileged finite sequen= ces instead of privileged finite subsets.I think that the opposition betwee= n non-directed and directed notions should be kept clear. > Thanks again and best wishes Marco >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]