From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6891 Path: news.gmane.org!not-for-mail From: Marco Grandis Newsgroups: gmane.science.mathematics.categories Subject: Simplicial versus (cubical with connections) Date: Tue, 13 Sep 2011 17:12:17 +0200 Message-ID: References: Reply-To: Marco Grandis NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1315963119 17295 80.91.229.12 (14 Sep 2011 01:18:39 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 14 Sep 2011 01:18:39 +0000 (UTC) To: categories@mta.ca, Original-X-From: majordomo@mlist.mta.ca Wed Sep 14 03:18:35 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1R3e7W-0002zL-JX for gsmc-categories@m.gmane.org; Wed, 14 Sep 2011 03:18:34 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36882) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R3e6G-000798-3A; Tue, 13 Sep 2011 22:17:16 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R3d0e-0007z2-F1 for categories-list@mlist.mta.ca; Tue, 13 Sep 2011 21:07:24 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6891 Archived-At: Dear categorists, I would like to comment on Ronnie Brown's message, copied below, insisting on a parallelism that is not often acknowledged, and may =20 'clarify' - for instance - why simplicial groups somehow behave as 'cubical groups with connections' (see Tonks' paper cited by RB), rather than as 'ordinary cubical groups'. The degeneracies of a simplicial object correspond to the =20 connections (or higher degeneracies) of a cubical one, introduced by Brown =20 and Higgins, more than to the ordinary degeneracies. Formally, this fact can be motivated as follows. Let us start from the cylinder endofunctor I(X) =3D X x [0, 1] of =20 topological spaces. Its main structure consists of natural transformations of powers of =20 I, derived from (part of) the lattice structure of [0, 1]: - two faces 1 --> I, sending x to (x, 0) OR (x, 1), - a degeneracy I --> 1, sending (x, t) to x, - two connections I^2 --> I, sending (x, t, t') to (x, max(t, =20 t')) OR (x, min(t, t')). Then we collapse the higher face of I (for instance), and we get a =20 cone functor C, with a monad structure: - the lower face of I gives the unit 1 --> C, - the lower connection gives the multiplication C^2 --> C, - the other transformations (including the degeneracy of I) induce =20 nothing. Now the cylinder I, with the above structure (which i [myself, not =20 the cylinder] call a 'diad'), operating on any space, gives a cocubical object with connections, while the monad C gives an augmented cosimplicial object. [[ Addendum. If one wants to take on the parallelism to the singular cubical/=20 simplicial set of a space X, the construction becomes more involved. One should start from: - the cocubical space I* (with connections) of all standard cubes, =20 produced by the cylinder I on the singleton space; - the augmented cosimplicial space Delta* produced by C on the empty =20 space 0 (taking care that C(0), defined as a pushout, is the singleton, =20 and C^n(0) is the standard simplex of dimension n-1). Then one applies to these structures the contravariant functor Top(-, =20= X) and gets the singular cubical set of X (with connections) OR the singular =20 simplicial set of X (augmented). ]] With best regards Marco Grandis On 12 Sep 2011, at 11:35, Ronnie Brown wrote: > The reference is included in this review *MR1173825 *of the cubical =20= > case. > > Tonks, A. P. mathscinet/search/publications.html?pg1=3DIID&s1=3D325533>(4-NWAL) =20 > search/institution.html?code=3D4_NWAL> > Cubical groups which are Kan. > /J. Pure Appl. Algebra/ bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/journaldoc.html?=20 > cn=3DJ_Pure_Appl_Algebra> 81 bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html?=20= > pg1=3DISSI&s1=3D118323>(1992), bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html?=20= > pg1=3DISSI&s1=3D118323>no. 1, bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html?=20= > pg1=3DISSI&s1=3D118323> 83=9687. bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/mscdoc.html?=20 > code=3D55U10,%2818D35,18G30%29> unicat.bangor.ac.uk:4550/resserv', 'AMS:MathSciNet', 'atitle=3DCubical=20= > %20groups%20which%20are%=20 > 20Kan&aufirst=3DA.&auinit=3DAP&auinit1=3DA&auinitm=3DP&aulast=3DTonks&co= den=3DJPAA=20 > A2&date=3D1992&epage=3D87&genre=3Darticle&issn=3D0022-4049&issue=3D1&pag= es=3D83-87=20 > &spage=3D83&stitle=3DJ.%20Pure%20Appl.%20Algebra&title=3DJournal%20of%=20= > 20Pure%20and%20Applied%20Algebra&volume=3D81')> > > The author shows that group objects in the category of cubical sets =20= > with connections [R. Brown and P. J. Higgins, J. Pure Appl. Algebra =20= > 21 (1981), no. 3, 233--260; MR0617135 (82m:55015a) ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/=20 > publdoc.html?r=3D1&pg1=3DCNO&s1=3D617135&loc=3Dfromrevtext>] satisfy = the =20 > Kan extension condition. This is a very nice correspondence with =20 > the simplicial case [J. C. Moore, in S=E9minaire Henri Cartan de =20 > l'Ecole Normale Sup=E9rieure, 1954/1955, Exp. No. 18, Secr=E9tariat =20= > Math., Paris, 1955; see MR0087934 (19,438e) bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publdoc.html?=20 > r=3D1&pg1=3DCNO&s1=3D87934&loc=3Dfromrevtext>]. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]