From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3407 Path: news.gmane.org!not-for-mail From: "Stephen Lack" Newsgroups: gmane.science.mathematics.categories Subject: Re: classifying functor and colimits Date: Tue, 29 Aug 2006 07:45:46 +1000 Message-ID: NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241019286 8600 80.91.229.2 (29 Apr 2009 15:34:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:46 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Aug 28 21:56:35 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 28 Aug 2006 21:56:35 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GHrqf-0001ov-64 for categories-list@mta.ca; Mon, 28 Aug 2006 21:53:01 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 40 Original-Lines: 107 Xref: news.gmane.org gmane.science.mathematics.categories:3407 Archived-At: Dear David, The classifying space functor is the composite of the nerve functor N:Cat-->SSet and the geometric realization functor SSet-->Top, and it makes sense to consider them separately. The nerve functor preserves all limits (since it is a right adjoint) but not all colimits. It does preserve=20 sequential colimits, since it is given by homming out of the finite ordinals, which are finitely presentable.=20 More explicitly, the nerve NC of a category C is the simplicial set whose set (NC)_n of n-simplices is the set of all functors from [n] to C, where [n] is the category {0<1<...Top) and=20 so preserves all colimits, but relatively few limits. It does, as Tom Leinster observes, preserve finite products. Putting these facts together, one sees that the classifying=20 space functor preserves coproducts, filtered colimits, and finite = products. Steve Lack. -----Original Message----- From: cat-dist@mta.ca on behalf of Tom Leinster Sent: Mon 8/28/2006 10:45 PM To: categories@mta.ca Subject: categories: Re: classifying functor and colimits =20 Dear David, > I have been plagued by the following question: does the classifying > space functor commute with (co)limits? The classifying space functor (from Cat to Top) does preserve finite products. It doesn't preserve all infinite products, e.g. let A be the discrete category with two objects and consider the product of infinitely many copies of A. Nor does it preserve all colimits, as the following example shows. Let 1 be the terminal category, 2 the category consisting of a single arrow, and 3 the category consisting of a commutative triangle: 1 =3D . 2 =3D . --> . 3 =3D . --> . --> . Take the two different functors from 1 to 2. The pushout of the diagram in Cat formed by these functors is 3, and B3 is Delta^2, the standard topological 2-simplex. However, B1 is the one-point space and B2 is the unit interval, so the pushout of B1 and B2 is an interval of length 2, which is not homeomorphic to Delta^2. This doesn't answer your question about sequential colimits, but maybe it gives some helpful context. Best wishes, Tom > > In particular, I have a system of compact topological groups G_i > indexed by the natural numbers, and a whole lot of inclusions. > > Is B colim G_i homotopic to colim BG_i ? > > I have a hint that this should be so in my particular situation (in a > letter of Serre to Grothendieck), but I'd like to know how the > general case goes. > > Cheers, > > = ------------------------------------------------------------------------ > -- > David Roberts > School of Mathematical Sciences > University of Adelaide SA 5005 > = ------------------------------------------------------------------------ > -- > droberts@maths.adelaide.edu.au > www.maths.adelaide.edu.au/~droberts > www.trf.org.au > > > > > > --=20 Tom Leinster