From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/268 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Re: Classifying Space... Date: Fri, 24 Apr 2009 20:25:21 +0100 (BST) Message-ID: Reply-To: Tom Leinster NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1240615116 9018 80.91.229.12 (24 Apr 2009 23:18:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 24 Apr 2009 23:18:36 +0000 (UTC) To: Hugo.BACARD@unice.fr, categories@mta.ca Original-X-From: categories@mta.ca Sat Apr 25 01:19:55 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LxUgT-0007q0-S4 for gsmc-categories@m.gmane.org; Sat, 25 Apr 2009 01:19:54 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LxUGJ-0004g2-Ph for categories-list@mta.ca; Fri, 24 Apr 2009 19:52:51 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:268 Archived-At: Dear Hugo, Your question involves the functors N | | Cat -----> SSet ------> Top (nerve and geometric realization) and their composite, the classifying space functor B. 1. The nerve functor N has a left adjoint, so in particular it preserves finite products. Hence if M is a monoid in Cat (i.e. a strict monoidal category) then N(M) is, in a natural way, a monoid in SSet. 2. It's also true, though not totally obvious, that the geometric realization functor | | preserves finite products. So if X is a monoid in SSet then |X| is a topological monoid. 3. Putting these together, if M is a strict monoidal category then its classifying space B(M) is a topological monoid. If M is a non-strict monoidal category then B(M) is not necessarily a topological monoid in a natural way, but it is a "homotopy topological monoid" in any of several accepted senses. For instance, it is a Delta-space in the sense of Segal, and an A_infinity-space in the sense of Stasheff (although that doesn't deal satisfactorily with the unit). Similarly, if M is a symmetric monoidal category then B(M) is a "homotopy topological commutative monoid", e.g. a Gamma-space or an E_infinity space. Best wishes, Tom On Thu, 23 Apr 2009, Hugo.BACARD@unice.fr wrote: > > > Dear category theorists, > > > Sorry for my following stupid questions , but i would like: > > -Given a monoidal category M , for first assumed to be strict, What kind of > thing do we obtain when we take it's classifying space ?: we take the nerve of M > and then realising the simplicial sets obtained > > Explicitely there are theses questions : > > 1) Does the nerve of M "preserve" (or "reflect") the monoidal structure of M ? > > -Is it a monoid in the category of simplicial sets ? > -If yes , can we have conditions on a monoid of sSet to be the nerve of a > monoidal category ? I mean, does some kind of "segal condition" ? > > > > 2) And What kind of topological spaces of the realization of the nerve > -Is it a topological monoid , with some extra structure ? > > > 3) And what hapen if M is not strict, or is symmetric, or braided , etc... > > > > > > Thank you and sorry if these are completely stupid questions > > > > > > > > >