From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5174 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Comma categories Date: Fri, 25 Sep 2009 08:37:52 +1000 Message-ID: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1253924271 4314 80.91.229.12 (26 Sep 2009 00:17:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 26 Sep 2009 00:17:51 +0000 (UTC) To: Tony Meman , categories Original-X-From: categories@mta.ca Sat Sep 26 02:17:44 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 1MrKyu-0004AU-3e for gsmc-categories@m.gmane.org; Sat, 26 Sep 2009 02:17:44 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MrKVF-0005Jf-3k for categories-list@mta.ca; Fri, 25 Sep 2009 20:47:05 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5174 Archived-At: On 25/09/09 6:23 AM, "Tony Meman" wrote: > Dear category theorists, > I have two questions concerning comma categories. > > If C is a category with a terminal object *, is the comma category (C,*) > consisting of arrows from C to * isomorphic to the category C itself? If > this is true, the same should apply to the dual case with an initial object. > Dear Tony, Yes, this is true. You could even take it as a definition of terminal object. > The category sSet of simplicial sets is the category of functors from the > opposite delta category Delta^op to Set. The category of pointed simplicial > sets sSet* is defined as the comma category (delta0, sSet), where > delta0=hom(-,[0]). Is sSet* isomorphic to the category of functors from > ([0],Delta)^op to Set? > No, this is not true. The category sSet* is pointed (it has a terminal object which is also initial), while the category of functors from ([0],Delta)^op to Set is not. I'm not sure if there was supposed to be a connection between the two questions, but just in case, I might point out that [0] is not initial in Delta (in fact it is terminal). Steve Lack. > Thank you in advance for any help. > Tony > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]