From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3863 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: subdivision Date: Sat, 25 Aug 2007 17:27:49 +1000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019567 10620 80.91.229.2 (29 Apr 2009 15:39:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:27 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Sat Aug 25 15:04:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 25 Aug 2007 15:04:15 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IOzpr-000730-HY for categories-list@mta.ca; Sat, 25 Aug 2007 14:54:11 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 14 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:3863 Archived-At: Dear Jim The answer is No, I believe, but the only thinking I have done about the question is as follows. In his paper on adjoint functors, Kan constructed a category %A for each category A. He used this to construct what we now call Kan extensions. It provides the domain of a diagram whose limit gives the end of a functor T : A^op x A --> X. At Bowdoin College in 1969, Mac Lane called %A the Kan subdivision category. It is a bit like barycentric subdivision in that each arrow f (edge) of A becomes an object (vertex) [f] of %A; the only non-identity arrows of %A are formally adjoined as in the situation [a] --> [f] <-- [b] where f : a --> b and a and b are identified with their identity arrows. There are not many composable pairs in %A so it seems that N%A is not the barycenric subdivision of the nerve of A. At this point I dug out my old copy of Kan's 1957 paper "On c.s.s. complexes" on which I scribbled some notes back in the late 1960s. If S is the category of finite sets and P : S --> Ord is the covariant powerset functor into ordered sets, we can define a functor D : S --> Simp into the category of simplicial sets (= css complexes) by (Ds)_q = Ord([q] , Ps). Kan's functor "Delta prime" is the restriction of D to the (topologists') simplicial category. Then Sd : Simp --> Simp is the extension along the Yoneda embedding of "Delta prime" to a colimit preserving functor. This yields the formula Sd(X)_q = coend^[n] X_n x Ord([q] , P[n]), but I see no way of using this when X is the nerve of a category A. Maybe there is more chance in the case of groupoids. (The left adjoint to Sd is Kan's functor Ex which starts a simplicial set on its way to becoming a Kan complex.) What made you suspect the existence of such a subdivision of categories? Ross On 16/08/2007, at 3:35 AM, James Stasheff wrote: > Is there a `barycentric' subdivision operator Sd on categories > such that with N = nerve > SdN = NSd > ?