From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2953 Path: news.gmane.org!not-for-mail From: Joachim Kock Newsgroups: gmane.science.mathematics.categories Subject: Re: nerves Date: Sun, 18 Dec 2005 00:09:31 +0100 Message-ID: References: <43906B25.9020802@math.upenn.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: QUOTED-PRINTABLE X-Trace: ger.gmane.org 1241019003 6534 80.91.229.2 (29 Apr 2009 15:30:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:03 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Dec 18 12:30:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 18 Dec 2005 12:30:27 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Eo1FU-00046e-Sl for categories-list@mta.ca; Sun, 18 Dec 2005 12:19:00 -0400 In-reply-to: <43906B25.9020802@math.upenn.edu> X-Sender: kock@127.0.0.1:55110 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 61 Xref: news.gmane.org gmane.science.mathematics.categories:2953 Archived-At: > Does anyone know a reference for > grothedieck's characterization for when a simplicial set > is the nerve of a groupoid > or if anyone observed it earlier? >=20 > jim The earliest written testemony I know of is in Seminaire Bourbaki, 19= 61: A. Grothendieck, Technique de descente et theoremes d'existence en= =20 geometrie algebrique, III: Preschemas quotients. Seminaire Bourbak= i=20 t.12, 1960/61, exp. 212. In Section 4, categories are characterised as presheaves on Delta tha= t take amalgamated sums over [0] to fibre products. (Delta =3D skeleto= n of finite non-empty ordinals). Groupoids are characterised as presheave= s on Phi taking amalgamated sums to fibre products, where Phi is the symme= tric version of Delta, i.e. skeleton for non-empty phinite sets and any ma= ps. (Delta and Phi are not Grothendieck's notation.) Also the terminology 'nerve of a category' is usually attributed to Grothendieck, but it is actually not used in the above Expose'. (Of course, 'nerve of a covering' goes much further back -- I think t= o=20 Cech in the 1930s.) The details probably should have been in SGA1 expose' VII (which was = never written), and appeared instead in Section 2 of J. Giraud, Methode de = la descente, Bull. Soc. Math. France Mem., 1964. (Giraud was supposed t= o write Expose' VII, but the manuscript got longer and longer, less and= =20 less geometric, and was delayed for these reasons, and finally he dec= ided=20 to publish it separately instead. (He says something like this in th= e=20 introduction to the long memoir.)) Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock Departament de Matem=E0tiques -- Universitat Aut=F2noma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 32 50 Fax: +34 93 581 27 90 http://mat.uab.es/~kock/ ----------------------------------------------------------------