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?
jim
The earliest written testemony I know of is in Seminaire Bourbaki, 1961: A. Grothendieck, Technique de descente et theoremes d'existence en geometrie algebrique, III: Preschemas quotients. Seminaire Bourbaki t.12, 1960/61, exp. 212. In Section 4, categories are characterised as presheaves on Delta that take amalgamated sums over [0] to fibre products. (Delta = skeleton of finite non-empty ordinals). Groupoids are characterised as presheaves on Phi taking amalgamated sums to fibre products, where Phi is the symmetric version of Delta, i.e. skeleton for non-empty phinite sets and any maps. (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 to 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 to write Expose' VII, but the manuscript got longer and longer, less and less geometric, and was delayed for these reasons, and finally he decided to publish it separately instead. (He says something like this in the introduction to the long memoir.)) Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock <kock@mat.uab.es> Departament de Matemà tiques -- Universitat Autònoma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 32 50 Fax: +34 93 581 27 90 <A HREF="http://mat.uab.es/~kock/">http://mat.uab.es/~kock/</A> ----------------------------------------------------------------