* Re: nerves
2005-12-02 15:41 nerves jim stasheff
@ 2005-12-17 23:09 ` Joachim Kock
0 siblings, 0 replies; 2+ messages in thread
From: Joachim Kock @ 2005-12-17 23:09 UTC (permalink / raw)
To: categories
> 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 Matemàtiques -- 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>
----------------------------------------------------------------
^ permalink raw reply [flat|nested] 2+ messages in thread