Subject: categories: nerves

From: jim stasheff <[email-masked]>
Date: Fri, 02 Dec 2005 10:41:25 -0500

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

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From: Joachim Kock <[email-masked]>
Date: Sun, 18 Dec 2005 00:09:31 +0100

> or if anyone observed it earlier?

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.

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Joachim Kock <[email-masked]>
Departament de Matemàtiques -- Universitat Autònoma de Barcelona
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