From: Tibor Beke <beke@math.uu.nl>
Subject: Re: response to completion question
Date: Fri, 7 May 1999 19:32:47 +0200 (MET DST)

> Another, by Quillen, is listed as "privately circulated MS - not to
> appear".

Quillen's influential manuscript has been made public meanwhile, as part
of:

Friedlander, Eric M.; Mazur, Barry: Filtrations on the homology of
algebraic varieties. With an appendix by Daniel Quillen. Mem. Amer. Math.
Soc. 110 (1994), no. 529

(This was also pointed out by Ulrike Tillman..........DMD)

Warning: what's below is not meant to be a direct response to Sanderson's
question, since (a) it's situated in the world of simplicial sets (b)
involves "group completion" in a sense that DIFFERS from that of
Barratt-May-Moerdijk-Priddy-Quillen-Segal.  There may be a connection,
though, and I think the question is reasonably pretty and un-looked-at for
the past 40 years (could be wrong!)

Puppe ("Semi-simplicial monoid complexes", Annals of Math vol 70, 1959,
pp.379-394) attributes the following conjecture to J.C. Moore:

Let U be the Grothendieck functor Monoids --> Groups ("group completion").
Let M be a simplicial monoid whose \pi_0 is a group.  Then the natural
inclusion u: M --> U(M) is a weak equivalence.

He proves it in a number of cases; for example, if M is free (which shows
that one can take the free *monoid* in Milnor's FK construction) and when M
is degreewise Ore (funnily, this is the condition on \pi_0 that shows up in
Segal's proof of homological group completion).

When \pi_0 is not necessarily a group, Puppe attributes the following
conjecture (which is a generalization) to Moore and Kan:

\bar{W}(u) is a weak equivalence for any simplicial monoid.

\bar{W} is the base of the standard twisted cartesian product (see e.g.
May's book).

(Let me emphasize that I include these conjectures as curiosities!  I do
not know what their status is.)

T. Beke