Subject: Simplicial sets of finite type (a forum question)
Date: Mon, 30 Jun 2003 19:26:09 +0400 (MSD)
From: ssp@pdmi.ras.ru
To: dmd1@lehigh.edu

Simplicial sets of finite type

Given a simply connected CW-space
having finitely many n-cells for every n (for example, CP^infty),
is it homotopy equivalent to the geometric realization
of a simplicial set having finitely many n-simplices for every n?

The answer "yes" is claimed
in Bousfield's - Kan's "Homotopy limits..." (Ch. V, Lemma 7.5);
the proof refers to the Wall's paper
"Finiteness conditions for CW-complexes" published in Annals of Math.
This reference seems to be incorrect,
just because Wall does not consider simplicial sets.

Does anybody know a reliable reference for this assertion?
Is it true?