Matthias Künzer matthias.kuenzer at math.rwth-aachen.de
Wed Oct 8 15:23:09 EDT 2008

On Wednesday 08 October 2008 14:04, you wrote:
> Dear Algtop-list,
>
> > by Whitehead, a weak homotopy equivalence between connected Kan
> > simplicial sets is a simplicial homotopy equivalence. Does there exist
> > a construction (involving choices) of the inverse-up-to-homotopy?
> >
> > (We are particularly interested in the case of a weak homotopy
> > equivalence in the image of \bar W: simp. gps -> simp. sets.)
> >
> > Best
>   Matthias Künzer
> ========
> Well, what about he paper:
> -
> MR0279808 (43 #5529) Curtis, Edward B. Simplicial homotopy theory.
> Advances in Math. 6 1971 107--209 (1971). (Reviewer: A. L. Bousfield)
> 55.40
>
> Marek G.

Dear Marek,

Theorem (2.20) of Curtis' paper is based on Theorem (1.23): a Kan complex
has a minimal one as a deformation retract. The proof of (1.23) looks in fact
sufficiently constructive to be useful for our purposes.

Thanks!

Matthias