#
Décalage and Kan's simplicial loop group functor

##
Danny Stevenson

Given a bisimplicial set, there are two ways to extract from it a
simplicial set: the diagonal simplicial set and the less well known
total simplicial set of Artin and Mazur. There is a natural
comparison map between these simplicial sets, and it is a theorem due
to Cegarra and Remedios and independently Joyal and Tierney, that
this comparison map is a weak homotopy equivalence for any
bisimplicial set. In this paper we will give a new, elementary proof
of this result. As an application, we will revisit Kan's simplicial
loop group functor $G$. We will give a simple formula for this
functor, which is based on a factorization, due to Duskin, of
Eilenberg and Mac Lane's classifying complex functor $\overline{W}$.
We will give a new, short, proof of Kan's result that the unit
map for the adjunction $G\dashv \overline{W}$ is a weak homotopy
equivalence for reduced simplicial sets.

Keywords:
simplicial loop group, d\'{e}calage, Artin-Mazur total simplicial
set

2010 MSC:
18G30, 55U10

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 28, pp 768-787.

Published 2012-12-13.

http://www.tac.mta.ca/tac/volumes/26/28/26-28.dvi

http://www.tac.mta.ca/tac/volumes/26/28/26-28.ps

http://www.tac.mta.ca/tac/volumes/26/28/26-28.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/28/26-28.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/28/26-28.ps

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