An $E^2$-type closed model category for bisimplicial groups

Alexander Nofech

anofech@shaw.ca

A closed model category structure is defined
on the category of bisimplicial groups 
in which the weak equivalences are isomorphisms 
on bigraded homotopy groups $\pi_{k,l}$
and at the same time isomorphisms on the $E^2$ term 
of the Quillen spectral sequence. There is an
analogue of the spiral exact sequence of Dwyer-Kan-Stover.
 
One of the reasons for looking specifically at groups rather than at 
a general construction of a $E^2$-type model category is that it is
easier to find the abelianization of a cofibrant group.
This structure is considered as a convenient setting for a 
study of the relation between bigraded homotopy and hyperhomology.