Applications and Examples in Delicate Homotopy Theories of Simplicial Objects

Abstract

In this chapter we consider homotopy theories of simplicial objects which resemble the homotopy theory of simplicial groups. It is well known that in the Quillen model category of simplicial groups ΔGr all objects are fibrant; i.e. all simplicial groups satisfy the Kan extension condition. Moreover the free simplicial groups form a sufficiently large class of cofibrant objects in the sense that the homotopy category of free simplicial groups is equivalent to the homotopy category Ho(ΔGr) defined by localization with respect to weak equivalences. Since free groups are cogroups we see that free simplicial groups are simplicial objects in a special theory T of cogroups. In this chapter we study the homotopy theory of “free” simplicial objects in any theory of cogroups, or more generally in any theory of coactions. Such homotopy theories are canonical generalizations of the homotopy theory of simplicial groups.