Functors for ordinary and stable simplicial groups : a connection to Conduché's stable crossed modules

Abstract

I. A simplicial group models a pointed connected topological space up to homotopy. We may truncate a simplicial group in such a way that the homotopy groups in positions n and 0 are preserved. In this way, we obtain an [n,0]-simplicial group. This process gives a truncation functor from the category of simplicial groups to the category of [n,0]-simplicial groups. We construct the right-adjoint to this truncation functor that preserves homotopy groups, using methods from Conduché. II. A stable simplicial group, also called group spectrum or Kan spectrum, models a topological spectrum up to homotopy. We construct adjoint functors between the category of stable simplicial groups and the category of [n,-∞]-stable simplicial groups that respect homotopy groups. The category of [1,0]-stable simplicial groups is defined as a full subcategory of [1,-∞]-stable simplicial groups. We show that the category of [1,0]-stable simplicial groups is equivalent to the category of stable crossed modules in the sense of Conduché, using a construction of countably iterated semidirect products.