Simplicial groups

Abstract

This is a somewhat complex chapter on the homotopy theory of simplicial groups and groupoids, divided into seven sections. Many ideas are involved. Here is a thumbnail outline:

• Section 1 Skeleta: Skeleta for simplicial sets were introduced briefly in Chapter I, and then discussed more fully in the context of the Reedy closed model structure for bisimplicial sets in Section IV.3.2. Skeleta are mose precisely described as Kan extensions of truncated simplicial sets. The current section gives a general description of such Kan extensions in a more general category C, followed by a particular application to a description of the skeleta of almost free morphisms of simplicial groups. The presentation of this theory is loosely based on the Artin-Mazur treatment of hypercovers of simplicial schemes [3], but the main result for applications that appear in later sections is Propositopn 1.9. This result is used to show in Section 5 that the loop group construction outputs cofibrant simplicial groups.