Homotopy theory and simplicial groupoids

Abstract

The homotopy theory of simplical groups is well known [2, Ch. VI] to be equivalent to the pointed homotopy theory of reduced (i.e. only one vertex) simplicial sets (by means of a pair of adjoint functors G and W̄. The aim of this note is to show that similarly, the homotopy theory of simplical groupoids is equivalent to the (unpointed) homotopy theory of (all) simplical sets. This we do by 1. (i) showing that the category of simplicial groupoids admits a closed model catagory structure in the sense of Quillen [3], and 2. (ii) extending the functors G and W̄ to pair of adjoint functors G: (simplicial sets)↔(simplicial groupoids): W̄ which induce the desired equivalence of homotopy theories. We also show that the category of simplical groupoids admits a simplical structure which produces “function complexes” and “simplical monoids of self homotopy equivalences” of the correct homotopy types.