On the homology and homotopy of commutative shuffle algebras

For commutative algebras there are three important homology theories, Harrison homology, André-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.