  This is the second in a series of papers laying the foundations for a
differential graded approach to derived differential geometry (and other
geometries in characteristic zero). In this paper, we extend the classical
notion of a dg-algebra to define, in particular, the notion of a differential
graded algebra in the world of C-infinity rings. The opposite of the category
of differential graded C-infinity algebras contains the category of
differential graded manifolds as a full subcategory. More generally, this
notion of differential graded algebra makes sense for algebras over any (super)
Fermat theory, and hence one also arrives at the definition of a differential
graded algebra appropriate for the study of derived real and complex analytic
manifolds and other variants. We go on to show that, for any super Fermat
theory S which admits integration, a concept we define and show is satisfied by
all important examples, the category of differential graded S-algebras supports
a Quillen model structure naturally extending the classical one on differential
graded algebras, both in the bounded and unbounded case (as well as
differential algebras with no grading). Finally, we show that, under the same
assumptions, any of these categories of differential graded S-algebras have a
simplicial enrichment, compatible in a suitable sense with the model structure.
