Differential Graded Schemes II: The 2-category of Differential Graded   Schemes
  We construct a 2-category of differential graded schemes. The local affine models in this theory are differential graded algebras, which are graded commutative with unit over a field of characteristic zero, are concentrated in non-positive degrees and have perfect cotangent complex. Quasi-isomorphic differential graded algebras give rise to 2-isomorphic differential graded schemes and a differential graded algebra can be recovered up to quasi-isomorphism from the differential graded scheme it defines. Differential graded schemes can be glued with respect to an etale topology and fibered products of differential graded schemes correspond on the algebra level to derived tensor products. 
Kai Behrend