higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The notion of dg-scheme is a generalization of the notion of scheme to dg-geometry.
Kai Behrend, Differential graded schemes I: prefect resolving algebras (arXiv:0212225)
Kai Behrend, Differential Graded Schemes II: The 2-category of Differential Graded Schemes (arXiv:0212226)
I. Ciocan-Fontanine, Mikhail Kapranov, Derived Quot schemes, Ann. Sci. ´ Ecole Norm. Sup. (4) 34(3), 403–440 (2001).