higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Derived differential geometry is higher differential geometry in an ambient (∞,1)-topos which is not 1-localic. This is the derived geometry corresponding to differential geometry. Typically this is specifically taken to be the derived geometry induced by the Lawvere theory for smooth algebras ($C^\infty$-rings):
See
Urs Schreiber, Seminar on derived differential geometry
Dominic Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry (pdf)