higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The term $\mathcal{D}$-geometry refers to a synthetic formulation of differential geometry with a focus on the geometry of de Rham spaces. A quasicoherent sheaf of modules over a de Rham space is called a D-module, short for module over the algebra of differential operators, which is where the term “D-geometry” derives from.
regular differential operator, ordinary differential equation
connection, flat connection, local system, D-module, perverse sheaf
holonomic D-module, meromorphic connection, characteristic variety
Kähler differential, tangent category, tangent (∞,1)-category, cotangent complex, deformation theory, formally smooth morphism, jet bundle, jet space
Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules (pdf)