This entry is about the concept in the algebraic geometry of jet bundles and differential equations. For the concept of geometry as seen by D-branes see at D-brane geometry.
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
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
Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras
chapter 2, Geometry of D-schemes (pdf)
Jacob Lurie, Notes on crystals and algebraic $\mathcal{D}$-modules, 2009 (pdf)