synthetic differential geometry, deformation theory
infinitesimally thickened point
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Mikhail Kapranov introduced a framework of noncommutative algebraic geometry, which could be interpreted as the study of infinitesimal thickenings of commutative schemes within bigger noncommutative spaces. The idea is to look at commutative schemes together with a sheaf of nilpotent noncommutative O-algebras; the formalism utilizes the filtrations relating to the commutator expansions from Maslov-Feynman calculus.
M. Kapranov, Noncommutative geometry based on commutator expansions, J. reine und angew. Math. 505 (1998), 73-118, math.AG/9802041.
lecture at msri 2000: Noncommutative neighborhoods and noncommutative Fourier transform, link
Guillermo Cortiñas, The structure of smooth algebras in Kapranov’s framework for noncommutative geometry, math.RA/0002177.
G. Cortiñas, De Rham and infinitesimal cohomology in Kapranov’s model for noncommutative algebraic geometry, Compositio Mathematica 136, 171-208, 2003, math.AG/0102133.
Shilin Yu, Dolbeault dga of a formal neighborhood, arxiv/1206.5155; The Dolbeault dga of the formal neighborhood of a diagonal, arxiv/1211.1567