nLab
noncommutative thin scheme

Given a commutative field $k$, a noncommutative thin $k$-scheme (in the terminology of Maxim Kontsevich) is a left exact functor from the category ${\mathrm{Alg}}_k^{\mathrm{fd}}$ of finite dimensional $k$-algebras to Set, or equivalently by duality, from ${\mathrm{Coalg}}_k^{\mathrm{fd}}{}^{\mathrm{op}}$ to $\mathrm{Set}$. Every such functor is representable by a $k$-coalgebra.

• L. le Bruyn, Noncommutative geometry and dual coalgebras, arXiv:0805.2377

• M. Kontsevich, Y. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, math.RA/0606241

• blog entry

category: algebraic geometry, noncommutative geometry