nLab
noncommutative thin scheme

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