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