Finitely coordinated and finitely copresentable affine algebraic sets

In order to describe the finite character of geometrical objects, the notions of finitely coordinated and finitely copresentable objects are introduced. It is shown that they work well in geometrical categories of affine algebraic sets. Their connections with the finitely generated and finitely presentable objects of Gabriel–Ulmer are established. The noetherian case, where finitely coordinated objects are identical to finitely copresentable ones, is studied.