A Categorical Formulation of Algebraic Geometry

We construct a category, $\mathrm{\Omega }$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a “spec datum” is introduced, as a certain relation between categories, of which one has been given a Grothendieck topology. A “geometry” is interpreted as a sub-category of $\mathrm{\Omega }$, and a formalism is given by which such a subcategory is to be associated to a spec datum, reflecting the standard construction of the category of schemes from the category of rings by affine charts.