A Categorical Formulation of Algebraic Geometry

TL;DR

This paper introduces a categorical framework for algebraic geometry, defining a category of pointed categories and correspondences, and formalizing the construction of schemes from algebraic data.

Contribution

It presents a novel categorical formalism for algebraic geometry, including the notion of spec data and a method to associate geometries with categorical data.

Findings

Defined the category $\Omega$ of pointed categories and correspondences.

Introduced the concept of a spec datum with Grothendieck topology.

Provided a formalism linking categorical data to geometric structures like schemes.

Abstract

We construct a category, $\Omega$, 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 $\Omega$, 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.