Toolkit for the algebraic geometer

TL;DR

This paper develops a unifying formalism for various geometric structures using sheaf theory, enabling a generalized approach to schemes and semiring schemes.

Contribution

It introduces a theory of schemes associated with sites, generalizing multiple geometries and clarifying the role of semiring schemes.

Findings

Unified framework for different geometries via sheaf theory

Application to semiring schemes and their foundational role

Recovery of underlying topological spaces from abstract constructions

Abstract

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model spaces via sheaf theory and recover a posteriori the underlying topological spaces that are often present in the construction of such geometric objects. We apply this formalism to semiring schemes and reason why the usual definition of semiring schemes has to be considered as the good approach to the geometry of semirings.