Gabriel-Ulmer Duality for Topoi, An Introduction

TL;DR

This paper explores the Gabriel-Ulmer duality within the context of topoi, connecting algebraic and geometric perspectives through sheaf theory, providing a foundational and accessible introduction to the subject.

Contribution

It relates Gabriel-Ulmer duality to the geometry of topoi, emphasizing the role of left exact functors and sheaf theory, offering a self-contained introductory perspective.

Findings

Clarifies the connection between Gabriel-Ulmer duality and topoi.

Highlights the importance of left exact functors in the duality.

Provides an accessible introduction to the subject.

Abstract

The aim of this paper is to relate the classical result of Gabriel-Ulmer to the geometry of topoi. The usage of the attribute 'left exact' when dealing with functors involved in this duality is indeed not casual and it is related to the geometrical side of the story, i.e. sheaf theory, as in \cite{SGA}. Thought to be a very basic introduction to the subject, it is mostly self-contained. The reader is assumed to be familiar with the fundamentals of category theory, no further prerequisite knowledge is required. The proof of Gabriel-Ulmer duality follows \cite{MP}.