Idempotents and the points of the topos of M-sets

TL;DR

This paper classifies the points and localizing subcategories of the topos of M-sets for a finite monoid M, using idempotents and two-sided idempotent ideals, and introduces a new topology on these points.

Contribution

It provides a complete classification of points and localizing subcategories of the topos of M-sets via idempotents and two-sided ideals, with a novel topology.

Findings

Points correspond to idempotents of M

Localizing subcategories correspond to two-sided idempotent ideals

Introduces a new topology on the points of the topos

Abstract

The aim of this paper is to study the points and localising subcategories of the topos of $M$-sets, for a finite monoid $M$. We show that the points of this topos can be fully classified using the idempotents of $M$. We introduce a topology on the iso-classes of these points, which differs from the classical topology introduced in SGA4. Likewise, the localised subcategories of the topos $M$-sets correspond to the set of all two-sided idempotent Ideals of $M$.