Idempotents and the points of the topos of M-sets

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 $M$.