The points and localisations of the topos of $M$-sets

Abstract

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme $X$ from the topos $\mathsf{Qcoh}(X)$. This was achieved using topos points and localisations of $\mathsf{Qcoh}(X)$. With this philosophy in mind, the aim of this paper is to study the topos of $M$-sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of $M$-sets for finite monoids in terms of the idempotent elements of $M$ and idempotent ideals of $M$, respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction.