Localization of monoids and topos theory

Abstract

Let $M$ be a monoid that is embeddable in a group. We consider the topos $\mathbf{PSh}(M)$ of sets equipped with a right $M$-action, and we study the subtoposes that are of monoid type, i.e. the subtoposes that are again of the form $\mathbf{PSh}(N)$ for $N$ a monoid. Our main result is that every subtopos of monoid type can be obtained by localization at a prime ideal of $M$. Conversely, we show that localization at a prime ideal produces a subtopos if and only if $M$ has the right Ore property with respect to the complement of the prime ideal. We demonstrate our calculations in some examples: free monoids, two monoids related to the Connes-Consani Arithmetic Site, and torus knot monoids.