The underlying monoid action of a module

Here is another class of monoid actions that you can add to your repertoire. Let $F : Ob(Ring) \to Arrows(Cat)$ be a functor-valued function of rings. Let $R$ be a commutative ring with identity. Then $F$ outputs a forgetful functor from the abelian category of $R$-modules to the topos of multiplicative actions of $R$. \[ F(R) : mod(R) \to act(R_*) \] This produces the monoid action associated to any module. As R-modules are concrete categories, these functors are output action predecessors of the faithful functor to the topos of sets. The faithful functor to the topos of sets can then be recovered by the functor to the topos of $R_*$ monoid actions. As can be seen here, while most categories have a functor to category of sets, in some cases like with R-modules there are functors to other interesting topoi.

A special case is vector spaces, there we see that the action monoid is a group with zero. The topoi of actions of a group with zero is bivalent, but it is not classical because a group with zero is still not a group. It follows that the image of $F$ of the class of fields consists entirely of bivalent topoi. We can consider different topoi properties like these to better understand categories of modules over a ring.