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Let $X$ be a monoid scheme. We will show that the stalk at any point of $X$defines a point of the topos $\Qc(X)$ of quasi-coherent sheaves over $X$. As itturns out, every topos point of $\Qc(X)$ is of this form if $X$ satisfies somefiniteness conditions. In particular, it suffices for $M/M^\times$ to befinitely generated when $X$ is affine, where $M^\times$ is the group ofinvertible elements.  This allows us to prove that two quasi-projective monoid schemes $X$ and $Y$are isomorphic if and only if $\Qc(X)$ and $\Qc(Y)$ are equivalent.  The finiteness conditions are essential, as one can already conclude by thework of A. Connes and C. Consani \cite{cc1}. We will study the topos points offree commutative monoids and show that already for $\mathbb{N}^\infty$, thereare `hidden' points. That is to say, there are topos points which are notcoming from prime ideals. This observation reveals that there might be a moreinteresting `geometry of monoids'.

Topos points of quasi-coherent sheaves over monoid schemes