Are affine binoid schemes equivalent to binoids? ($\mathbb{F}_{1}$-geometry) This is the same question as this one, but for binoids.

One of the most mysterious objects in mathematics is the elusive "field with one element" and one particular model for geometry over $\mathbb{F}_1$, is that of binoids. Here are some definitions:

*

*A binoid is a commutative monoid $M$ together with an absorbing element $0$.

*An ideal of $M$ is a subset $I$ such that

*

*$0\in I$.

*If $a,b\in I$, then $ab\in I$.

*If $a\in I$ and $r\in M$, then $ra\in I$.



*An ideal $I$ of $M$ is prime if whenever $ab\in I$ then $a\in I$ or $b\in I$.

*The spectrum $\mathrm{Spec}(M)$ of a binoid $M$ is the set of all prime ideals of $M$.

*The Zariski topology on $\mathrm{Spec}(M)$ is the topology generated by the collection $\{D(I)\}$ with $D(I)=\mathrm{Spec}(M)\setminus V(I)$, where
$$V(I)=\{\mathfrak{p}\in\mathrm{Spec}(M):I\subset\mathfrak{p}\}.$$

*A distinguished open of $\mathrm{Spec}(M)$ is a set of the form $D_f=D(\{f\})$ for some $f\in A$. These form a basis for the Zariski topology on $\mathrm{Spec}(M)$.

*A binoidal space is a pair $(X,\mathcal{O}_X)$ with $X$ a topological space and $\mathcal{O}_X$ a sheaf of binoids on $X$.

*An affine binoid scheme is a binoidal space of the form $(\mathrm{Spec}(M),\mathcal{O}_{M})$, where $\mathcal{O}_{M}$ is defined on the distinguished opens by
$$\mathcal{O}_{M}(D_f)=M_f.$$
Now, the theory of affine schemes is the same as that of rings in that we have a contravariant equivalence of categories
$$\mathrm{Spec}:\mathrm{Rings}\cong\mathrm{AffSch}^\circ_\mathbb{Z}:\Gamma.$$
Does the functor $\mathrm{Spec}:\mathrm{Binoids}\to\mathrm{AffineBinoidSchemes}^\circ$ also determine a contravariant equivalence of categories, where its inverse functor is given by global sections?
 A: Yes, the association $M \mapsto \text{Spec}(M)$ is a fully faithful functor from the category $\text{Alg}_{\mathbb{F}_1} = \text{Mon}_*$ of pointed monoids (what you call binoids) to the category of monoid spaces $\text{MonSpaces}$. Since every pointed monoid is local, morphisms of monoid spaces already include the condition that the induced morphisms on stalks are local. Therefore one should view monoid spaces as the analogue of locally ringed spaces and not just ringed spaces. If we now restrict the target category to affine monoid schemes, the functor $\text{Spec}$ becomes essentially surjective by definition and hence defines an equivalence of categories. Since it is contravariant, we thus have an equivalence $\text{Aff}_{\mathbb{F}_1}^{\text{op}} \simeq \text{Alg}_{\mathbb{F}_1} = \text{Mon}_*$.
And yes, the inverse is then given by taking global sections just as for affine schemes.
I think a reference would be this preprint by Cortinas, Haesemeyer, Walker and Weibel, which sets up the basic theory of monoid schemes, but I have not checked to be honest.
