In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a lot and I get the feeling of it being the 'right' approach to $\mathbb F_1$.

A blueprint is a commutative monoid $B$ with zero, together with a pre-addition, which is -the way he defines it- an additive, multiplicative equivalence relation on $\mathbb N(B)$ respecting the $0$, i.e. ($0_\mathbb N \equiv 0_B$) and reflecting equality on simple terms $a \equiv b \implies a = b$. A morphism of blueprints is a homomorphism of monoids respecting $0$ and the pre-addition.

The idea is that blueprints should be $\mathbb F_1$-algebras, with $\mathbb F_1$ being given by $\left\{0,1\right\}$ with standard multiplication and trivial pre-addition. $\mathbb F_1$ is then the initial object in the category of blueprints.

Blueprints generalize semirings and commutative monoids in a nice way ($\text{SemiRing}$ and $\text{CommMonoids}$ are full reflective subcategories of the category of blueprints).

They carry just enough structure to talk about ideals. He then defines blueprinted spaces and locally blueprinted spaces in much the way you'd expect, and from that also 'blue schemes' (as locally blue spaces being locally isomorphic to $\text{Spec}(B)$ for a blueprint $B$).

Now, I'm sure the answer is simple, but are 'Blueprints' a geometric theory? In particular, what would be a nice way to write down the axioms? Let me phrase it like this

Is there a geometric theory whose $\text{Set}$ based models are blueprints? Is it coherent? Can it be extended to a theory of $\textit{local}$ blueprints?

It is a standard fact that the classifying topos of the coherent theory of local $R-$algebras is the gros zariski topos of schemes over $Spec(R)$. Given that the answer to the above question is yes, is the following true?

A blue scheme (as defined by Lorscheid) is the same as an object of the classifying topos of the theory of local blueprints.

In the same style of thinking, the topos of sheaves on the spectrum of a given blueprint $B$ should classify the 'theory of prime filters on $B$' in the same way as the topos of sheaves on the spectrum of a commutative ring classifies its theory of prime filters.

I'm still a novice in topos theory, which is why I'm asking: Does any of this make sense?

In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a lot and I get the feeling of it being the 'right' approach to $\mathbb F_1$.

A blueprint is a commutative monoid $B$ with zero, together with a pre-addition, which is -the way he defines it- an additive, multiplicative equivalence relation on $\mathbb N(B)$ respecting the $0$, i.e. ($0_\mathbb N \equiv 0_B$) and reflecting equality on simple terms $a \equiv b \implies a = b$. A morphism of blueprints is a homomorphism of monoids respecting $0$ and the pre-addition.

The idea is that blueprints should be $\mathbb F_1$-algebras, with $\mathbb F_1$ being given by $\left\{0,1\right\}$ with standard multiplication and trivial pre-addition. $\mathbb F_1$ is then the initial object in the category of blueprints.

Blueprints generalize semirings and commutative monoids in a nice way ($\text{SemiRing}$ and $\text{CommMonoids}$ are full reflective subcategories of the category of blueprints).

They carry just enough structure to talk about ideals. He then defines blueprinted spaces and locally blueprinted spaces in much the way you'd expect, and from that also 'blue schemes' (as locally blue spaces being locally isomorphic to $\text{Spec}(B)$ for a blueprint $B$).

Now, I'm sure the answer is simple, but are 'Blueprints' a geometric theory? In particular, what would be a nice way to write down the axioms? Let me phrase it like this

Is there a geometric theory whose $\text{Set}$ based models are blueprints? Is it coherent? Can it be extended to a theory of $\textit{local}$ blueprints?

It is a standard fact that the classifying topos of the coherent theory of local $R-$algebras is the gros zariski topos over $Spec(R)$. Given that the answer to the above question is yes, is the following true?

A blue scheme (as defined by Lorscheid) can be regarded as an object of the classifying topos of the theory of local blueprints.

In the same style of thinking, the topos of sheaves on the spectrum of a given blueprint $B$ should classify the 'theory of prime filters on $B$' in the same way as the topos of sheaves on the spectrum of a commutative ring classifies its theory of prime filters.

I'm still a novice in topos theory, which is why I'm asking: Does any of this make sense?