The broadest version of my question is the following:

Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R_{\geq 0}$ in which there is no subtraction?

The reason I think that such a theory might have been developed is that I know that such "rings" appear in algebraic geometry: in tropical geometry, in the study of semialgebraic sets, and so on.

Some specific questions that I hope such literature might answer (but probably this is too optimistic):

Is there an "etale site" over $\mathbb R_{\geq 0}$?

 

What is the "etale homotopy type" or "Galois group" of $\mathbb R_{\geq 0}$?

I'm under the vague impression that $\mathbb R_{\geq 0}$ is an approximation of the absolute field $\mathbb F_1$. I also speculate that $\mathbb R_{\geq 0} \hookrightarrow \mathbb R$ is "etale" but not a cover, so that $\mathbb R_{\geq 0}$ has multiple components in etale topology. This is in spite of the fact that $\mathbb R_{\geq 0}$ seems like a "field".

The broadest version of my question is the following:

Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R_{\geq 0}$ in which there is no subtraction?

The reason I think that such a theory might have been developed is that I know that such "rings" appear in algebraic geometry: in tropical geometry, in the study of semialgebraic sets, and so on.

Some specific questions that I hope such literature might answer (but probably this is too optimistic):

Is there an "etale site" over $\mathbb R_{\geq 0}$?

What is the "etale homotopy type" or "Galois group" of $\mathbb R_{\geq 0}$?

I'm under the vague impression that $\mathbb R_{\geq 0}$ is an approximation of the absolute field $\mathbb F_1$. I also speculate that $\mathbb R_{\geq 0} \hookrightarrow \mathbb R$ is "etale" but not a cover, so that $\mathbb R_{\geq 0}$ has multiple components in etale topology. This is in spite of the fact that $\mathbb R_{\geq 0}$ seems like a "field".