F1-schemes and Toric Varieties

There are by now several attempts to make the theory of the field of one element F1 rigorous. In [10] the authors formalize the transition from rings to schemes on a categorial level and apply this machinery to the category of sets to obtain the category of F1-schemes as in [1]. In [3] and [5] the authors extend the definition of rings in order to capture a structure that deserves to be called F1. In [1] the author tried instead to fix the minimum properties any of these theories must share. The current paper extends this line of thought. We use terminology of [1] and [2].