Some remarks on blueprints and \(\mathbb{F}_1\)-schemes
The authors review in this paper with great care 3 different approaches to \(\mathbb F_1\)-geometry, which are Toen-Vaquie's relative schemes, Lorscheid's blue schemes and Connes-Consani's \(\mathbb F_1\)-schemes. They explain certain adjunctions between monoids, rings and blueprints in detail and extend them to the corresponding categories of sheaves. These adjunctions allows them to merge blue schemes with Connes and Consani's notion of \(\mathbb F_1\)-schemes, which results in the category of so-called \(\mathbb F_1\)-schemes with relations as a common generalization.
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