On the notion of geometry over $\F_1$
  We refine the notion of variety over the "field with one element" developed by C. Soul\'e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul\'e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above "field". 
Alain Connes, Caterina Consani