The Weil-étale fundamental group of a number field I

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Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta-function ζₓ(s) at s=0. In this paper we consider the category of sheaves Xₗ on this conjectural site for X=Spec(OF) the spectrum of a number ring. We show that Xₗ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov Picard group of F. This leads us to give a list of topological properties that should be satisfied by Xₗ. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of étale sheaves computing the expected Lichtenbaum cohomology.