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Lichtenbaum has conjectured the existence of a Grothendieck topology for anarithmetic scheme $X$ such that the Euler characteristic of the cohomologygroups of the constant sheaf $\mathbb{Z}$ with compact support at infinitygives, up to sign, the leading term of the zeta-function $\zeta_X(s)$ at $s=0$.In this paper we consider the category of sheaves $\bar{X}_L$ on thisconjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. Weshow that $\bar{X}_L$ has, under natural topological assumptions, a welldefined fundamental group whose abelianization is isomorphic, as a topologicalgroup, to the Arakelov Picard group of $F$. This leads us to give a list oftopological properties that should be satisfied by $\bar{X}_L$. Theseproperties 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 tocomplexes of \'etale sheaves computing the expected Lichtenbaum cohomology.

The Weil-\'etale fundamental group of a number field I