  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 $\mathbb{Z}$ with compact support at infinity
gives, 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 this
conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We
show that $\bar{X}_L$ 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 $\bar{X}_L$. 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 \'etale sheaves computing the expected Lichtenbaum cohomology.
