  We study the proalgebraic space which is the inverse limit of all finite
branched covers over a normal toric variety with branching set the invariant
divisor under the algebraic torus action. These are completions
(compactifications) of the adelic abelian algebraic group which is the
profinite completion of the algebraic torus. We prove that the vector bundle
category of the proalgebraic toric completion of a toric variety is the direct
limit of the respective categories of the finite toric varieties coverings
defining the completion. In the case of the complex projective line we obtain
as proalgebraic completion the adelic projective line P. We define holomorphic
vector bundles over P. We also introduce the smooth, Sobolev and Wiener adelic
loop groups and the corresponding Grassmannans; we describe their properties
and prove Birkhoff's factorization for these groups. We prove that the adelic
Picard group of holomorphic line bundles is isomorphic to the rationals and
prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of
higher rank over P.
