  We define and study the overconvergent site of an algebraic variety, the
sheaf of overconvergent functions on this site and show that the modules of
finite presentations correspond to Berthelot's overconvergent isocrystals. We
work with Berkovich theory instead of rigid analytic geometry and do not use
any of Berthelot's results. This gives a complete alternative approach to rigid
cohomology.
