  Inspired by a theorem of Bhatt-Morrow-Scholze, we develop a stacky approach
to crystals and isocrystals on "Frobenius-smooth" schemes over F_p . This class
of schemes goes back to Berthelot-Messing and contains all smooth schemes over
perfect fields of characteristic p.
  To treat isocrystals, we prove some descent theorems for sheaves of Banachian
modules, which could be interesting in their own right.
