  Given a graded monoid A with 1, one can construct a projective monoid scheme
MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with
the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove
several basic results regarding these. We show that:
  1.) Every quasicoherent sheaf F on MProj(A) can be constructed from a graded
A--set in analogy with the construction of quasicoherent sheaves on Proj(R)
from graded R--modules.
  2.) High enough twists of coherent sheaves are generated by finitely many
global sections, hence that every coherent sheaf is a quotient of a locally
free sheaf.
  3.) Coherent sheaves have finite spaces of global sections.
  The last part of the paper is devoted to classifying coherent sheaves on P^1
in terms of certain directed graphs and gluing data. The classification of
these over F_1 is shown to be much richer and combinatorially interesting than
in the case of ordinary P^1, and several new phenomena emerge.
