  By analogy with algebraic geometry, we define a category of non-linear
sheaves (quasi-coherent homotopy-sheaves of topological spaces) on projective
toric varieties and prove a splitting result for its algebraic K-theory,
generalising earlier results for projective spaces. The splitting is expressed
in terms of the number of interior lattice points of dilations of a polytope
associated to the variety. The proof uses combinatorial and geometrical results
on polytopal complexes. The same methods also give an elementary explicit
calculation of the cohomology groups of a projective toric variety over any
commutative ring.
