  In this paper we give an inherently toric description of a special class of
sheaves (known as equivariant sheaves) over toric varieties, due in part to A.
A. Klyachko. We apply this technology to heterotic compactifications, in
particular to the (0,2) models of Distler, Kachru, and also discuss how
knowledge of equivariant sheaves can be used to reconstruct information about
an entire moduli space of sheaves. Many results relevant to heterotic
compactifications previously known only to mathematicians are collected here --
for example, results concerning whether the restriction of a stable sheaf to a
Calabi-Yau hypersurface remains stable are stated. We also describe
substructure in the Kahler cone, in which moduli spaces of sheaves are
independent of Kahler class only within any one subcone. We study F theory
compactifications in light of this fact, and also discuss how it can be seen in
the context of equivariant sheaves on toric varieties. Finally we briefly
speculate on the application of these results to (0,2) mirror symmetry.
