  Viewing a fan as a partially ordered set (of cones) we consider a category of
sheaves on the fan which corresponds to a category of equivariant sheaves on
the corresponding toric variety if the fan is rational. In this category we
define an object which corresponds to the equivariant intersection cohomology
complex. Our first main result is the ``elementary'' decomposition theorem for
the direct image under subdivision of fans We also develop the Borel-Moore-
Verdier duality in the derived category of sheaves on the fan.
