  In this paper, we construct a sheaf-based topos quantum theory. It is well
known that a topos quantum theory can be constructed on the topos of presheaves
on the category of commutative von Neumann algebras of bounded operators on a
Hilbert space. Also, it is already known that quantization naturally induces a
Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum
theory akin to the presheaf-based one can be constructed on sheaves defined by
the quantization-induced Lawvere-Tierney topology. That is, starting from the
spectral sheaf as a state space of a given quantum system, we construct
sheaf-based expressions of physical propositions and truth objects, and thereby
give a method of truth-value assignment to the propositions. Furthermore, we
clarify the relationship to the presheaf-based quantum theory. We give
translation rules between the sheaf-based ingredients and the corresponding
presheaf-based ones. The translation rules have `coarse-graining' effects on
the spaces of the presheaf-based ingredients; a lot of different proposition
presheaves, truth presheaves, and presheaf-based truth-values are translated to
a proposition sheaf, a truth sheaf, and a sheaf-based truth-value,
respectively. We examine the extent of the coarse-graining made by translation.
