  Isham's topos-theoretic perspective on the logic of the consistent-histories
theory is extended in two ways. First, the presheaves of consistent sets of
history propositions in the topos proposed by Isham are endowed with a
Vietoris-type of topology and subsequently they are sheafified with respect to
it. The category resulting from this sheafification procedure is the topos of
sheaves of sets varying continuously over the Vietoris-topologized base poset
category of Boolean subalgebras of the universal orthoalgebra of quantum
history propositions. The second extension of Isham's topos consists in
endowing the stalks of the aforementioned sheaves, which were originally
inhabited by structureless sets, with further algebraic structure, that also
enjoys a quantum causal interpretation, so as to arrive at the topos of
consistent-histories of quantum causal sets. The resulting quantum causal
histories topos is compared with Markopoulou's quantum causal histories, with
Mallios and this author's finitary spacetime sheaves of quantum causal sets, as
well as with Butterfield and Isham's topos perspective on the Kochen-Specker
theorem of quantum logic.
