Sheafifying Consistent Histories

Abstract

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.