Sheafifying Consistent Histories

Isham’s topos-theoretic perspective on the logic of the consistent-histories theory [34] is extended in two ways. First, the presheaves of consistent sets of history propositions in their corresponding topos originally proposed in [34] are endowed with a Vietoristype 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 UP of quantum history propositions. The second extension of the topos in [34] 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 à la [51, 42, 53, 55, 56] so as to arrive at the topos of consistent-histories of quantum causal sets. Not being able to resist the temptation, we speculate on a possible application of such topos-theoretic models to the problem of quantum gravity (ie, when spacetime structure, causality and its dynamics are supposed to be treated quantum mechanically)—an application that has been anticipated on general grounds by Butterfield and Isham [14] and partly worked out in a special finitary algebraic, sheaf-theoretic and categorical setting by this author [51, 52, 42, 53, 54]. In particular, rather general, but striking, similarities between the topos of consistent-histories of quantum causal sets that arises from our second extension of [34], the topos of finitary spacetime sheaves of non-abelian incidence algebras modeling a dynamical and locally finite quantum causality and its associated non-commutative topology in [42, 53, 54], as well as the recently proposed quantum spacetime scenario based on the so-called quantum causal histories of Markopoulou [43], are exposed. The paper closes with this author’s personal views, anticipations and speculations about the future of the general research program of applying sheaf and topos-theoretic ideas primarily to quantum gravity and then to quantum logic. Theoretical Physics Group, Blackett Laboratory of Physics, Imperial College of Science, Technology and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK; e-mail: [email protected]