We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences - such as covariance - follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1

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