Observables IV: The presheaf perspective

In this fourth of our series of papers on observables we show that one can associate to each von Neumann algebra R a pair of isomorphic presheaves, the upper presheaf O^{+}_{R} and the lower presheaf O^{-}_{R}, on the category of abelian von Neumann subalgebras of R. Each $A \in R_{sa}$ induces a global section of O^{+}_{R} and of O^{-}_{R} respectively. We call them \emph {contextual observables}. But we show that, in general, not every global section of these presheaves arises in this way. Moreover, we discuss states of a von Neumann algebra in the presheaf context.