On the Pre- and Promonoidal Structure of Spacetime

The notion of a joint system, as captured by the monoidal (a.k.a. tensor) product, is fundamental to compositional, process-theoretic approach physical theories. Promonoidal categories generalise replacing functors normally used form systems with profunctors. Intuitively, this allows formation which may not always give system again, but instead generalised given presheaf. This extra freedom gives new, richer that can be applied categorical formulations spacetime. Whereas previous have relied on partial structure only defined pairs independent (i.e. spacelike separated) systems, here we concrete formulation spacetime where for any pair representable presheaves correspond precisely those actual arise from combining whereas more general virtual inherit some logical/compositional properties their ``actual'' counterparts. We show there are two ways doing this, corresponding roughly relativistic versions conjunction and disjunction. former endows category slices in Lorentzian manifold promonoidal structure, latter augments an (even more) way combine fails interchange law.