Material histories and proarrow equipments
Symmetric monoidal categories can be understood as theories of resource convertibility. Objects denote collections of resources and arrows denote the possible transformations thereof. Viewed another way, we can think of arrows in symmetric monoidal categories as possible material histories, describing a sequence of events from the point of view of the resource theory in question. In particular we will view these material histories as the result of some process.
From every resource theory we may freely construct a single object proarrow equipment (double category with companion and conjoint structure) which contains the original resource theory as its category of vertical cells. The resource-theoretic interpretation of morphisms extends to an interpretation of cells, allowing material histories to be decomposed into concurrent components. In this way, proarrow equipments capture something fundamental about the nature of concurrent processes.
I will demonstrate all of this, with a particular emphasis on the category of horizontal cells of these freely generated proarrow equipments -- a planar monoidal category with interesting structure. Time permitting, I will present a more direct axiomatisation of the category of horizontal cells.