Space and time in monoidal categories
Enrique Moliner, Pau
The use of categorical methods is becoming more prominent and successful in both physics and computer science. The basic idea is that objects of a category can represent systems, and morphisms can model the processes that transform those systems. We can see parts of computational protocols or physical processes as morphisms, which, when appropriately combined using tensor products and categorical composition, model the protocol or process as a whole. However, in doing so, some information about the protocols or processes is forgotten, namely in what location of spacetime did the events involved take place, and what was the causal structure among them. The goal of this thesis is to explore how these categorical models can be enhanced to include information on the spacetime location and causal structure of events. First, we introduce the theory of subunits, which are subobjects of the monoidal unit for which a canonical isomorphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules, and under mild conditions they endow any monoidal category with a topological intuition. We introduce and study well-behaved notions of restriction, localisation, and support. Subunits in general form only a semilattice, but we develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame. Afterwards, we introduce a number of constructions to explore how the theory of subunits can be used in practice. Inspired by logical clocks, we define a diagrammatic category where we can capture simple protocols and their causal structure. To progress towards more detailed spacetime and causal information, we define the category of protocols, which formalises the idea of letting a morphism from a category be supported in a different category. This allows us to have one category to model the systems and processes and another one to model spacetime. In particular, we can treat both toy models of spacetime and more realistic ones in the same mathematical footing. A notion of causal structure is defined for monoidal categories, and a generalisation of the usual causal analysis in physics for points to arbitrary regions is provided. We give examples of protocols seen as diagrams and as objects in the category of protocols, both with toy models of spacetime as well as with more realistic ones.