Extending Baez-Stay's "Rosetta Stone"

In 2009, Baez and Stay uploaded "Physics, Topology, Logic and Computation: A Rosetta Stone", a paper that has puzzled me since I first read it in 2010.

In that article, Baez and Stay build a bridge between various notions, notably relating physics and a view of computation coming from linear logic. My understanding is that the interaction flow in a (categorical) proof net of linear logic can be modelled via the same algebraic or topological structures a theoretical physicist would use to model particles.

Via Curry-Howard, we get here a very nice and deep correspondence between trajectories of particles in physics, and trajectories of resources during computation --- in a linear setting, for the computation side: no duplication nor erasure of resources, behaviours that are inherently not matching with physics.

Such bridges, in my opinion, have the following usefulness: they pave the way for thought experiments in which one would transfer observed behaviours and intuitions from one side to the other.

In a sense, the mathematical structure under play being the same, similar behaviours should be observed provided that they can be realised by each side.

I'll get back to this bridge principle quite often here.

Mathematical modelling relies on over-approximation of situations; and if two situations A and B inject in the same mathematical model M, then there is a bridge; but you can only use it to transfer from A situations that do not violate the extra constraints of B that were erased when abstracting it as M.

My PhD journey (2012-2016) has introduced me to various styles of fixed points, but also to the categorical notion of traces. When modelling programs, traces allow to model loops (for, while). And loops are everywhere -- notably to explore graphs, or discrete spaces.

Baez and Stay's Rosetta Stone does not introduce trace operators, so that there is no bridge for these traces at the moment. My understanding is that, on the physics side, trace operators would capture things akin to loops in Feynman diagrams.

Let's think of what such an extension of this Rosetta Stone to trace operators could bridge.

Could we transfer intuitions from algorithmic loops, used for instance to visit graphs, to the physics side?

Could we think that particles can explore the local space before choosing the direction to take? So that each particle has its own time, and the one that we observe "collapsed" this extra exploration time?

Could we see here an explanation to the least action principle?