Diagram Categories for Physics

Evan Patterson, Andrew Baas, Timothy Hosgood, and James Fairbanks (the Decapodes line of work) make the physicists’ informal Tonti diagrams into honest mathematics. A physical theory becomes a diagram in a category, and there are morphisms between diagrams.

A diagram in a category is a functor from an indexing shape . The category of diagrams takes these functors as objects, and its morphisms are reindexings of the shape together with natural comparisons of the functors. Because the diagrams themselves are the objects of a category, separately specified pieces of physics can be glued, refined, and compared rather than merely drawn.

The relevant is a category of de Rham objects: differential forms with the exterior derivative , the Hodge star , and the time derivative . A concrete PDE such as advection–diffusion is assembled by an undirected wiring diagram that shares physical quantities as wires among these operators, and the composite diagram is then discretised on a mesh for simulation.

Example, advection–diffusion by wiring

The scalar transport law is presented as a wiring diagram whose boxes are , , and and whose shared wires are the concentration form and its flux. Swapping the diffusion box for a nonlinear one is a morphism of diagrams, not a rewrite from scratch.

Remark, de Rham structure

The resulting discrete exterior calculus respects by construction, since the objects are the forms with and , and the diagrams inherit the cochain structure underlying de Rham cohomology.