Processes interact by passing information at boundaries. Non-interacting processes compose in series (one after another) or in parallel (side by side). When they interact, they exchange data at meeting points, creating a two-dimensional structure: horizontal composition is sequential, vertical is parallel, and the corners are where interaction happens.
This paper presents a calculus that models process interaction by taking an arbitrary multicategory of non-interactive processes and adding rewriting rules for their interactions. The rewrite system is confluent (different reduction orders reach the same result) and terminating (every computation finishes). Terms modulo convertibility form a virtual double category — the two-dimensional structure that process interaction demands.
The connection to the free cornering of a monoidal category provides the semantics. The cornering construction takes a monoidal category (processes with parallel composition) and freely adds the ability for processes to interact at corners. The functor from the calculus into the free cornering gives a sound denotational semantics: programs have well-defined meanings in the double-categorical model. If you think of the calculus as an operational semantics (how programs run) and the free cornering as a denotational semantics (what programs mean), the functor proves they agree.