The central features of concurrent, parallel and distributed computation are captured in the settings of monoidal categories with finite biproducts, countable coproducts, and distributive laws. Such categories are not only thoroughly researched, but central to any study of dynamics. In these categories the interplay of monoids (algebras) and comonoids (coalgebras) play an important role in the analysis of dynamics. A traditional usage is in Hopf algebras, the central features of which abstract to bimonoids (bialgebras). Processes which make progress by interaction require further research into a different interplay between monoids and comonoids. Call these dimonoids (dialgebras). In addition, careful attention must be paid to the role of silent moves. The combination of dimonoid structure with nontrivial grade zero objects requires careful study. The research will proceed first at the level of automata (semiautomata, labeled transition systems) and then abstract to equivalence classes of automata functorially.
