Traces for coalgebraic components

This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.