Polynomial monads as context–Cnt constructions

Demonstrate that polynomial monads, in the sense of Gambino–Kock, arise as instances of the Cnt construction: construct an appropriate contentad (a wreath around an arrow pseudomonad) whose Cnt wreath product yields a double category capturing polynomial monads, and prove the resulting identification analogously to how spans arise from the Ctx construction.

Background

The paper develops the Ctx construction as a wreath product in the Kleisli tricompletion of spans, showing that familiar structures—such as Kleisli categories, Para, and Span—emerge from suitable contextads. In the dual setting, the authors introduce Cnt and contentads, suggesting that certain effectful or output-augmenting structures can be obtained via mixed distributive laws and wreath products.

In their concluding discussion on duality and distributive laws, the authors propose a parallel to the spans-as-Ctx paradigm: namely, that polynomial monads (as studied by Gambino and Kock) should be obtainable via the Cnt construction. Establishing this would clarify how polynomial data types/effects fit into the broader wreath-based framework and how context and content interact categorically.