  We present categories of open dynamical systems with general time evolution
as categories of coalgebras opindexed by polynomial interfaces, and show how
this extends the coalgebraic framework to capture common scientific
applications such as ordinary differential equations, open Markov processes,
and random dynamical systems. We then extend Spivak's operad Org to this
setting, and construct associated monoidal categories whose morphisms represent
hierarchical open systems; when their interfaces are simple, these categories
supply canonical comonoid structures. We exemplify these constructions using
the 'Laplace doctrine', which provides dynamical semantics for active
inference, and indicate some connections to Bayesian inversion and coalgebraic
logic.
