  We define indexed categories of (open) dynamical system and random dynamical
system over polynomial interfaces, where time is given by an arbitrary monoid
$\mathbb{T}$. We consider the case of open random dynamical systems over both
open and closed noise sources, and the case where the interface of the random
system is `nested' over the interface of its noise source. We show that, in
discrete time, our categories of dynamical systems over polynomial interfaces
$p$ are equivalent to Spivak's categories $p$-$\mathbf{Coalg}$ of
$p$-coalgebras. We then define a notion of generalized $pT$-coalgebra for a
monad $T$, thereby extending the coalgebraic notion of dynamical system to
general time, and show that this construction bestows a notion of open Markov
process when the monad $T$ is a probability monad. Finally, we list some
further connections and open questions.
