Some Notions of (Open) Dynamical System on Polynomial Interfaces

TL;DR

This paper develops a categorical framework for open dynamical and random systems over polynomial interfaces, extending coalgebraic models to general time and probabilistic contexts, with equivalences and new constructions.

Contribution

It introduces categories of open dynamical systems over polynomial interfaces, extends coalgebraic models to arbitrary monoids and probability monads, and explores their properties and connections.

Findings

Categories of open dynamical systems are equivalent to $p$-coalgebras.

Extended coalgebraic notions to general monads and time.

Connected open Markov processes with probabilistic monads.

Abstract

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.