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.

翻译：我们提出了一般时间演化的开放动力系统范畴，这些范畴被定义为由多项式接口索引的余代数范畴，并展示了这如何扩展余代数框架以涵盖常见的科学应用，如常微分方程、开放马尔可夫过程和随机动力系统。随后我们将Spivak的操作子Org扩展到该设定中，并构造了相应的幺半范畴，其态射表示分层开放系统；当它们的接口简单时，这些范畴提供了典型的余幺半群结构。我们通过“拉普拉斯理论”来示例这些构造，该理论为主动推理提供了动力学语义，并指出了与贝叶斯反演和余代数逻辑的一些联系。