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:我们把开放的动态系统分类为一般时间演化的开源动力系统,作为由多元交界面共同索引的热点数的分类,并展示这如何扩大煤热源框架,以捕捉普通差异方程式、开放的Markov进程和随机的动态系统等共同的科学应用。 然后我们将Spivak的歌剧Org推广到这一环境,并构建相关的单亚化分类,其形态表现为等级开放系统;当它们的界面简单时,这些类别提供可控共质结构。我们用“Laplace 理论”来举例说明这些构造,该理论为主动推断提供了动态的语义,并指明了与Bayesian的反向和煤热镜逻辑的一些连接。