Dynamical Systems in Categories

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• 作者：Mike Behrisch ; Sebastian Kerkhoff ; Reinhard Pöschel…
• 关键词：Dynamical system ; Theory of systems ; Finite product category ; Coalgebra
• 刊名：Applied Categorical Structures
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：25
• 期：1
• 页码：29-57
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematical Logic and Foundations; Theory of Computation; Convex and Discrete Geometry; Geometry;
• 出版者：Springer Netherlands
• ISSN：1572-9095
• 卷排序：25

文摘

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.