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 substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, 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.

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 substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, 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.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Preliminaries related to topology and measure theory . . . . . . . . 4 2.2 Basic notions from category theory . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Classical dynamical systems theory . . . . . . . . . . . . . . . . . . . . . . 23 3 Dynamical Systems in Abstract Categories . . . . . . . . . . . . . . . . . . 30 3.1 Monoids and monoid actions in abstract categories . . . . . . . . . . 31 3.2 Abstract dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Nonautonomous dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Dynamical Systems as Algebras and Coalgebras . . . . . . . . . . . . . .38 4.1 From monoids to monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 From abstract dynamical systems to monadic algebras . . . . . . . 48 4.3 Connections to coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Exponential objects in Top for locally compact Hausdorff spaces . . 52 4.5 (Co)Monadic (co)algebras and adjoint functors . . . . . . . . . . . . . .56

Cette thèse est structurée autour de deux axes d’études : (1) développer des logiques épistémiques formalisant la prise en compte de nouvelles données en présence d'informations incomplètes ou incohérentes ; (2) caractériser les notions de bisimulation sur les modèles de ces nouvelles logiques. Les logiques modales utilisées pour formaliser des raisonnements dans le cadre d’informations incomplètes et incohérentes, telle que la logique modale de contingence, sont généralement plus faibles que les logiques modales standards. Nos travaux se basent sur des méthodes logiques, algébriques et co-algébriques / In this Ph.D. dissertation we investigate reasoning about information change in the presence of incomplete or inconsistent information, and the characterisation of notions of bisimulation on models encoding such reasoning patterns. Modal logics for incomplete and inconsistent information are typically weaker than the standard modal logics, such as the modal logic of contingency. We use logical, algebraic and co-algebraic methods to achieve our aims. The dissertation consists of two main parts. The first part focusses on reasoning about information change, and the second part focusses on expressivity and bisimulation. In the following, we give an overview of the contents of this dissertation