- AutorIn
- Dr. Mike Behrisch
- Dr. Sebastian Kerkhoff
- Prof. Dr. Reinhard Pöschel
- Dr. Friedrich Martin Schneider
- Prof. Dr. Stefan Siegmund
- Titel
- Dynamical Systems in Categories
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-129909
- Übersetzter Titel (DE)
- Dynamische Systeme in Kategorien
- Schriftenreihe
- MATH-AL
- Bandnummer
- 2013, 05
- Erstveröffentlichung
- 2013
- Abstract (EN)
- 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.
- Freie Schlagwörter (DE)
- Dynamische Systeme, Kategorien mit endlichen Produkten, Koalgebren
- Freie Schlagwörter (EN)
- Dynamical system, theory of systems, finite product category, coalgebra
- Klassifikation (DDC)
- 510
- Klassifikation (RVK)
- SI 8420, SK 350
- Publizierende Institution
- Technische Universität Dresden, Dresden
- Sonstige beteiligte Institution
- Technische Universität Dresden, Dresden
- URN Qucosa
- urn:nbn:de:bsz:14-qucosa-129909
- Veröffentlichungsdatum Qucosa
- 09.12.2013
- Dokumenttyp
- Preprint
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
- Inhaltsverzeichnis
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