80,125 research outputs found
Dynamical Systems in Categories
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 . . . . . . . . . . . . . .5
Galois differential algebras and categorical discretization of dynamical systems
A categorical theory for the discretization of a large class of dynamical
systems with variable coefficients is proposed. It is based on the existence of
covariant functors between the Rota category of Galois differential algebras
and suitable categories of abstract dynamical systems. The integrable maps
obtained share with their continuous counterparts a large class of solutions
and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added
Naturality and Induced Representations
We show that induction of covariant representations for C*-dynamical systems
is natural in the sense that it gives a natural transformation between certain
crossed-product functors. This involves setting up suitable categories of
C*-algebras and dynamical systems, and extending the usual constructions of
crossed products to define the appropriate functors. From this point of view,
Green's Imprimitivity Theorem identifies the functors for which induction is a
natural equivalence. Various spcecial cases of these results have previously
been obtained on an ad hoc basis.Comment: LaTeX-2e, 24 pages, uses package pb-diagra
Open Dynamical Systems as Coalgebras for Polynomial Functors, with Application to Predictive Processing
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.Comment: In Proceedings ACT 2022, arXiv:2307.1551
Realization of minimal C*-dynamical systems in terms of Cuntz-Pimsner algebras
In the present paper we study tensor C*-categories with non-simple unit
realised as C*-dynamical systems (F,G,\beta) with a compact (non-Abelian) group
G and fixed point algebra A := F^G. We consider C*-dynamical systems with
minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center
of A which we assume to be nontrivial. We give first several constructions of
minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra
associated to a suitable Z-bimodule. These examples are labelled by the action
of a discrete Abelian group (which we call the chain group) on Z and by the
choice of a suitable class of finite dimensional representations of G. Second,
we present a construction of a minimal C*-dynamical system with nontrivial Z
that also encodes the representation category of G. In this case the C*-algebra
F is generated by a family of Cuntz-Pimsner algebras, where the product of the
elements in different algebras is twisted by the chain group action. We apply
these constructions to the group G = SU(N).Comment: 34 pages; References updated and typos corrected. To appear in
International Journal of Mathematic
Dynamical systems and categories
We study questions motivated by results in the classical theory of dynamical
systems in the context of triangulated and A-infinity categories. First,
entropy is defined for exact endofunctors and computed in a variety of
examples. In particular, the classical entropy of a pseudo-Anosov map is
recovered from the induced functor on the Fukaya category. Second, the density
of the set of phases of a Bridgeland stability condition is studied and a
complete answer is given in the case of bounded derived categories of quivers.
Certain exceptional pairs in triangulated categories, which we call Kronecker
pairs, are used to construct stability conditions with density of phases. Some
open questions and further directions are outlined as well.Comment: 35 page
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