Double Categories of Open Dynamical Systems (Extended Abstract)

Myers, David Jaz

doi:10.4204/eptcs.333.11

Cited by 15 publications

(25 citation statements)

References 3 publications

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“…In [21,15,7] the directed syntax for composing dynamical systems is defined by the operad underlying Lens FinSet op , +, 0 0 , often referred to as the operad of wiring diagrams. Following the Catlab implementation, we instead focus on operad underlying Lens Cospan(FinSet op ) , +, 0 0 .…”

Section: Example 23 (Operad Of Directed Wiring Diagrams)mentioning

confidence: 99%

“…The framework for directed composition of dynamical systems relies heavily on the generalized lens construction defined in [19]. Although this theory is robustly developed in [7], here we present a variant that aligns with our Julia implementation. In particular, we restrict our attention to dynamical systems defined on Euclidean spaces.…”

Section: Directed Compositionmentioning

confidence: 99%

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Operadic Modeling of Dynamical Systems: Mathematics and Computation

Libkind¹,

Baas²,

Patterson³

et al. 2021

Preprint

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Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of -sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.

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Section: Example 23 (Operad Of Directed Wiring Diagrams)mentioning

confidence: 99%

Section: Directed Compositionmentioning

confidence: 99%

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Libkind¹,

Baas²,

Patterson³

et al. 2021

Preprint

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“…However, it is useful to first put this structure in context: why are double fibrations an important structure to consider? We will discuss three reasons for looking at double fibrations: (i) they form a common generalization for recent work on monoidal fibrations [MV20], discrete double fibrations [Lam21], and 2-fibrations [Buc14] (ii) recent work in applied category theory [Mye21] has demonstrated a need for double fibrations; (iii) to further the study of double-categorical logic, a precise theory of double fibrations is required. Let us consider each of these in more detail.…”

Section: Introductionmentioning

confidence: 99%

“…Coming from a different angle, [Mye21] has shown how double fibrations can be useful in investigating aspects of applied category theory. In particular, the paper considers double categories associated to certain types of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Double Fibrations

Cruttwell¹,

Lambert²,

Pronk³

et al. 2022

Preprint

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“…The purpose of this search is, on a mathematical level, to unify Myers and Spivak's 'F -lenses' [Spi19] with mixed optics [CEG+20]. The first provide a very flexible and well-motivated framework for dealing with bidirectional systems, further corroborated by their applications in [Mye20;Mye21]. The second generalize lenses in many directions, providing a way to speak of bidirectional data accessing in the presence of side-effects and non-cartesian structure, as well as providing a convincing operational semantics for feedback systems [CGHR21].…”

Section: Introductionmentioning

confidence: 99%

Seeing double through dependent optics

Matteo¹

2022

Preprint

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Tambara modules are strong profunctors between monoidal categories. They've been defined by Tambara in the context of representation theory [Tam06], but quickly found their way in applications when it was understood Tambara modules provide a useful encoding of modular data accessors known as mixed optics. To suit the needs of these applications, Tambara theory has been extended to profunctors between categories receiving an action of a monoidal category.Motivated by the generalization of optics to dependently-typed contexts, we sketch a further extension of the theory of Tambara modules in the setting of actions of double categories (thus doubly indexed categories), by defining them as horizontal natural transformations. The theorems and constructions in [PS08] relevant to profunctor representation theorem for mixed optics [CEG+20] are reobtained in this context. This reproduces the definition of dependent optics recently put forward by Vertechi [Ver22] and Milewski [Mil22], and hinted at in [BCG+21].

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Double Categories of Open Dynamical Systems (Extended Abstract)

Cited by 15 publications

References 3 publications

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Double Fibrations

Seeing double through dependent optics

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