• Corpus ID: 838018

The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes

@article{Rupel2013TheOO,
  title={The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes},
  author={Dylan Rupel and David I. Spivak},
  journal={ArXiv},
  year={2013},
  volume={abs/1307.6894}
}
We investigate the hierarchical structure of processes using the mathematical theory of operads. Information or material enters a given process as a stream of inputs, and the process converts it to a stream of outputs. Output streams can then be supplied to other processes in an organized manner, and the resulting system of interconnected processes can itself be considered a macro process. To model the inherent structure in this kind of system, we define an operad $\mathcal{W}$ of black boxes… 
Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
TLDR
This paper uses the language of operads to study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones, and defines two W-algebras, G and L, which associate semantic content to the structures in W.
Operads of Wiring Diagrams
This monograph is a comprehensive study of the combinatorial structure of various operads of wiring diagrams and undirected wiring diagrams. Our first main objective is to prove a finite presentation
Wiring diagrams as normal forms for computing in symmetric monoidal categories
TLDR
An "unbiased" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams, is presented, because the interchange law and other laws of a SMC hold identically in a wiring diagram.
Nesting of dynamical systems and mode-dependent networks
TLDR
This paper provides a formal semantics, using the category-theoretic framework of operads and their algebras, to capture the nesting property and dynamics of mode-dependent networks.
Categorical Data Structures for Technical Computing
TLDR
This paper develops the mathematical theory of acsets and describes a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.
An algebra of open continuous time dynamical systems and networks
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a
Dynamical Systems and Sheaves
TLDR
A categorical framework for modeling and analyzing systems in a broad sense, which includes lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.
Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An
Networks of open systems
  • E. Lerman
  • Mathematics
    Journal of Geometry and Physics
  • 2018
Morphisms of Networks of Hybrid Open Systems
TLDR
It is shown that a collection of relations holding among pairs of systems induces a relation between interconnected systems, and the procedure for building networks preserves morphisms of open systems, which both justifies the formalism and concretizes the intuition that a network is aCollection of systems pieced together in a certain way.
...
...

References

SHOWING 1-10 OF 25 REFERENCES
The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits
TLDR
It is shown that wiring diagrams form the morphisms of an operad $\mcT$, capturing this self-similarity, and is moved on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well.
The Art of the Propagator
TLDR
A programming model built on the idea that the basic computational elements are autonomous machines interconnected by shared cells through which they communicate that makes it easy to smoothly combine expressionoriented and constraint-based programming.
Causal Theories: A Categorical Perspective on Bayesian Networks
In this dissertation we develop a new formal graphical framework for causal reasoning. Starting with a review of monoidal categories and their associated graphical languages, we then revisit
Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation
TLDR
Focussing on transitive networks that have only one type of cell (identical cell networks), this work addresses three questions relating the network structure to dynamics, and investigates how the dynamics of coupled cell networks with different structures and numbers of cells can be related.
Lecture Notes in Mathematics
Vol. 72: The Syntax and Semantics of lnfimtary Languages. Edited by J. Barwtse. IV, 268 pages. 1968. DM 18,I $ 5.00 Vol. 73: P. E. Conner, Lectures on the Action of a Finite Group. IV, 123 pages.
The Geometry of Iterated Loop Spaces
i Preface This it the first of a series of papers devoted to the study of iterated loop spaces. Our goal is to develop a simple coherent theory which encompasses most of the known results about such
A Course on Quantum Techniques for Stochastic Mechanics
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of ‘chemical reaction networks’, which
Cycles of time : an extraordinary new view of the universe
Roger Penrose's groundbreaking and bestselling "The Road to Reality" provided a comprehensive yet readable guide to our present understanding of the laws that are currently believed to govern our
Dynamics on Networks of Manifolds
TLDR
It is proved that the appropriate maps of graphs called graph brations give rise to maps of dynamical systems, which gives rise to invariant subsystems and injective graph fibrationsGive rise to projections of dynamicals systems.
Higher Operads, Higher Categories
Part I. Background: 1. Classical categorical structures 2. Classical operads and multicategories 3. Notions of monoidal category Part II. Operads. 4. Generalized operads and multicategories: basics
...
...