• 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… 

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References

SHOWING 1-10 OF 23 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

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.

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

Théorie des nombres

Let R be a c o m m u t a t i v e r ing with 1. À bilinear fo rm over R is a pa i r ( M , b) where M is a finitely gene ra t ed pro jec t ive A m o d u l e and b : M x M —• R is s y m m e t r i c

The geometry of iterated loop spaces

Operads and -spaces.- Operads and monads.- A? and E? operads.- The little cubes operads .- Iterated loop spaces and the .- The approximation theorem.- Cofibrations and quasi-fibrations.- The smash