© 2018

Operads of Wiring Diagrams

  • Provides a self-contained introduction to wiring diagrams, operads, and operad algebras

  • Includes over 100 illustrations and a chapter of practice problems

  • Presents generators and relations for both the operads and algebras of wiring diagrams introduced by David Spivak


Part of the Lecture Notes in Mathematics book series (LNM, volume 2192)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Donald Yau
    Pages 1-9
  3. Part I

    1. Front Matter
      Pages 11-12
    2. Donald Yau
      Pages 13-37
    3. Donald Yau
      Pages 39-61
    4. Donald Yau
      Pages 63-87
    5. Donald Yau
      Pages 89-105
  4. Part II

    1. Front Matter
      Pages 157-157
    2. Donald Yau
      Pages 159-174
    3. Donald Yau
      Pages 175-189
    4. Donald Yau
      Pages 221-239
  5. Part III

    1. Front Matter
      Pages 241-242
    2. Donald Yau
      Pages 287-289
  6. Back Matter
    Pages 291-308

About this book


Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads.

The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of the relational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images.

Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.


MSC (2010): 18D50, 37A60, 55P48, 81P45, 94C15 wiring diagrams undirected wiring diagrams operads colored operads operad algebras operad maps propagator algebra discrete systems open dynamical systems relational algebra information and communication, circuits

Authors and affiliations

  1. 1.Department of MathematicsThe Ohio State University at NewarkOhioUSA

About the authors

Donald Yau is Professor of Mathematics at The Ohio State University at Newark. He obtained his PhD at MIT and held a post-doctoral position at the University of Illinois at Urbana-Champaign. His research focuses on algebraic topology. He has so far authored five books and over 30 articles.

Bibliographic information