• 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

Book

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

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
6. Donald Yau
Pages 107-155
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 191-209
5. Donald Yau
Pages 211-219
6. Donald Yau
Pages 221-239
5. ### Part III

1. Front Matter
Pages 241-242
2. Donald Yau
Pages 243-267
3. Donald Yau
Pages 269-285
4. Donald Yau
Pages 287-289
6. Back Matter
Pages 291-308

### Introduction

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.

### Keywords

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

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

• Book Title Operads of Wiring Diagrams
• Authors Donald Yau
• Series Title Lecture Notes in Mathematics
• Series Abbreviated Title Lect.Notes Mathematics
• DOI https://doi.org/10.1007/978-3-319-95001-3
• Copyright Information Springer Nature Switzerland AG 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-95000-6
• eBook ISBN 978-3-319-95001-3
• Series ISSN 0075-8434
• Series E-ISSN 1617-9692
• Edition Number 1
• Number of Pages XI, 308
• Number of Illustrations 437 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site