Operads of Wiring Diagrams

Description

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. Front Matter ....Pages i-xi Introduction (Donald Yau)....Pages 1-9 Front Matter ....Pages 11-12 Wiring Diagrams (Donald Yau)....Pages 13-37 Generators and Relations (Donald Yau)....Pages 39-61 Decomposition of Wiring Diagrams (Donald Yau)....Pages 63-87 Finite Presentation (Donald Yau)....Pages 89-105 Finite Presentation for Algebras over Wiring Diagrams (Donald Yau)....Pages 107-155 Front Matter ....Pages 157-157 Undirected Wiring Diagrams (Donald Yau)....Pages 159-174 Generators and Relations (Donald Yau)....Pages 175-189 Decomposition of Undirected Wiring Diagrams (Donald Yau)....Pages 191-209 Finite Presentation for Undirected Wiring Diagrams (Donald Yau)....Pages 211-219 Algebras of Undirected Wiring Diagrams (Donald Yau)....Pages 221-239 Front Matter ....Pages 241-242 A Map from Normal to Undirected Wiring Diagrams (Donald Yau)....Pages 243-267 A Map from Wiring Diagrams to Undirected Wiring Diagrams (Donald Yau)....Pages 269-285 Problems (Donald Yau)....Pages 287-289 Back Matter ....Pages 291-308