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 theorem for each operad of wiring diagrams, describing each one in terms of just a few operadic generators and a small number of generating relations. For example, the operad of wiring diagrams has 8 generators and 28 generating relations, while the operad of undirected wiring diagrams has 6 generators and 17 generating relations. Our second main objective is to prove a corresponding finite presentation theorem for algebras over each operad of wiring diagrams. As applications we provide finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the (typed) relational algebra. We also provide a partial verification of Spivak's conjecture regarding the quotient-freeness of the relational algebra. Our third main objective is to construct explicit operad maps among the several operads of wiring diagrams. In particular, there is a surjective operad map from the operad of all wiring diagrams, including delay nodes, to the operad of undirected wiring diagrams. This monograph is intended for graduate students, mathematicians, scientists, and engineers interested in operads and wiring diagrams. Assuming no prior knowledge of categories, operads, and wiring diagrams, this monograph is self-contained and can be used as a supplement in a graduate course and for independent study. There are over 100 graphical illustrations and a chapter with a list of problems.