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We investigate the hierarchical structure of processes using the mathematicaltheory of operads. Information or material enters a given process as a streamof inputs, and the process converts it to a stream of outputs. Output streamscan then be supplied to other processes in an organized manner, and theresulting system of interconnected processes can itself be considered a macroprocess. To model the inherent structure in this kind of system, we define anoperad $\mathcal{W}$ of black boxes and directed wiring diagrams, and we definea $\mathcal{W}$-algebra $\mathcal{P}$ of processes (which we call propagators,after Radul and Sussman). Previous operadic models of wiring diagrams useundirected wires without length, useful for modeling static systems ofconstraints, whereas we use directed wires with length, useful for modelingdynamic flows of information. We give multiple examples throughout to groundthe ideas.

The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes