In this book, the authors give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI-modules.

The usefulness of this approach is that it allows the authors to handle all the classical as well as more esoteric structures under a common framework and treat all the situations simultaneously. Many of the known constructions simply become Kan extensions. In this common framework, the authors also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras, as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, the authors construct the relevant model category structures. This produces many new examples.