Modular models of monoids with operations by lifting functors along fibrations

Inspired by Plotkin and Power's algebraic treatment of computational effects and the principle of notions of computations as monoids, we propose a categorical framework for equational theories and models of monoids equipped with operations. This framework generalises Plotkin and Power's algebraic treatment of effectful operations taking or returning values as input or output to operations that may take or return computations as input or output. Additionally, to give semantic models of computational effects in a modular way, we introduce a formal theory of modular constructions of algebraic structures based on the framework of lifting functors along fibrations.