Globular PROs and the weak ω-categorification of algebraic theories

Batanin and Leinster’s work on globular operads has provided one of many potential definitions of a weak ω-category. Through the language of globular operads they construct a monad whose algebras encode weak ω-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak ω-categorifications of any equational algebraic theory. We first review the classical theory of operads and PROs. We then present how Leinster’s globular operads can be extended to a theory of globular PROs via categorical enrichment over the category of collections. It is then shown how a process called globularization allows us to construct from a classical PRO P a globular PRO whose algebras are those algebras for P which are internal to the category of strict ω-categories and strict ω-functors. Leinster’s notion of a contraction structure on a globular operad is then extended to this setting of globular PROs in order to build a monad whose algebras are globular PROs with contraction over the globularization of the classical PRO P. Among these PROs with contraction over P is the globular PRO whose algebras are by construction the fully weakened ω-categorifications of the algebraic theory encoded by P