The $\omega$-categorification of Algebraic Theories

TL;DR

This paper develops a monadic framework for weak $oldsymbol{oldsymbol{oldsymbol{ extomega}}}$-categorifications of algebraic theories using globular operads and PROs, extending classical operad theory to higher categories.

Contribution

It introduces a method to construct a monad for weak $oldsymbol{oldsymbol{oldsymbol{ extomega}}}$-categorifications of any algebraic theory via globular PROs and globularization, extending Leinster's globular operads.

Findings

Constructed a monad for weak $oldsymbol{ extomega}$-categorifications.

Extended globular operads to globular PROs via categorical enrichment.

Provided a process to obtain fully weakened $oldsymbol{ extomega}$-categorifications.

Abstract

Batanin and Leinster's work on globular operads has provided one of many potential defnitions of a weak $\omega$-category. Through the language of globular operads they construct a monad whose algebras encode weak $\omega$-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak $\omega$-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 $\omega$-categories and strict $\omega$-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 weakenings of the globularization of the classica PRO P. Among these weakenings is the initial weakining whose algebras are by construction the fully weakened $\omega$-categorifications of the algebraic theory encoded by P.