Globular PROs and the weak ω-categorification of algebraic theories

Date

2019-08-01

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Abstract

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.

Description

Keywords

Globular PRO, Higher category theory, Globular operad, Duoidal categories, Enriched monoidal categories, Monads

Graduation Month

August

Degree

Doctor of Philosophy

Department

Department of Mathematics

Major Professor

David Yetter

Date

2019

Type

Dissertation

Citation