Operads in Higher-Dimensional Category Theory

TL;DR

This paper develops a generalized operad framework to define and analyze weak n-categories, providing a new language for higher-dimensional algebra and simplifying existing approaches.

Contribution

It introduces a theory of generalized operads and multicategories to define weak n-categories, offering a new perspective and tools in higher-dimensional category theory.

Findings

Proposes a new definition of weak n-category.

Explains the case when n=2.

Shows how operads simplify higher-dimensional algebra.

Abstract

The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n=2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.