A generalization of categorification, and higher "theory" of algebras

TL;DR

This paper introduces 'higher theories' of algebras, a new mathematical framework that generalizes categorification and connects to higher categories, aiming to deepen understanding of algebraic structures.

Contribution

It defines the concept of theorization, a process that generalizes categorification, and develops a hierarchy of higher theories extending higher categories.

Findings

Introduction of higher theories as iterated generalizations of operads

Establishment of theorization as a unifying process

Connection of hierarchies of theories to higher categories

Abstract

We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of "algebraic theory". Indeed, our "theories" are `higher order' generalizations of coloured operad or multicategory, where we see an operad as analogous to Lawvere's theory. Higher theories are obtained by iterating a certain process, which we call "theorization", generalizing categorification in the sense of Louis Crane. The hierarchy of all iterated theorizations contains in particular, the hierarchy of all higher categories. As an expanded introduction to the mentioned work, we here introduce the notion of theorization, discuss basic ideas, notions, examples, facts and problems about theorization, and describe how these lead to our work, and what will be achieved.