An extension of Batanin's approach to globular algebras

TL;DR

This paper extends Batanin's monadic approach to globular algebras, introducing new constructions, properties, and criteria to better understand higher categories and their algebraic structures.

Contribution

It refines Batanin's framework by defining cellular extensions, free constructions, and criteria for globular monads, enhancing the understanding of algebraic globular higher categories.

Findings

Defined the notion of cellular extension and its free construction.

Established an adjunction between globular algebras and polygraphs.

Introduced criteria to simplify the use of constructions without explicit globular monad descriptions.

Abstract

In earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced by Street and Burroni for strict categories, to all algebraic globular higher categories. In this work, we refine this perspective and introduce new constructions and properties for this class of higher categories. In particular, we define the notion of cellular extension and its associated free construction, from which we obtain another definition of polygraphs and the adjunction between globular algebras and polygraphs. We moreover introduce two criteria allowing one to use most of the constructions of this article without having to describe explicitly the underlying globular monad.