Involutive Weak Globular Higher Categories

TL;DR

This paper explores involutive weak globular $oldsymbol{ extomega}$-categories using Jacque Penon's framework, constructing free structures, defining monadic concepts, and providing examples to advance understanding of higher category theory.

Contribution

It introduces a monadic framework for involutive weak globular $oldsymbol{ extomega}$-categories, including free constructions and explicit examples, expanding the theoretical foundation.

Findings

Constructed free self-dual globular $oldsymbol{ extomega}$-magmas.

Defined monadic structures for involutive weak globular $oldsymbol{ extomega}$-categories.

Provided concrete examples illustrating the concepts.

Abstract

We investigate the notion of involutive weak globular $\omega$-categories via Jacque Penon's approach. In particular, we give the constructions of a free self-dual globular $\omega$-magma, of a free strict involutive globular $\omega$-category, over an $\omega$-globular set, and a contraction between them. The monadic definition of involutive weak globular $\omega$-categories is given as usual via algebras for the monad induced by a certain adjunction. In our case, the adjunction is obtained from the "free functor" that associates to every $\omega$-globular set the above contraction. Some examples of involutive weak globular $\omega$-categories are also provided.