Algebraic Definition of weak ($\infty$; n)-Categories

TL;DR

This paper introduces a sequence of monads on infinity-graphs to algebraically model weak (; n)-categories and weak -groupoids, proposing conjectures linking algebraic structures to higher category theory.

Contribution

It defines new monads on infinity-graphs and conjectures their algebras model weak -groupoids and (; n)-categories in an algebraic framework.

Findings

Defines monads (;n) on --graphs.

Proposes conjecture linking algebraic algebras to weak -groupoids.

Proposes conjecture linking algebraic algebras to weak (; n)-categories.

Abstract

In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely algebraic setting, are models of weak $\infty$-groupoids. And for all $n>1$ we conjecture that algebras for $\mathbb{T}^(\infty;n)$ which are defined in a purely algebraic setting, are models of weak $(\infty; n)$-categories.