An algebraic definition of ($\infty$,n)-categories

Camell Kachour

In this paper we define a sequence of monads $T^{(\infty,n)} (n\in\mathbb{N})$ on the category $\infty-Gr$ of $\infty$-graphs. We conjecture that algebras for $\T^{(\infty,0)}$, which are defined in a purely algebraic setting, are models of $\infty$-groupoids. More generally, we conjecture that $T^{(\infty,n)}$-algebras are models for $(\infty,n)$-categories. We prove that our $(\infty,0)$-categories are bigroupoids when truncated at level 2.

Keywords: ($\infty$,n)-categories, weak $\infty$-groupoids, homotopy types

2010 MSC: 18B40,18C15, 18C20, 18G55, 20L99, 55U35, 55P15

Theory and Applications of Categories, Vol. 30, 2015, No. 22, pp 751-774.

Published 2015-06-01.

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