Operadic definitions of weak n-category: coherence and comparisons

Thomas Cottrell

This paper concerns the relationships between notions of weak n-category defined as algebras for n-globular operads, as well as their coherence properties. We focus primarily on the definitions due to Batanin and Leinster.

A correspondence between the contractions and systems of compositions used in Batanin's definition, and the unbiased contractions used in Leinster's definition, has long been suspected, and we prove a conjecture of Leinster that shows that the two notions are in some sense equivalent. We then prove several coherence theorems which apply to algebras for any operad with a contraction and system of compositions or with an unbiased contraction; these coherence theorems thus apply to weak $n$-categories in the senses of Batanin, Leinster, Penon and Trimble.

We then take some steps towards a comparison between Batanin weak n-categories and Leinster weak n-categories. We describe a canonical adjunction between the categories of these, giving a construction of the left adjoint, which is applicable in more generality to a class of functors induced by monad morphisms. We conclude with some preliminary statements about a possible weak equivalence of some sort between these categories.

Keywords: n-category, operad, higher-dimensional category

2010 MSC: 18D05, 18D50, 18C15

Theory and Applications of Categories, Vol. 30, 2015, No. 13, pp 433-488.

Published 2015-04-08.


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