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Operadic definitions of weak n-category: coherence and comparisons

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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.

http://www.tac.mta.ca/tac/volumes/30/13/30-13.pdf

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