We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We first provide a generalisation of Trimble's original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by a V-operad P. We define weak n-categories by iterated weak enrichment using a series of parametrising operads Pi. We then show how to construct from such a theory an n-dimensional globular operad for each n ≥ 0 whose algebras are precisely the weak n-categories, and we show that the resulting globular operad is contractible precisely when the operads Pi are contractible. We then show how the globular operad associated with Trimble's topological definition is related to the globular operad used by Batanin to define fundamental n-groupoids of spaces.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 2, pp.217-248.