Comparing operadic theories of n-category

We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories 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 will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the ncategories we started with. 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 an operad P in the category V. 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 n-categories we constructed by iterated weak enrichment, 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. This has the potential to provide an easier method for proving Batanin’s Homotopy Hypothesis than the partial proofs currently in the literature. The general theory provides a framework for using everything known about classical operads to study higher-dimensional categories; it also has the potential to provide a useful class of semi-strict n-categories for studying coherence.