  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 will show how to take
a theory of n-categories of the former kind and produce a globular operad whose
algebras are the n-categories 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 iterating the weak enrichment
construction using a series of parametrising operads P_i. We then show how to
construct from such a theory an n-dimensional globular operad for each $n \geq
0$ 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 P_i 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.
