Hi Jamie, You said : "Batanin, Leinster and other have presented related definitions of weak n-groupoid in terms of contractible globular operads.". I personally find these definitions of "contractible n-groupoids" extremely beautiful. To be more precise they gave an operadic approach of weak higher categories with which we can extract a definition of weak n-groupoids and can say : a weak n-groupoid is a specific algebra for the operad K of weak higher categories (build first by Batanin). However it is important to know that neither Batanin or Leinster have defined a monad, specific to higher groupoids, which algebras are models of globular weak higher groupoids. However this was done in my work here : http://www.tac.mta.ca/tac/volumes/30/22/30-22.pdf where in particular I proved that my models of weak higher groupoids are also algebras for the operad K of Batanin (which algebras are his definition of weak higher categories). Remark : And with similar methods we can go beyond, and build cubical and multiple weak higher groupoids, but this is an other story ... (see my arxived work ...) The homotopy hypothesis for these globular weak higher groupoids (those defined by Batanin in 1998, or the definition of Grothendieck-Maltsiniotis, or my approach), seems to be a difficult problem (for that it is good to see the work of Ara (thesis), Tuy=C3=A9ras (thesis) and Simon Henry), and it is not evident at all that the homotopy hypothesis is in fact true. However we suspect it to be true only based on the fact that Kan-complexes models homotopy of spaces, and we suspect that there is a Quillen model structure on the category of weak globular higher groupoids which is Quillen equivalent to the category of Kan-complexes equipped with the induced model structure on the category of simplicial sets. In fact, in=C2=A0http://www.tac.mta.ca/tac/volumes/30/22/30-22.pdf, I said = that we have a generalized version of the homotopy hypothesis of Grothendieck, which is the statement that the category of globular weak (infinity,N)-categories (which is the category of algebras for a fixed monad, for each integer N; and these algebras are still algebras for the operad K of Batanin !), should be Quillen equivalent to the category of other simplicial models of (infinity,N)-categories. Best, Camell. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]