globular nerves of simplicially enriched groupoids there is as far as i know essentially just one reasonable concept of the "globular nerve" of a simplicially enriched groupoid. perhaps the easiest if not the most obvious way to describe this concept is as follows: define the "simplicial j-globe" g_j to be the simplicial set obtained as the image of the simplicial map from the j-simplex to the "co-discrete" simplicial set on 2 vertexes v0 and v1 (= simplicial nerve of the "co-discrete" or "chaotic" category with 2 objects v0 and v1), assigning v0 to the even-numbered vertexes of the j-simplex and v1 to the odd-numbered vertexes. the good news is that the geometric realization of the simplicial j-globe g_j really does look like a "j-globe" (that is a j-dimensional "disk" or "ball"), with it's boundary decomposed into 2 [j-1]-globe "hemispheres", and so on. the bad news is that the sequence g_j, j=0,1,... doesn't form a co-globular object (in batanin's sense) in the category of simplicial sets, because the orientations of the hemispheres don't match up the right way; instead of: leading hemisphere of boundary of leading hemisphere of boundary = leading hemisphere of boundary of trailing hemisphere of boundary and: trailing hemisphere of boundary of leading hemisphere of boundary = trailing hemisphere of boundary of trailing hemisphere of boundary, we have: leading hemisphere of boundary of leading hemisphere of boundary = trailing hemisphere of boundary of trailing hemisphere of boundary and: trailing hemisphere of boundary of leading hemisphere of boundary = leading hemisphere of boundary of trailing hemisphere of boundary. thus for example the boundary of the simplicial 2-globe g_2 looks like: . ->. <- instead of like: . -> . -> as we would prefer it. when simplicial sets are functorially mapped to bi-pointed simplicially enriched groupoids by the left adjoint f of the functor "hom(v0,v1)", however, the existence of inverses in a simplicially enriched groupoid provides enough flexibility to repair the orientation mismatch problem. i won't try here to give a precise formula for the boundary-facial inclusion maps that make the sequence f(g_j), j=0,1,... into a co-globular object in the category of bi-pointed simplicially enriched groupoids, but it can be done. notice that a "co-globular bi-pointed x" can be re-interpreted as a "co-globular x", promoting the model globes up one dimension and taking the point as the new model 0-globe (if the category of x's is sufficiently nice as in the case here). once we have the desired co-globular object g in the category of simplicially enriched groupoids, we can define the "globular nerve" functor from simplicially enriched groupoids to globular sets as the right adjoint of it's co-continuous extension, or as composing the yoneda embedding with restriction of pre-sheafs along g. there are various possibly interesting things to try to do with this concept of globular nerve of a simplicially enriched groupoid. (of course the fact that simplicially enriched groupoids are an especially nice way of representing homotopy types is what drives much of the interest here.) one obvious possibility is to study the higher-dimensional operad o obtained as "the natural theory of the globular nerve of a simplicially enriched groupoid" (when this latter concept is formalized in any of a number of more or less equivalent ways). o is a contractible higher-dimensional operad, and probably interesting in a lot of ways, but on the other hand it's also somewhat unpleasantly big. for example, for the 2-stage tree: /\ / \, there are an infinite number of operations of this type in o, which is rather a lot compared to, for example, the just two such operations in the "natural theory of the underlying globular set of a gray-category". a possibly more interesting thing to try is to look for a sub-operad o' of o such that simplicially enriched groupoids are essentially o'-algebras with the extra property that all globes in them are strictly invertible (in a sense that can easily be made precise in this context). if such an operad o' exists then it's probably a good candidate for "the theory of semi-strict infinity-categories". i would like to resolve the question of whether such an operad o' exists.