  In this paper we apply some tools developed in our previous work on
Grothendieck $\infty$-groupoids to the finite-dimensional case of weak
3-groupoids.
  We obtain a semi-model structure on the category of Grothendieck 3-groupoids
of suitable type, thanks to the construction of an endofunctor $\mathbb{P}$
that has enough structure to behave like a path object. This makes use of a
recognition principle we prove here that characterizes globular theories whose
models can be viewed as Grothendieck $n$-groupoids (for $0\leq n \leq \infty$).
Finally, we prove that the obstruction in arbitrary dimension (possibly
infinite) only resides in the construction of (slightly less than) a path
object on a suitable category of Grothendieck (weak) $n$-categories with weak
inverses. This also gives a sufficient condition for endowing an $n$-groupoid
\`a la Batanin with the structure of a Grothendieck $n$-groupoid.
