n-category = (n,n)-category
n-groupoid = (n,0)-category
A strict -groupoid is an algebraic model for certain simple homotopy types/∞-groupoids based on globular sets. It is almost like a chain complex of abelian groups (under Dold-Kan correspondence) except that the fundamental group is allowed, more generally, to be non-abelian and to act on all the other homotopy groups. In fact, strict -groupoids are equivalent to crossed complexes.
Following work of J. H. C. Whitehead, in (Brown-Higgins) it is shown that the 1-category of strict -groupoids is equivalent to that of crossed complexes. This equivalence is a generalization of the Dold-Kan correspondence to which it reduces when restricted to crossed complexes whose fundamental group is abelian and acts trivially. More details in this are at Nonabelian Algebraic Topology.
There is a model structure on strict ∞-groupoids.
A textbook reference is
The equivalence of strict -groupoids and crossed complexes is discussed in
Notice that this article says ”-groupoid” for strict globular -groupoid and ”-groupoid” for strict cubical -groupoid, and also contains definitions of -fold categories, and of what are now called globular sets.