n-category = (n,n)-category
n-groupoid = (n,0)-category
There is quite a difference between the Kan complex structure
In the first, if we are given a -horn, then there is exactly one -simplex in , since the -horn has a chain of -composable arrows of in it (at least unless or , which cases are slightly different) and that chain gives the required -simplex. In other words, there is a ‘canonical’ filler for any horn. In , there will usually be many fillers; however the fact that this simplicial set is Kan is a property of retractions on standard simplices, and is not specifically a property of the space - that is the basic intuition.
Abstracting, in part, from this idea, Brown and Higgins developed the idea of a cubical T-complex. This was a cubical set with in each dimension , a subset of the -cubes being declared ‘thin’. The term was adopted to indicate that they, somehow, were of lower dimension than they looked to be. The theory was initiated in a simplicial context in the 1977 Bangor thesis of Keith Dakin listed below, and used by Brown and Higgins who showed that cubical -complexes were equivalent to crossed complexes. The corresponding simplicial -complex theory was further developed in the 1978 Bangor thesis of Nick Ashley, (see below for publication).
Relevant references for simplicial T-complexes include:
M.K. Dakin, Kan complexes and multiple groupoid structures, Ph.D Thesis, University of Wales, Bangor, 1977. Esquisses Math. (1983) 32 Part 2
N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Ph.D Thesis University of Wales, Bangor, 1978; Dissertationes Math., 165, (1989), 11 – 58. Esquisses Math. (1983) 32 Part 3
G. Nan Tie, A Dold-Kan theorem for crossed complexes, J. Pure Appl. Alg., 56, (1989.), 177 -– 194.
G. Nan Tie, Iterated W and T-groupoids, J. Pure Appl. Alg., 56, (1989), 195 -– 209.
Ronnie Brown, and P.J. Higgins, On the algebra of cubes J. Pure Appl. Algebra 21 (1981) 233–260.
R. Brown, and P.J. Higgins, The equivalence of -groupoids and cubical -complexes Cahiers Topologie G'eom. Diff'erentielle 22 (1981) 349–370.