n-category = (n,n)-category
n-groupoid = (n,0)-category
A cubical -complex is a cubical set which is also a special kind of Kan complex, in that it has a family (for ) of thin elements with the properties, (due first to Keith Dakin, for the simplicial case, in his doctoral thesis):
T1) Degenerate elements are thin.
T2) Every box in has a unique thin filler.
T2) If all but one face of a thin element are thin, then so also is the remaining face.
A modification of axiom T2) to be more like an axiom for a quasi-category is suggested in the paper by Steiner referenced below.
T-complexes first appeared in:
A main application of the idea of thin elements in the cubical case was to define the notion of commutative cube. In the case of cubical -groupoids with connection, the boundary of a cube is commutative if and only if the boundary has a thin filler. One consequence is that any well defined composition of thin elements is thin. This is a key element of the proof of the Higher Homotopy van Kampen theorem in the colimits paper, which thus nicely generalises the proof of the 1-dimensional van Kampen theorem for the fundamental groupoid on a set of base points.
Cubical T-complexes are thus a key concept in the long series of articles by Brown and Higgins on strict omega-groupoids. For instance
Ronnie Brown, P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233-260.
R. Brown, P.J. Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11-41.
R. Brown, P.J. Higgins, Tensor products and homotopies for -groupoids and crossed complexes, J. Pure Appl. Alg. 47 (1987) 1-33.
R. Brown, R. Street, Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product, Cah. Top. G'eom. Diff. Cat., 52 (2011) 188-208.
For more on ‘thinness’ in a cubical context, see:
P.J. Higgins, Thin elements and commutative shells in cubical -categories, Theory Appl. Categ., 14 (2005) 60–74.
R. Steiner, Thin fillers in the cubical nerves of omega-categories, Theory Appl. Categ. 16 (2006), 144–173.