n-category = (n,n)-category
n-groupoid = (n,0)-category
A -admissible -simplex in a stratified simplicial set is a map of stratified simplicial sets . Explicitly, this consists of a (thin) -simplex such that is thin for every whose image contains , , and .
A complicial set is a stratified simplicial set satisfying the following three axioms:
if is a -admissible simplex whose th and th faces are thin, then its th face is also thin;
there is a unique thin filler for all -dimensional inner -horns whose faces are -admissible and whose faces are -admissible (nb: there is no condition on );
all thin 1-simplices are degenerate.
Equivalently, a complicial set is a stratified simplicial set that is (right) orthogonal to each of the following classes of stratified maps:
the primitive -extensions , where has all -simplices except thin, has all -simplices thin, and both stratified sets have any simplex with im thin;
the inclusions for all , ;
the unique surjection , where every 1-simplex in is thin.