A stratified simplicial set is a simplicial set equipped with information about which of its simplices are to be regarded as being thin in that they are like identities or at least like equivalences in a higher category.
The theory of simplicial weak ω-categories is based on stratified simplicial sets.
A stratification of a simplicial set is a subset of its set of simplices (not in general a simplicial subset!) such that
no 0-simplex of is in ;
every degenerate simplex in is in .
A stratified simplicial set is a pair consisting of a simplicial set and a stratification of .
The elements of are called the thin simplices of .
For and stratified simplicial sets, a morphism of simplicial sets is set to be a stratified map if it respects thin cells in that
The category of stratified simplicial sets and stratified maps between them is usually denoted .
Every simplicial set gives rise to a stratified simplicial set
using the maximal stratification: all simplices of dimension >0 are regarded as thin;
using the minimal stratification: only degenerate simplices are thin.
The standard thin -simplex is obtained from by making its only non-degenerate -simplex thin.
The th standard admissible -simplex , defined for , , is obtained from by making all simplices with im thin.
The standard admissible -dimensional -horn , defined for , , is the pullback of the stratified simplicial set .
A complicial set is a stratified simplicial set satisfying certain extra conditions. Complicial sets are precisely those simplicial sets which arise (up to isomorphism) as the ω-nerve of a strict ω-category , where the thin cells are the images of the identity cells of .
Consider the monoidal category where is the Verity-Gray tensor product.
(Notice that this is not closed, as far as I understand.)
A useful quick introduction is the beginning of these slides: