nLab
stratified simplicial set

Idea

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 ∞-categories is based on stratified simplicial sets.

Definition

A stratification of a simplicial set X:Δ opSetX : \Delta^{op} \to Set is a subset tX [n]X nt X \subset \coprod_{[n]} X_n of its set of simplices (not in general a simplicial subset!) such that

A stratified simplicial set is a pair (X,tX)(X, t X) consisting of a simplicial set XX and a stratification tXt X of XX.

The elements of tXt X are called the thin simplices of XX.

For (X,tX)(X, t X) and (Y,tY)(Y, t Y) stratified simplicial sets, a morphism f:XYf : X \to Y of simplicial sets is set to be a stratified map if it respects thin cells in that

f(tX)tY. f(t X ) \subset t Y \,.

The category of stratified simplicial sets and stratified maps between them is usually denoted StratStrat.

This category is a quasitopos. Hence, in particular, it is cartesian closed.

Examples

The category of stratified simplicial sets

There are several tensor products on the category StratStrat of stratified simplicial sets that make it a monoidal category.

Strat with the Verity-Gray tensor product

Consider the monoidal category (Strat,)(Strat, \otimes) where \otimes is the Verity-Gray tensor product.

(Notice that this is not closed, as far as I understand.)

Using the canonical stratification of ∞-nerves on strict ∞-categories as complicial sets, the ω\omega-nerve is a functor

N:StrωCatStrat. N : Str \omega Cat \to Strat \,.
Proposition

The functor N:StrωCatStratN : Str \omega Cat \to Strat has a left adjoint F:StratStrωCatF : Strat \to Str \omega Cat which is a strong monoidal functor.

Or so it is claimed on slide 60 of Ver07

References

A useful quick introduction is the beginning of these slides: