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 identies or at least like equivalences in a higher category.

The theory of simplicial weak ω-categories is based on stratified simplicial sets.

Definition

A stratification of a simplicial set $X:{\Delta }^{\mathrm{op}}\to \mathrm{Set}$ is a subset $tX\subset {\coprod }_{[n]}{X}_n$ of its set of simplices (not in general a simplicial subset!) such that

• no 0-simplex of $X$ is in $tX$;

• every degenerate simplex in $X$ is in $tX$.

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

The elements of $tX$ are called the thin simplices of $X$.

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

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

The category of startified simplicial sets and stratified maps between them is usually denoted $\mathrm{Strat}$.

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

Examples

• Every simplicial set gives rise to a stratified simplicial set

  • using the maximal stratification: all simplices are regarded as thin;

  • using the minimal stratification: only degeneracies are thin.

  These two stratifications give left and right adjoints to the forgetful functor from stratified simplicial sets to simplicial sets.

• The standard thin $n$-simplex is obtained from $\Delta [n]$ my making its only non-degenerate $n$-simplex thin.

• The $k$th standard admissible $n$-simplex ${\Delta }_k^a[n]$, defined for $n\ge 2$, $0<k<n$, is obtained from $\Delta [n]$ by making all simplices $\alpha :[m]\to [n]$ with $k-1,k,k+1\in$ im$(\alpha )$ thin.

• The standard admissible $(n-1)$-dimensional $k$-horn ${\Lambda }_k^a[n]$, defined for $n\ge 2$, $0<k<n$, is the pullback of the stratified simplicial set ${\Delta }_k^a[n]$.

• 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 $N(C)$ of a strict ω-category $C$, where the thin cells are the images of the identity cells of $C$.

• A simplicial set is a Kan complex precisely if its maximal stratification makes it a weak complicial set.

The category of stratified simplicial sets

There are several tensor products on the category $\mathrm{Strat}$ of stratified simplicial sets that make it a monoidal category.

Strat with the Verity-Gray tensor product

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

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

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

$$N:\mathrm{Str}\omega \mathrm{Cat}\to \mathrm{Strat}.$$N : Str \omega Cat \to Strat \,.

Proposition

The functor $N:\mathrm{Str}\omega \mathrm{Cat}\to \mathrm{Strat}$ has a left adjoint $F:\mathrm{Strat}\to \mathrm{Str}\omega \mathrm{Cat}$ 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:

• Dominic Verity, Weak Complicial sets and internal quasi-categories (pdf slides)