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 ω-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

  • no 0-simplex of XX is in tXt X;

  • every degenerate simplex in XX is in tXt X.

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

  • 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 nn-simplex is obtained from Δ[n]\Delta[n] my making its only non-degenerate nn-simplex thin.

  • The kkth standard admissible nn-simplex Δ k a[n]\Delta^a_k[n], defined for n2n \geq 2, 0<k<n0 \lt k \lt n, is obtained from Δ[n]\Delta[n] by making all simplices α:[m][n]\alpha \colon [m] \to [n] with k1,k,k+1k-1,k,k+1 \in im(α)(\alpha) thin.

  • The standard admissible (n1)(n-1)-dimensional kk-horn Λ k a[n]\Lambda^a_k[n], defined for n2n \geq 2, 0<k<n0 \lt k \lt n, is the pullback of the stratified simplicial set Δ k a[n]\Delta^a_k[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)N(C) of a strict ω-category CC, where the thin cells are the images of the identity cells of CC.

  • 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 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 ω-nerves on strict ω-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:

Revised on December 7, 2013 22:14:47 by Tim Porter (2.26.35.180)