We have the category, $S$, of simplicial sets and $SS$, symmetric simplicial sets (whose simplices are unordered). There are functors

$H:S\to SS$ forgetting the ordering on simplices and

$L:SS\to S$ mapping an unordered simplex to all possible ordered simplices that could correspond to it (so an unordered $n$-simplex maps to $n!$ ordered simplices).

A monograph of Denis-Charles Cisinski defined a model-structure on $SS$ and showed that these functors define a Quillen equivalence with the usual model structure on $S$.

For any simplicial set $X$, there is a canonical cofibration

$\eta_X:X\to LHX$ --- every simplix of $X$ is augmented by all possible other orderings of its vertices.

Question: Is there a class of simplicial set (like fibrant ones, for instance) for which $\eta_X$ is a weak equivalence?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:

$H:S\to SS$ forgetting the ordering on simplices and

$L:SS\to S$ mapping an unordered simplex to all possible ordered simplices that could correspond to it (so an unordered $n$-simplex maps to $n!$ ordered simplices).

A monograph of Denis-Charles Cisinski defined a model-structure on $SS$ and showed that these functors define a Quillen equivalence with the usual model structure on $S$.

For any simplicial set $X$, there is a canonical cofibration

$\eta_X:X\to LHX$ --- every simplex of $X$ is augmented by all possible other orderings of its vertices.

Question: Is there a class of simplicial sets (like fibrant ones, for instance) for which $\eta_X$ is a weak equivalence?