Simplicial set of permutations Let $S_k$ be  the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow  S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone reordering and degeneracy
$s_i :S_k \rightarrow S_{k+1} $ by adding 1 to all elements with $\eta(j)>\eta(i)$  and incerting into the result a new element $\eta(i)+1$ right after $\eta(i)$ on $i+1$ place. It is a simplicial set, contractible and classifies reorderings of simplicial sets. 
Is it known? May be in higher symmetric something?
(Update) Boris Tsygan pointed the right direction in Facebook duscussion
The object is classical and it has a name 
"Symmetric crossed simplicial group”.
It was introduced almost simultaneously 
in 
Appendix A10, page 191
“Symmetric objects” 
B. L. Feigin and B. L. Tsygan
 “Additive K-theory”
 1987
 K-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86
LNM 1289
Krasauskas, R.
"Skew-simplicial groups",
Lithuanian Mathematical Journal,
Jan 1987
vol 27 issue 1
p. 47--54
And independently
Zbigniew Fiedorowicz and Jean-Louis Loday “Crossed simplicial groups and their associated homology”
Trans. Amer. Math. Soc. 326 (1991), 57-87 
It has big value in everything symmetric. Geometric realization $|S_\bullet|$ is the  topological group structure on infinite dimensional sphere. 
 A: I think something equivalent (or at least closely related) to this has been studied in the combinatorics literature.  
A CW complex of course has a poset of faces.  In this case, this poset is obtained by ordering permutations by subword inclusion up to deletion and monotone reordering.  The keyword used in the combinatorics literature for this sort of subword inclusion is permutation patterns.
Now, if a CW complex is regular, then the order complex of the face poset is homeomorphic to the complex.  As you point out in the comments, the simplicial set has $n!$ faces of dimension $n$, and in particular has a single vertex.  So the simplicial set isn't regular, as I'd initially thought it might be, and your question doesn't reduce directly to this poset.  It certainly seems like the two objects should be closely related, however.
In any case, the lattice of permutations ordered by pattern containment has been studied by Jason Smith.  See, for example, the paper
Smith, Jason P., A formula for the Möbius function of the permutation poset based on a topological decomposition, Adv. Appl. Math. 91, 98-114 (2017). ZBL1370.05227.
That paper cites also his earlier papers on the topic.
