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.

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