$K(Z,2)$ out of circular permutations

This is a small addition to the note arXiv:1908.04029v2 [math.GT]. We briefly discuss $SC$, a simplicial homotopy model of $K(Z,2)$ constructed from circular permutations. In any dimension, the number of simplices in the model is finite. The complex $SC$ naturally manifests as a simplicial set representing "minimally" triangulated circle bundles over simplicial bases. On the other hand, the homotopy |SC| ≈ B(U(1)) ≈ K(Z,2) appears to be a canonical fact from the foundations of crossed simplicial groups theory.