This is the simplicial set / complex constructed by Volodin, using a construction similar to that of the Vietoris complex. It is the Volodin space of the family of subgroups of the stable general linear group described as follows:

We let $T_n^\sigma(R)$ be the subgroup of $G\ell_n(R)$ formed by the $\sigma$-triangular matrices, (discussed at higher generation by subgroups), and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell(R)$ as the colimit of the nested sequence of all the $G\ell_n(R)$, take $G = G\ell(R)$. Considering the family, $\mathcal{H}$, of all the $T_n^\sigma(R)$, form the corresponding Volodin space.

References

A. A. Suslin and M. Wodzicki, Excision in algebraic K-theory, The Annals of Mathematics, 136, (1992), 51 – 122.

I. Volodin, Algebraic K-theory as extraordinary homology theory on the category of associative