Combinatorics of K(Z,2)?

I have some idea, but I'm not sure it works well.

First I give a triangulation to $CP^1$: let's say that $CP^1$ is $C$ with a point $\infty$ added. Then we can fix 6 vertices, $0,1,-1,i,-i,\infty$ and divide $CP^1$ in 6 vertices, 12 edges and 8 triangles. Each complex number belongs to one of the 8 triangles according to the sign of its real part, the sign of its imaginary part and whether its modulus is greater or lower than 1.

So there are 26 types of points in $CP^1$: 3 choices for the sign (or zero) for the imaginary part, 3 choices for the sign (or zero) for the real part, 3 choices for the sign of the logarithm of the modulus would yield 27, but it is not possible to have 0 real, 0 imaginary and 1 modulus (we say that $\infty$ has modulus greater than 1, but its imaginary and real parts are 0). Note that a complex number (not $\infty$) has only 25 types: the ones different from the $\infty$ type.

More generally, every homogeneous $n+1$-tuple $[z_0,\dots, z_n]$ in $CP^n$ has exactly one representative whose first nonzero coordinate is 1. The following coordinates are just complex numbers, so each of them is in one of the 25 types stated before. So we associate to a point in $CP^n$ the following data: 1) The position of the first nonzero coordinate, which is some integer $0\leq k\leq n+1$; 2) the $n-k$-tuple of the types of complex numbers in the preferred representative.

Points with the same data are in the same internal part of the same simplex, and the dimension of a simplex is given by the number of < and > appearing in point 2 (I have no time now to explain in detail, but I hope it's quite clear).

For example, vertices are points of the form $[0,0,0,1,i,0,-i,0,1,-1,0]$, that is, $n+1$-tuplesmade just of $0,1,-1,i,-i$ with the first nonzero coordinate equal to 1.

Here is my model https://arxiv.org/abs/1908.04029 called in the paper $\pmb SC$. The homotopy issue only mentioned and postponed to more general later writings but it is exactly what was mentioned by @AndréHenriques — factor of symmetric cross-simplicial group $\pmb S$ (which is contractible) by free right acton of Connes’ cyclic group