Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this does not look providing stable regular answer. 


(update)
Many thanks for comments.


My motivation for the question is following As a current work lemma i have got such a model, hopefully (not all written yet). And i am wondering how much is known in such a surface level question.


The model is a semi-simplicial set. As k simplexes we take the set of all circular permutations of k+1 elements which are the same as oriented necklaces with k+1 beads colored by different ordered colors 0,...,k. The face map is deletion i-th colored bead. This is K(Z,2) and a sort of tame semi-simplicial model for Connes' cyclic simplex. My current simple proof is that this space classifies circle bundles. The related way of proof is like in comment of André Henriques. Semi-simplicial set of permutations you can see as a contractible semi-cyclic set and have to say words about trivial role of degenerations here. 


Interestingly that here only restricting the complex up to dimension 2 you will got a deformed sphere (a couple of dunce-huts glued by "boundary circle"), but further restrictions up to dimension 2n will not provide models of $CP^n$. 



Combinatorics of K(Z,2)?
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


