Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$-simplex in $K$; face maps are defined in the obvious way. One can add in degeneracies by hand to make this a simplicial set.

In general, $|F(K)|$ is not homotopy equivalent to $K$; for example, $|F(\Delta^n)|$ is a wedge of $n$-spheres. (By contrast, the "singular simplicial set" where we don't require the $(n+1)$-tuple to form a $n$-simplex but merely any simplex, is homotopy equivalent and even homeomorphic in the geometric realization to $|K|$: see this question) This I learned from the discussion here. However, I wanted to know what information in the literature exists about the relationship between $|K|$ and $|F(K)|$ in general: for example, are there some restrictions on the homology or homotopy of $|F(K)|$ based on the homology or homotopy of $|K|$?