I asked myself exactly this question the other day (while looking back at Björner's handbook article), and I poked around in Björner's papers looking for an answer. My guess is that Björner was referring to the argument, attributed to Quillen, that is found on p. 92 of his article [Homotopy type of posets and lattice complementation][1].

Quillen's argument is a great little microcosm of homotopy theoretical ideas. While reminiscent of the one in Björner's Handbook article, instead of using a map that goes directly between the original complex and the nerve, Quillen cooks up a third space that maps to both by homotopy equivalences (a tried and true technique). The third space consists of all pairs $(\sigma, F)$ where $\sigma$ is a simplex in $K$ and $F$ is an element of the nerve containing $\sigma$ in its intersection (that is, $F$ is a finite subset of $I$, and $\sigma\in Y_i$ for all $i\in I$); this is a poset, via the natural orderings on $K$ and on $\mathcal{N}$ by set-theoretic inclusion. Note that this poset is kind of like the graph of a multifunction version of the map Björner uses in the Handbook article: Björner shows that the map sending $\sigma$ to the largest such subset $F$ is a homotopy equivalence, but when no largest $F$ exists, instead of making a choice, Quillen just accepts the full swath of choices. As is so often the case in homotopy theory, the indeterminacy is in some sense contractible (the poset consisting of all $F$ containing $\sigma$ is the face poset of an infinite simplex), and so everything works out. Of course Quillen needs a version of his Fiber Theorem (aka Theorem A).

I asked myself exactly this question the other day (while looking back at Björner's handbook article), and I poked around in Björner's papers looking for an answer. My guess is that Björner was referring to the argument, attributed to Quillen, that is found on p. 92 of his article Homotopy type of posets and lattice complementation.

Quillen's argument is a great little microcosm of homotopy theoretical ideas. While reminiscent of the one in Björner's Handbook article, instead of using a map that goes directly between the original complex and the nerve, Quillen cooks up a third space that maps to both by homotopy equivalences (a tried and true technique). The third space consists of all pairs $(\sigma, F)$ where $\sigma$ is a simplex in $K$ and $F$ is an element of the nerve containing $\sigma$ in its intersection (that is, $F$ is a finite subset of $I$, and $\sigma\in Y_i$ for all $i\in I$); this is a poset, via the natural orderings on $K$ and on $\mathcal{N}$ by set-theoretic inclusion. Note that this poset is kind of like the graph of a multifunction version of the map Björner uses in the Handbook article: Björner shows that the map sending $\sigma$ to the largest such subset $F$ is a homotopy equivalence, but when no largest $F$ exists, instead of making a choice, Quillen just accepts the full swath of choices. As is so often the case in homotopy theory, the indeterminacy is in some sense contractible (the poset consisting of all $F$ containing $\sigma$ is the face poset of an infinite simplex), and so everything works out. Of course Quillen needs a version of his Fiber Theorem (aka Theorem A).