Nerve theorem for locally infinite covers by subcomplexes

Question

Let $Y$ be a simplicial complex and let $\{Y_i\}_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup_{i\in I}Y_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each finite $J\subset I$, we have that $\bigcap_{j\in J}Y_j$ is either empty or contractible.

One version of the nerve theorem says that, in the above situation, $Y$ is homotopy equivalent to $\mathcal N$.

I'm interested in the proof in the specific situation where we cannot assume that each simplex of $Y$ is contained in finitely many of the $Y_i$.

The theorem is stated in the above generality in [1], as Theorem 10.6. In the proof given there, it is assumed "for convenience" that the preceding local finiteness assumption holds, and that in the general case one uses a "slightly different" argument.

I tried a bit to deduce it from other versions of the nerve theorem (see e.g. [2]), say by trying to replace $\{Y_i\}$ by an open covering with a similar nerve. (I had some difficulty finding a more detailed proof in the literature, and I would like to see one to see if it's possible to modify it slightly.)

What is the "slightly different" argument mentioned in [1], or where can it be found?

[1]: Björner, Anders, Topological methods, Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 1819-1872 (1995). ZBL0851.52016.

[2]: Proposition 4G.2 and Corollary 4G.3 in:

Hatcher, Allen, Algebraic topology, Cambridge: Cambridge University Press (ISBN 0-521-79540-0/pbk). xii, 544 p. (2002). ZBL1044.55001.

Answer 1

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).