at.algebraic topology - Homotopy pullbacks of simplicial spaces, and Bousfield-Friedlander

The following represents what I know about this; I don't know of a published reference.

Given a map $f:Z_bulletto Y_bullet$ of simplicial "spaces" (to make things easy, assume spaces are simplicial sets), let's call it a realization quasi-fibration (RQF) if for every $U_bulletto Y_bullet$, the homotopy pullback of the geometric realizations is weakly equivalent to the realizations of the levelwise homotopy pullbacks. The Bousfield-Friendlander theorem gives a sufficient condition for $f$ to be a RQF, in terms of the dreaded $pi_*$-Kan condition.

Some facts:

• The pullback of an RQF $f$ along any $U_bulletto Y_bullet$ is itself an RQF.




• Let $F[n]_{bullet}$ be the simplicial space which is free on a point in degree $n$. Then $f$ is an RQF if and only if its pullback along all $g: F[n]_bullet to Y_bullet$, for all $n$, is an RQF.

These two facts are consequences of something that is sometimes called "descent"; basically, the facts that homotopy colimits distribute over homotopy pullbacks, and compatible homotopy pullbacks assembled by a homotopy colimit result in a homotopy pullback.

So the above gives exact criteria for $f$ to be an RQF. Whether the pullback of an RQF $f$ along a map $g$ is again an RQF only depends on the homotopy class of $g$. So if $f:Z_bulletto Y_bullet$ is any map, let
$pi_0Y$ be the simplicial set whose $k$-simplices are $pi_0(Y_k)$, which is to say all homotopy classes of maps $F[k]_bulletto Y_bullet$. Let $RQF(f)subseteq pi_0Y$ be the sub-simplicial set whose $k$-simplices correspond to $g:F[k]_bulletto Y_bullet$ such that the pullback of $f$ along $g$ is an RQF.

So the criterion is: $f$ is an RQF iff $RQF(f)=pi_0Y$.

It turns out that since geometric realization always preserves products, any map $Z_bullet to point_bullet$ is an RQF. Thus $RQF(f)$ contains all $0$-simplices of $pi_0Y$. Thus, if all $Y_k$ are connected, $f$ is an RQF, which implies the result you describe.