TITLE: Can homotopy pullbacks of spaces be checked on fibers?
QUESTION [6 upvotes]: As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak equivalence. More precisely, consider the following diagram:
$$\begin{array}{c}&&&&& A&  \longrightarrow & B\\
&&&&&\downarrow && \downarrow \\
&&&&& C &\longrightarrow & D \\
&&&&\nearrow & &\nearrow\\
&&&1  & \longrightarrow & 1  \end{array}$$
Suppose that, after having taken homotopy pullbacks on both sides, the resulting front face is a homotopy pullback (i.e. the top horizontal arrow [between homotopy fibers] is a weak equivalence), and that this happens for any vertex of $C$. Is it true that the square involving $A,B,C,D$ is a homotopy pullback?
This seems to appear quite often but I couldn't find a precise reference with a proof.
Thanks in advance for your help.

REPLY [2 votes]: If you are looking for a reference, Propostion 3.3.18 of Munson Volic's "Cubical homotopy theory" (available at http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf) is doing what you want.