# Do homotopy pullbacks preserve weak homotopy equivalences?

Suppose we have maps of spaces (however nice you want) $g_1:X_1\to A$, $g_2:X_2\to A$ and a map $f:Y\to A$. From these, we can form the homotopy pullbacks $Y\times^{h}_{A}X_1$ and $Y\times^{h}_{A}X_2$. A map $f:X_1\to X_2$ such that $g_2\circ f=g_1$ induces a map $$Y\times^{h}_{A}X_1\to Y\times^{h}_{A}X_2.$$

If $f$ is a weak homotopy equivalence, is this induced map also a weak homotopy equivalence?

Can someone suggest a nice place to read about these sorts of details of homotopy pullbacks?

Yes, in general the claim is as follows: given a weak equivalence of spaces $X \to X'$ under $A$, and given a fibration $B \to A$, then $X \times_A B \to X' \times_A B$ is a weak equivalence. This is a consequence of the right properness of the usual model structure on topological spaces (well, actually, it is essentially the same thing; the right properness can be seen because all objects are fibrant).
R. Brown and P.R.Heath, Coglueing homotopy equivalences'', Math. Z. 113 (1970) 313-325.
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