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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?

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2 Answers

up vote 2 down vote accepted

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).

This is assuming you define homotopy pull-backs to mean that you replace one of the maps by a weakly equivalent fibration and form the pull-back with respect to that; in a general model category, you would have to replace all the maps by fibrations to get something invariant. (In fact, the whole point of homotopy limits and homotopy colimits is that ordinary limits and colimits are not homotopy invariant.)

A useful reference for this sort of thing is Dan Dugger's article on homotopy colimits. The second half of the book by Bousfield and Kan, "Homotopy colimits, completions, and localizations" also treats this subject.

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An early account of this type of result for equivalences rather than weak equivalences was in

R. Brown and P.R.Heath, ``Coglueing homotopy equivalences'', Math. Z. 113 (1970) 313-362.

which is available here. In that case, the type of proof there also gives more control of the homotopies involved than other proofs.

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