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Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it follow that the induced map $X \times_Z Y \to X' \times_{Z'} Y'$ between the pullbacks is also a homotopy equivalence? If not, what additional conditions (e.g., insisting that $X \to Z$ is a fibration, $Y' \to Z'$ is a cofibration, everything is a CW complex, etc.) are needed?

I'll also like to know the answer in the case of pushout squares, but I guess I can just dualize whatever the answer to the previous question turns out to be.

I tried to construct an inverse map directly using the homotopy inverses $X' \to X$, $Y' \to Y$, and $Z' \to Z$, but I could not guarantee that the resulting diagram commutes enough to produce a map $X' \times_{Z'} Y' \to X \times_Z Y$. Even then, I'm not certain that I can somehow glue the homotopies in a compatible way to prove that the constructed map is a homotopy inverse.

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  • $\begingroup$ No, there's no reason for this to be true with no hypotheses. You can write down easy counterexamples where $X$ and $X'$ are points. To fix this you replace pullbacks with homotopy pullbacks or assume that some of the maps are fibrations (which ensures that pullbacks are homotopy pullbacks). $\endgroup$ – Qiaochu Yuan Jul 18 '15 at 7:34
  • $\begingroup$ see hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf section 10 page 46 for some discussion on Qiaochu's comment. $\endgroup$ – Jack Davies Jul 18 '15 at 8:08
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I'll answer the question in the pushout case.

If one of the maps of each pushout square is a cofibration, the induced map will be a homotopy equivalence, see Proposition 5.3.4 of Tammo tom Dieck's "Algebraic Topology". You don't need any further conditions on the spaces for this, not even that they are compactly generated or of the homotopy type of a CW complex.

The reason for this being true is that the cofibration-condition guarantees that both pushouts will be homotopy pushouts.

As you guessed, the dual statement for pullbacks is also true.

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Results of this type for pushouts first appeared in the 1968 edition of the book which is now available as Topology and Groupoids. See also this stackexchange discussion, which gives a full statement of the pushout case, and for the pullback case see this paper. Certainly one needs cofibration or, dually, fibration, conditions.

The advantage of the proofs in these sources is that they give good control of the homotopies involved, which you do not get clearly from using homotopy pushouts or pullbacks.

As explained in the cited book, the result was found by generalising the result that a homotopy equivalence of spaces induces an isomorphism of homotopy groups: the point is that the homotopy equivalence may not be base point preserving.

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