This is probably trivial but I'm not the best with category theory.

Let $M$ be a right proper model category (that is pullbacks of weak equivalences along fibrations are weak equivalences). The claim is that a commutative square

$$ \require{AMScd} \begin{CD} X_1 @>{\alpha}>> X_2 \\ @VVV @VVV \\ Y_1 @>{\beta}>> Y_2 \end{CD}$$ is homotopy Cartesian if $\alpha$ and $\beta$ are weak equivalences.

My plan was to take pullbacks to get a commutative cube and use the cogluing lemma, as it's easy to make a another parallel weak equivalence to $\alpha$ and $\beta$, but I can't see how to get that $X_1\to U\times_{Y_2}X_2$ is a weak equivalence. Here I've factored $\beta$ through $U$ using a fibration and a trivial cofibration.

Am I on the right path? If not, what is the right approach?

  • $\begingroup$ It's actually true in any model category (or even any category with weak equivalences) but the proof would be different. $\endgroup$ – Zhen Lin Mar 14 '16 at 8:23

This actually is trivial. The factorization $X_2\to U\to Y_2$ with the first map a trivial cofibration and the second map a fibration does it. I don't know how to delete this, but I would be happy if someone who does did; I think it's not a valuable question.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.