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Hi,
I happened to notice today that there are problems with the proof of Lemma 1. Unfortunately, I do not have time to fix this myself, but I thought I’d let you know.
It can be fixed as follows.
1) Define $\hat{X}$ to be the pullback of $f \times id : X \times Y \rightarrow Y \times Y$ and the map $Y^{I} \rightarrow Y \times Y$ which is currently denoted (I would not use this notation myself!) by $(d_0,d_1)$.
2) The required map $p : Z \rightarrow Y$ is the composite of the map $Z \rightarrow X \times Y$ which is part of the pullback of 1), and the projection map $X \times Y \rightarrow Y$.
3) The required fibration $Z \rightarrow X$ arises in the same way as in 2), but composing with the the other projection map instead,
4) The required map $X \rightarrow Z$ arises via the universal property of the pullback by using the map $f \circ c : X \rightarrow Y^I$, where $c$ is the map $Y \rightarrow Y^I$ appearing in the factorisation which defines $Y^I$, and the map $id \times f : X \rightarrow X \times Y$.
Okay, so Chrome messes up the preview but is fine once the post is posted. Probably due to the interaction of javascripts between inserting the preview and re-running MathJaX on the page.
For my own future reference, the solution would appear to be documented here.
Also, the initial post in this thread is a copy of a post by Richard Williamson who reported the issue (so the “I” is not me but him!).
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