Ref request: making a homotopy equivalence fibrewise, in an abstract setting (eg fibration categories)

Question

The following useful lemma holds in a variety of settings:

Lemma. Let $p : Y_1 \to X$, $p_2 : Y_2 \to X$ be fibrations over a common base, and $f : Y_1 \to Y_2$ a map over $X$ that is a homotopy equivalence on total spaces. Then $f$ is a fibrewise homotopy equivalence.

Proposition 4.61 of Hatcher gives this in the topological setting. Essentially the same proof works in more abstract settings, e.g. Quillen model categories and fibration categories (aka Ken Brown’s categories of fibrant objects). Can anyone point to a good reference for the lemma in such a setting? Fibration categories would be ideal (that’s where I want to use it), but model categories or somewhere else similar would be fine too.

Answer 1

The dual theorem involving cofibrations is due to Dold and is discussed in an abstract setting in the book 'Abstract Homotopy and Simple Homotopy Theory' page 33. The proof given there dualises. It is in a cofibration category context à la Baues. You may also find a proof in his books but I am not sure where.

In the case you want, Peter, in fact more is true as there is a neat argument giving explicit homotopy coherence data for the inverse if I remember correctly.

Answer 2

Something even more general is true, where you replace "homotopy equivalence" by "local equivalence" for any left Bousfield localization. This is detailed in Dror Farjoun's book Cellular spaces, Null spaces, and Homotopy Localization, Section F.2. In particular, Theorem F.3 looks like the same picture you are drawing, if we take F to be the identity functor (so that local equivalences are just weak equivalences), and Theorem F.4 appears to be what you are asking for. The proof likely doesn't need the full model structure; just a way to get hold of homotopy colimits.

Answer 3

I was quite sure that this was explicitly written down by Rădulescu-Banu or by Hirschhorn, but I cannot find this exact statement.

However, this follows directly by combining Theorems 7.5.10 and 7.6.5 in Hirschhorn's Model Categories and Their Localizations which is a concise enough reference, I believe.

This gives a proof in model categories. Note that in fibration categories you need to also assume that $Y_1$ and $Y_2$ are cofibrant. Unfortunately, Hirschhorn's argument is phrased in such a way that it doesn't apply verbatim in this context.