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The proof of HTT Lemma A.3.6.17 does not seem right. I assume that the lemma is true, but the proof is all mangled, like it was somehow mixed up with an old version.
Oddities:
It asserts that the fibrant replacement functor induces a functor F:Uf→U∘ where Uf is the full subcategory of fibrant objects of U, and U∘ is the full subcategory of fibrant-cofibrant objects. Shouldn’t the map classifying the natural transformation Uf⊗[1]S→U∘ instead have target Uf?
W’_0 seems like it should be the class of w⊗id where w is a weak equivalence instead of the class of e⊗id where e is an equivalence.
Questions: Why is it enough to show that the upper and lower inclusion maps are weak equivalences near the end of the proof? Also, how does Corollary A.3.4.11 help in proving it at all?
A link please to what you are talking about, or is your message itself a palimpsest?
is all mangled, like it was somehow mixed up with an old version
I think a good term for this pheonomana in math writing is palimpsest where some old version has not been completely removed and rewritten.
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Bump? I did it because Urs and Todd are back from vacation, so I hope they’ll have a second to look at it.
First of all, I think you are correctly identifying the typos here.
I think instead of “fibrant replacement” we should read “cofibrant replacement”. And the “equivalence” that you refer to must be a “weak equivalence”, yes. In fact I think there is even more problem with the notation here. For instance id and F and hence α don’t really go between 𝒰f and 𝒰∘.
But as always, all these are pretty trivial things to fix. If one fixes it in the evident way then the main statement remains that we get the map h.
Let me see about the remaining question about the end of the proof…
Questions: Why is it enough to show that the upper and lower inclusion maps are weak equivalences near the end of the proof? Also, how does Corollary A.3.4.11 help in proving it at all?
The map h which we obtain from the fact that we have a functorial cofibrant replacement goes actually
h:(𝒰∘⊗[1]S)[W′−1]→𝒰∘[W−10](which incidentally makes the use of “equivalence” instead of “weak equivalence” in the sentence above correct after all), From Lemma A.3.5.14 we can extend this to a span
𝒰∘[W−10]⊗[1]S≃←(𝒰∘⊗[1]S)[W′−1]h→𝒰∘[W−10].To get a homotopy, we need to make this span into a genuine morphism,
𝒰∘[W−10]⊗[1]S→𝒰∘[W−10].i.e. invert the morphism on the left. This is supposedly accomplished by the remaining remarks. Let me see…
I’m pretty sure that X⊗[1]S is not a cylinder object for an S-category X in general because the tensor product does not preserve cofibrations, in this case, X∐X≅X⊗[1]∅S→X⊗[1]S is not necessarily a cofibration.
Also, the thing about equivalences is important for the proof of one direction: When we’re looking at the map Uf⊗[1]S→Uf, those elements of W’ have to be weak equivalences or we don’t even have a natural map Uf[W−1]→(Uf⊗[1]S)[W′−1] (on the top or bottom inclusion).
Also, lastly, A.3.5.14 says that we have a weak equivalence C[W−1]⊗D→(C⊗D)[W′−1] exactly when W′−1={x⊗idD:x∈W}. However, in the case of A.3.6.17, there are more morphisms in W′ than just those.
Hey Urs, I’ve been stuck for about a week on this (and this is the last barrier to me finishing chapter 2 of HTT). Not to try to guilt you into answering more quickly, but I would sincerely appreciate any sort of light you could shed on the matter.
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