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I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)

He uses this result to prove Jeff Smith's theorem, but the proof on the n lab (https://ncatlab.org/nlab/revision/combinatorial+model+category/59) does not seem to use such a technical preliminary result.

So I am wondering why does Lurie's proof is this much more complicated? And, if the added complexity is somehow necessary for the proof to work, could someone point out the "mistake" in the n lab and give reference?

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Lurie uses A.2.6.7 to prove the "easy" direction of Jeff Smith's theorem in A.2.6.8, namely that every combinatorial model category arises from the construction of the theorem. This part of the theorem is proven in the last sentence of the current revision on the nlab page ("To prove the converse,..."), where the facts of A.2.6.7 are indeed referred to, but they are regarded as being well-known. This seems reasonable to me because A.2.6.7 is a general fact about accessible categories, in no way specialized to the study of model categories. But it never hurts to be more explicit; one should always feel free to add details to an nlab page.

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  • $\begingroup$ You are definitely right, I read the n lab too fast. Do you have any insight on (the end of) the proof of lemma A.2.6.7? $\endgroup$ – user09127 Dec 31 '18 at 17:02
  • $\begingroup$ I just realized there is a discrepancy in the numbering between different versions of HTT. Are you referring to Corollary A.2.6.7 in the current version or Lemma A.2.6.7 in an older version? If you can clarify what you're asking about, I'd be happy to discuss it. BTW another source for the proof of Smith's theorem is the original write-up in Tibor Beke's thesis. $\endgroup$ – Tim Campion Jan 4 at 15:24
  • $\begingroup$ I was referring to the older one (I definitely should have clarified...) I'll have a look at Beke's thesis first, thanks for the reference! $\endgroup$ – user09127 Jan 5 at 14:52
  • $\begingroup$ I had a look again, and I must say that I still do not get why Lurie needs lemma 2.6.9 (newest HTT version), whereas the n lab does not (in the proof of the hard part of Jeff Smith's theorem.) I suspect the n lab is "cheating" somehow, but I do not get how... To be more precise, I suspect it has to do with the fact that the class of acyclic cofibrations can be generated by a set, but what would be the issue with the proof of the n lab? $\endgroup$ – user09127 Jan 11 at 14:01
  • $\begingroup$ As for what I do not understand in the proof of lemma 2.6.9 (again HTT new version), is the last paragraph. More precisely, I believe that lemma A.1.5.11 shows that the map $M\rightarrow N$ is in $S$, i.e. the last commutative square finishes completes the proof. If this is true, the end of the proof does what exactly? $\endgroup$ – user09127 Jan 11 at 14:45

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