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?



A question about combinatorial model categories
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.


