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Prop A.1.5.6 says that given an S-tree Y indexed by a well-founded poset A, and given two downward-closed subsets $A^{\prime\prime} \subseteq A^\prime \subseteq A$, we have that $Y_{A^{\prime\prime}}:=colim_{\alpha \in A^{\prime\prime}}Y_\alpha \to colim_{\alpha \in A^{\prime}}Y_\alpha=:Y_{A^\prime}$ is in the weakly saturated class of morphisms generated by $S$.
The beginning of the proof contains a reduction to the case where $A^{\prime\prime}=\emptyset$ and $A^\prime=A$, which uses remarks 1.5.5 and 1.5.3. However, it seems to me like 1.5.3 is completely irrelevant, and that the remark he wants to reference is 1.5.2.
The reason I think this is that we end up with a new S-tree with root $Y_{A^\prime\prime}$ after applying remark 1.5.5. I don’t see how pushing out over a morphism from the root is relevant at all. The reason why I think it should instead say 1.5.2 is that 1.5.2 states (among other things) that after reindexing according to 1.5.5, $Y_\emptyset=Y_{A^\prime\prime}$. This reduced to the case where $A^{\prime\prime}=\emptyset$, and the fact that we can take $A^\prime=A$ is just by restriction.
I don’t see how changing base (Remark 1.5.3) can actually be applied to the situation.
I think you are right, Harry. I think the paragraph between Notation A.1.5.2 and Remark A.1.5.3 was meant to be labeled a Remark, too, and the numbering got mixed up.
I also think if you call every typo an "error", you are bound for some surprises. There are many trivial typos, here and elsewhere.
On this page alone, for instance in the paragraph between Notation A.1.5.2 and Remark A.1.5.3 which I think you correctly identify as being referred to in A.1.5.6 , the last line is a typo: it should read $Y_B \simeq Y_\beta$ instead of $Y_B \simeq Y_\alpha$, I think. Then in the first line of Remark A.1.5.3 a backslash in the TeX-source is missing. In the first line of the proof of A.1.5.6 it says "enerality" instead of "generality", etc.
My advise: don’t think of such typos as "errors in proofs".
Well, for someone who’s learning the material for the first time without help from someone who’s already been initiated, I’m sure you can see how it would be frustrating.
Sure, I understand that it can be frustrating. But I see more frustration for you at the horizion if you channel your frustration into strong accusation of others. They can get frustrated (with you), too. ;-)
If you want an ’error in proof’ try Segal’s 1969 paper ’Equivariant contractibility of the general linear group of Hilbert space’ – the theorem is false as stated, but finally corrected in Appendix 3, as an afterthought, of Atiyah-Segal ’Twisted K-theory’, from 2004. They included the proof
to correct a number of misstatements by the second author and others which have often been repeated in the literature.
A slight understatement…
I wouldn’t go around blaming Segal for an error in proof, though, if it wasn’t for the above admission!
I feel like HTT is as bad as Lang’s algebra with the number of typos, at least in the earlier chapters.
I’m sure Jacob Lurie would like to hear of any errors… might be good to collect a batch and send them along.
There are plenty of minor typos of the kind we discussed above (exchanged symbols, wrong lemma-reference numbering, etc.). I think more of them towards the end, actually. In some parts in the appendix one can find remnants of what must have been earlier versions of the document that weren’t properly removed. I don’t find this surprising. In fact, I find it reassurring. ;-)
I was thinking that an obvious move would be to add a section Errata to the entry Higher Topos Theory. But so far I didn’t come across any typo that I personally found irritating enough to warrant going through the trouble. But if others here feel differently, like Harry does, then it might be a good idea .
Hey, speaking of the appendices, Urs, is there any chance you could do me a small favour? In a few proofs in the section on combinatorial model categories, Lurie refers to sections in the book that occur chronologically much later (chapters 4 and 5), where he proves things about accessible infinity-categories. Normally this wouldn’t be a problem, but the result on combinatorial model categories (the 3 criteria) is used heavily throughout the rest of the book.
Is there any way you could translate the proofs from oo-category language to ordinary ct-language? It’s only 3 or 4 short results.
Is there any way you could translate the proofs from oo-category language to ordinary ct-language?
In principle I suppose I could try to, but I am unlikely to have the time. But in case you don’t have access to the book by Adamek and Rosicky on locally presentable categories, send me an email.
I’ve got it right here (took it out of the library when I was reading the section on acc. categories). Could you tell me which results in ch. 4 and ch. 5 used in the proof of A.2.6 can be found in it?
What is “the proof of A.2.6”?
I thought you wanted general results on accessible and locally presentable categories.
No, there’s a section A.2.6 that builds up to the proof of theorem A.2.6.13, which is the “criteria for combinatorial model categories” (Jeff Smith’s theorem). The result is used practically everywhere from ch. 2.1.4 onwards. Lurie makes a lot of use of it throughout the rest of the book.
However, to prove this statement, he sends the reader on a wild-goose chase through chapters 4 and 5 (which use oo,1-categorical language) picking up results and using them (in addition to the results of A.1.5 on trees and accessibility). I was under the impression from what you just wrote that Adamek and Rosicky have all of these results stated for ordinary categories.
Ah, now I see what you mean. No, that stuff is not in AR.
The trouble here is that this theory of combinatorial model categories is supposedly currently and since a while ago being written up by Jeff Smith. But still not available.
Well, I guess I’ll do the work of translating the relevant proofs of ch.4 and ch.5 to proofs about ordinary categories, then put it on my personal web (since I can’t really figure out anywhere to put them right now (they’re pretty random propositions)). Even better, I’ll make a page called “notes on HTT” and just record errata, notes, etc. for future generations who are reading through it.
If you do errata, I suggest to put them into a section of that name at Higher Topos Theory.
Most propositions about accessible $(\infty,1)$-categories you could type into the entry accessible (infinity,1)-category or similar entries.
@Urs: I’m doing something similar to Bergman’s companion to Lang’s Algebra (math.berkeley.edu/~gbergman/.C.to.L/). If I moved the results to accessible (infinity,1)-category, it would be pointless, because anybody looking for them would not be able to find them. I could also put them there (even though they’d be scattered random propositions), but I’d like to write up a “guide” to go with the book (including suggested reading order and other things) because I’m reading the book right now, and I wish that I had a guide that does that.
Fine. Put it wherever you think is best suited. Just make sure to include hyperlinks back and forth afterwards, so that indeed people find it.
Absolutely! That’s the point of writing it up on the nLab instead of in a PDF.
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