This is just a short post letting you know that I have updated the Stacks project by working through all your comments once more. The last time I did this was on May 19 of this year. I try to respond to mathematical errors very quickly, but sometimes I do not realize a comment is pointing out an error until I really sit down and look carefully at the comment. If you’ve left a comment pointing out a mathematical error, feel free to also email. I remind you that it is very helpful to me if you suggest a fix or if you have a counter example to the statement you are objecting to or more generally if you discuss what you think went wrong.

Enjoy!

Yesterday I got an email asking about links between tags in proofs. Here is the question:

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On the page 00KD in the proof of Lemma 00KK the reader will find the strings “If (00KN)” or “Assume (00KP)”, provided you view the page in “tags” mode. However, these strings are not found in the pdf. You have to click on them to see what they mean, and it turns out to be the items (1), (2), … of Lemma 00KK. How does this work?

Related but different question: on the page 00KA shouldn’t there be a hyperlink to the definition of χ_M?

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The explanation is that in Lemma 00KK there is an itemized list in the underlying latex file. Then I decided to give latex labels to the items so I could refer to them in the proof. If you have a pdf reader which can deal with (internal) links, then you can click on the occurences of (1), (2), … in the proof of 00KK and you’ll be thrown to the corresponding statements in the proof of 00KK.

There are several other places in the Stacks project where the items of an itemized list have latex labels. But most of the time we don’t do this. (It turns out to work to be best for the structure of the Stacks project if each lemma has a single conclusion.) I would appreciate feedback on whether readers think it works well in the cases where we do do this.

I hope that this addresses the first question somewhat.

Second question. Ideally, mathematicians who help out with the Stacks project should only have to worry about helping make the pdfs readable and mathematically correct, etc. The underlying website code will hopefully display the mathematics in such a way that the experience is similar (or better than) reading the pdf. So if you want a “link” back to the definition of χ_M in Definition 00KA, then try to change or suggest a change in the latex so that it looks good in the pdf.

Specifically in Definition 00KA we could change the latex code to read

$\chi_M$ as defined in Definition \ref{definition-chi}.

except that then we would also need to insert, earlier, a definition environment where we define χ_M and φ_M. Perhaps this isn’t a bad idea.

Technically speaking (please ignore this): I do not want to add another layer to the underlying latex files. However, in the future we could have a “hover” functionality where hovering would show you the definitions of defined terms… I would want this to be implemented in such a way that no changes need to be made inside the latex files, but perhaps a separate file would be added (similar to the tags file being maintained outside the latex).

Kerodon is a site modeled after the Stacks project maintained by Jacob Lurie.

1. It uses the tags system for stable references as originally devised for the Stacks project by Cathy O’Neil.

This means that if you reference a tag in the Stacks project or Kerodon, you should make sure to specify which of these two you are referring to (if you use the “cite” links in either project and copy-paste from there this will work fine). All of the references I’ve seen to the Stacks project already do this, so I am not worried.

2. It is running the Gerby website infrastructure with some additional work by Opus Design for a distinct look.

Some of the work done for this by Pieter Belmans will also benefit the Stacks project (this will probably be mostly invisible to the user though).

3. It has a comment system, so please go over there and leave mathematical comments!

4. The mathematics in Kerodon is written and copyrighted by Jacob Lurie.

This is a difference in philosophy: (a) the contributors to the Stacks project collaboratively own the Stacks project and (b) you can directly access the underlying latex files to make changes to submit to the maintainer (me).

5. Currently the pdf version of Kerodon has 85 pages.

The plan is to add more over time. As you can see, the current material covers a tiny fraction of Jacob’s book entitled “Higher Topos Theory”. If you want to know more about what is planned in Kerodon, you’ll have to ask him. But for now, I am kind of taking this opportunity to read Kerodon the way you would read a webcomic. As new material gets added I will head over there and read it. I hope you will enjoy it as well.