4
$\begingroup$

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory. This MathOverflow question asked for a roadmap to Lurie's Higher Algebra. Still another question asked for a roadmap to learn derived algebraic geometry (DAG).

This question asks for a roadmap to spectral algebraic geometry (SAG), such as found in Lurie's Spectral Algebraic Geometry, which is more general than DAG, treating, for example, spectra of $\mathbb{E}_\infty$-rings (instead of only DG-rings).

Ideally, such a guide should include helpful sources other than Lurie's book (which are much rarer than references about DAG) and information about which parts should be skipped (at least on a first reading).


Edit: I just found out that this has already been asked at Quora (with one answer).

$\endgroup$
  • 3
    $\begingroup$ Just my two cents: I think of SAG as similar to EGA and SGA, and in that regard, I think Emerton's advice at terrytao.wordpress.com/career-advice/… is very useful. The context in which SAG could be applied depends, I think, rather heavily on what you're interested in. (E.g., if you're interested in applications to the chromatic world, then the Elliptic-n papers are also good reading, as is Lurie's survey of elliptic cohomology (the latter of which I'd recommend to anyone interested in the subject).) $\endgroup$ – skd Jan 17 at 18:20
  • 1
    $\begingroup$ @DenisNardin Thanks for pointing that! The following is a bit of a wild guess: it seems that much of SAG is already well-established enough for such a roadmap (even if it is a roadmap for a subset of SAG) to be helpful. (For example, DAG II was posted on the arXiv in the beginning of 2007) So, do you think restricting the question to a certain subset of SAG might salvage the question (if so what would be ideal?), or is it better to just wait for it to be finished, including the more specialized parts?) $\endgroup$ – Théo de Oliveira Santos Jan 17 at 22:50
  • 3
    $\begingroup$ @Untitled I think there's nothing wrong in wanting to read SAG. But asking for a "roadmap" is probably a bit too ambitious, even the "well established" parts are quite recent and have been rewritten significantly. For what is worth, if you know classical algebraic geometry and are able to read HTT and HA without too many problems (you don't need to have read all of HA, just that you are able to read it if need be), you don't need a roadmap for SAG. Just pick a section that interests you and start reading it. $\endgroup$ – Denis Nardin Jan 18 at 8:37
  • 2
    $\begingroup$ I agree with Denis. The point of HTT is to establish that "higher category theory behaves like ordinary category theory when done right", and lots of homotopy-coherence arguments are done in gory, simplicial detail so that they need not be done again. Then you notice that in HA, category theoretic arguments read very cleanly, the hard work having been done in HTT. Gory simplicial stuff reappears to establish that "higher algebra behaves like algebra when done right". But now, by the time we get to SAG, the hard homotopy-coherence work is already done... $\endgroup$ – Dylan Wilson Jan 19 at 18:50
  • 2
    $\begingroup$ (contd) as a result, the arguments in SAG are very pleasant to read, like "follow your categorical and algebraic intuition (but don't use 'elements' and stuff, use universal properties to build and verify things), then justify your manipulations with the hard-won theorems of HTT and HA" $\endgroup$ – Dylan Wilson Jan 19 at 18:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.