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A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT should be skipped (example guide).

This question asks for a similar guide for learning algebra in the context of $(\infty,1)$-categories, at the level of generality of Lurie's Higher Algebra.

More specifically, the focus should not be on DG-algebras, being instead about more general mathematical objects such as $\mathbb{E}_k$-rings.

Nice things to include in such a guide would be other helpful sources, parts that should be skipped, or concepts that are best treated as black boxes (at least when first approaching it).

Here is the sister question to this one, which asks for a roadmap to Lurie's Spectral Algebraic Geometry.

Emily
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  • Is this somehow related? – Giorgio Mossa Jan 17 '19 at 12:38
  • @GiorgioMossa Yes, it is definitely useful (as a “pre-guide”) for someone wanting to learn (the prerequisites to) his books. (But this question asks for what to do when one actually starts reading them.) – Emily Jan 17 '19 at 13:01
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    Before asking how one should approach reading Higher Algebra by Lurie, one must ask whether one should approach reading Higher Algebra by Lurie... (just kidding! just kidding!!) – Zach Teitler Jan 17 '19 at 23:05

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I think that reading the proof of straightening and unstraightening is probably a great way to get bogged down in the details and should be treated as a black box unless you interested in doing 'pure' higher category theory. The proof is nontrivial and also somewhat unenlightening (the reason that it works, especially in the marked case, involves some intuition that Lurie had about lax cones in $(\infty,2)$-categories that is not at all clear from the exposition, where things appear as if by magic).

I would also highly suggest Cisinski's new book as an introduction before reading HTT and HA. Some of the proofs in HTT are dated and can be done much more easily (not yet for marked straightening, though).

Harry Gindi
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  • Thanks for the answer! Do you mean Section 3.2 of HTT (or perhaps some sections of chapter 2 too)? I am reading Cisinski's book and it is (very!) very helpful indeed (as is Lurie's Kerodon). However what really troubles me is that there seem to be only a few references on HA (and SAG) on the level of generality of Lurie's work, whereas there are significantly more for DG-stuff and DAG. – Emily Jan 17 '19 at 13:47
  • @Untitled anything about straightening and unstraightening. The version in Lurie's book for unmarked straightening is now outdated, in my opinion, and the version of marked straightening is too hard to follow for a beginner. It's something worth reading if you're interested in maybe generalizing it, but otherwise, you can basically take it as axiomatic that this works. – Harry Gindi Jan 17 '19 at 13:57
  • I'll definitely avoid it then. By classifying straightening and unstraightening as “'pure' higher category theory”, do you mean they are technical work that needs to be done for quasicategories (i.e. they are purely model dependent and do not appear in other models for $\infty$-categories (or in Riehl-Verity's work on $\infty$-cosmoi))? – Emily Jan 17 '19 at 14:10
  • @Untitled Unfortunately, the papers of Riehl and Verity have not yet proven the full statement equivalent to 'marked straightening', although they promised it in the future as an application of Beck monadicity. However, their comprehension construction is closely related and slightly more 'conceptual', although they again chose for reasons of brevity to omit the lax $(\infty,2)$-categorical ideas and just explain things $(\infty,1)$-categorically. Riehl told me that the full version of the argument will eventually appear. I heard that she discussed it at a conf in Texas but I wasn't there. – Harry Gindi Jan 17 '19 at 14:16
  • Would that conference be this one? If so the slides of her talk are already available (with talk videos being posted once the conference ends) – Emily Jan 17 '19 at 14:26
  • @Untitled It was mentioned in the abstract, but not in the slides. I don't know if she got to it. "We conclude by reporting on an encounter with 2-complicial sets in the wild, where a suitably-defined fibration of 2-complicial sets enables the comprehension construction introduced in joint work with Verity. Special cases of the comprehension construction...." – Harry Gindi Jan 17 '19 at 14:31