# How should one approach reading Higher Algebra by Lurie?

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.

• Is this somehow related? – Giorgio Mossa Jan 17 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.) – Théo de Oliveira Santos Jan 17 at 13:01
• 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 at 23:05

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'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))? – Théo de Oliveira Santos Jan 17 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 at 14:16