higher algebra textbook

higher algebra textbook: temporary guide

This website contains information regarding my forthcoming textbook on higher algebra.

The book's tentative title is Higher Algebra: Chapter 0 (with apologies to Aluffi). A primary goal is that after reading it, one should be able to refer to Lurie's Higher Algebra more-or-less at will. It is contracted to be published by Cambridge University Press in their Studies in Advanced Mathematics series, hopefully sometime in 2024. This page should be seen as the "single source of truth" for the book; any updates will be posted here. [last update: 12/8/2023]

The book will be based on my lecture notes from teaching two iterations of homological algebra (Math 128) at Caltech: one in Winter '21 and one in Spring '23. Around 95% of the material that will be in the book is thusly already available, but it's slightly scattered and not all of it is in a polished form. The primary purpose of this webpage is to serve as a temporary guide to help interested readers navigate those two sets of lecture notes as easily as possible (while the book itself is not yet available), since neither set of lecture notes strictly supersedes the other.

The book will consist of two parts: "Higher Algebra" and "Higher Category Theory", in that order.

Part I is intended to serve as an introduction to higher algebra. In order to be as concrete and accessible as possible, it starts with a homotopical perspective on the notion of chain complexes of modules over a ring. It will consist of (a polished version of) §§1-7 and §11 of my Spring '23 notes. The material on stable ∞-categories and spectra is in §11 (which is particularly unpolished, but hopefully still useful). I think I've said more-or-less everything I wanted to about stable ∞-categories there, but the book will contain a bit more material on spectra.
Part II is intended to (among other things) cover most of the material in §§1-5 of Lurie's Higher Topos Theory (i.e. just skipping the actual ∞-topos theory in §§6-7). It will consist of (a polished version of) §§8-12 of my Winter '21 notes, although note that §12 is not yet complete.

The idea is that the two parts can be read essentially independently (and in particular the interested reader can jump directly to Part II), but (especially in its later sections) Part I actually relies on Part II for rigorous foundations. This is inspired by the following quote from Ravi Vakil on "backfilling" (taken from here), which I think is particularly applicable to the process of learning higher algebra.

Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

I am always grateful for comments, corrections, typos, etc.; none are too large or small! (Unless you find an inconsistency in the foundations of mathematics -- then, please keep it to yourself 😂)