Derived categories and $\infty$-categories necessary for condensed mathematics

Question

I am reading the three texts on condensed mathematics by Scholze and Clausen. I am also interested in paper "A $p$-adic 6-functor formalism in rigid-analytic geometry" by Lucas Mann.

To advance in the texts I will have to learn about derived categories and later about $\infty$-categories. In these texts the authors treat $\mathcal{D(A)}$ as a $\infty$-category.

Is there a text on derived categories using $\infty$-categories rather than just triangulated categories (that don't require me to study the entirety of higher topos theory to read it)? It is necessary for me to know the construction using triangulated categories first? How much of $\infty$-categories is necessary to read the three condensed texts? What is a good reference for it?

Answer 1

There are several questions (implicit) here.

1. In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed?

2. Does the development of condensed mathematics, and/or its use in applications, require knowledge of derived ($\infty$)-categories?

3. What are good references to learn about derived $\infty$-categories?

4. Must one learn the triangulated perspective first?

Let me try to say a few words about each.

1. Knowledge of derived categories as triangulated categories (and derived functors etc) is generally assumed throughout. $\infty$-categories are pretty much avoidable in "Condensed Mathematics" and the first half (until Lecture 9) of "Analytic Geometry", but become highly relevant both in the later parts of "Analytic Geometry" (about the formalism of analytic spaces) and in the lectures on "Complex Geometry". They are also absolutely indispensable for Lucas Mann's thesis (and I think reading his thesis will require familiarity with not just derived $\infty$-categories but very large chunks of Lurie's foundational works).

2. Mathematically speaking, no: There are many situations where condensed mathematics may be helpful and where nothing derived or higher categorical is relevant at all. I hope that in the future, there will be expositions of condensed mathematics that will be of a much more gentle nature. But in our texts, we optimized in the direction of generality and conciseness. The formalism of $\infty$-categories is a very convenient language for expressing many statements, and in particular for expressing them in their natural generality. But this does not mean that they are intrinsically necessary. As a concrete example, take the assertion that for a CW complex $X$, the derived solidification of $\mathbb Z[X]$ is computing the homology of $X$. Our preferred way of saying is that the map of condensed anima $X\to |X|$ (from $X$ to its homotopy type/anima $|X|$), coming from Lemma 11.9 of "Analytic Geometry", induces an isomorphism $$ \mathbb Z[X]^{L\blacksquare}\cong \mathbb Z[|X|]^{L\blacksquare}\cong \mathbb Z[|X|]$$ where the latter "is" the complex computing the homology of $X$. This is "better" as it produces a specific isomorphism, and is a statement not just about homology groups, but in the derived ($\infty$-)category (and also elucidates its functoriality in $X$). But there are weaker versions of this statement that can be formulated and proved using much more elementary technology.

3. Generally I would recommend Lurie's works on Higher Topos Theory and Higher Algebra, which contain everything you need, and get you set up with the right language and tools. Relevant here is just the first chapter of Higher Algebra, which is not too long. There may be other good references out there now, I hope somebody will point some out.

4. From a purely mathematical point of view, there's no logical necessity to first learn about triangulated categories; one can just go straight to stable $\infty$-categories (which are, to me at least, a much nicer notion). Whether this should be recommended is a different question that may lead to heated discussions...