Questions about SGA 4

Question

1. What is Deligne's motivation in Appendice 9 of Exposé VI to prove that every coherent topos has enough points? For instance, does that have applications in étale cohomology (or other parts of algebraic geometry)?
2. Which topics are discussed in the Exposés Vbis, VI, VIII, and IX? An answer to the question should provide either buzzwords or English literature covering these topics. What role do topoi play in these Exposés?
3. Here Joyal writes the following. What could he mean by that?

About half of the topos theory of SGA4 is devoted to categorical generalities. They are now subsumed by the modern theory of (locally) presentable categories.

Answer 1

1. I'm not aware of any applications of Deligne's result to algebraic geometry at the time. It does formally imply that certain topologies (e.g. fppf) have enough points, but this wasn't very useful because the actual points in many well-known topologies were only described by Gabber–Kelly and Schröer in 2014. On the other hand, points in the étale topology are just geometric points, so those are very useful in the theory, but you don't need Deligne's completeness theorem for this.

   Deligne's theorem does have formal consequences to the systematic study of coherent topoi, such as in Barwick–Glasman–Haine's Exodromy, where it is used in the proof of a base change theorem for oriented fibre products of bounded coherent $\infty$-topoi (Thm. 7.1.7).

2. This is many questions in one, but let me say some words.

   • V$^{\text{bis}}$ is the origin of hypercoverings and cohomological descent. This is a powerful tool that is used frequently in algebraic geometry, especially in combination with resolution of singularities or alterations to build up the cohomology of a variety from the cohomology of smooth projective varieties. This happens for example in Deligne's Hodge III paper, where he uses proper hypercoverings to construct the mixed Hodge structure on a singular or open variety.

     One alternative source for the material from V$^{\text{bis}}$ is the chapter on hypercoverings in the Stacks project, but I imagine to a category theorist this might look a bit dated.

     Descent with respect to arbitrary hypercoverings is also a topic that comes up in $\infty$-topoi, where this property is called hyperdescent. The relation between hypercompleteness and hyperdescent is explained in sections 6.5.3 of Higher Topos Theory, and §6.5.4 makes the case that you do not want to impose this as an axiom on an $\infty$-topos. Because hypercompleteness is related to Whitehead's theorem (which holds on the $\infty$-topos of a point), an $\infty$-topos can only have enough points (in the sense of question 1) if it is hypercomplete!

   • VI is about coherent topoi and is purely topos-theoretic in nature (although the definitions of quasi-compact and quasi-separated are inspired by the spaces occurring in algebraic geometry). I don't know secondary references for this, nor is it incredibly clear to me where it is used, so I will leave this to someone else to answer.

   • VIII is the first place where some actual computations in the étale cohomology happen. For example, it is explained that the étale site of a field $k$ with absolute Galois group $\Gamma_k$ is a $B\Gamma_k$ (at least if you understand $\Gamma_k$ as a profinite group and interpret $B\Gamma_k$ in that light).

     This material is all specific to the étale site in algebraic geometry, and strays away from general topos theory. Most of the results can be found in the Stacks Project, or Milne's book or notes on Étale Cohomology, or the book by Freitag and Kiehl, or ...

   • IX is about constructible sheaves and the cohomology on a curve. Constructible sheaves also exist in topology, but in algebraic geometry there are pervasive finiteness hypotheses for everything to work.

     More interesting is the computation of the cohomology of a curve. For example, you want to know that étale cohomology with finite coefficients vanishes above dimension $2$. In algebraic geometry, the only way to do this is to show that the Galois group of a function field of a curve has cohomological dimension $1$, and then use the Leray spectral sequence for $\operatorname{Spec} K(C) \hookrightarrow C$. The rest of the computation of the cohomology on a curve is based on its Picard group (or Jacobian).

3. This has been addressed by Tim Campion in the comments. As I noted, schemes don't start coming into SGA 4 until V$^{\text{bis}}$ (a little bit) and really starting at VII. All the material prior to IV (and even some of the material after that) really is general category theory, a substantial part of which consists of putting size bounds on everything to make sure all categories are small or at least locally small. This is exactly the study of accessible categories. The classical reference is probably Adámek–Rosicky's Locally presentable and accessible categories, and I suspect SGA 4 would be substantially shortened if one already has access to all that.