Set-theoretic size of cohomological Bousfield classes

Determine whether every well-generated tensor-triangulated category has only a set of cohomological Bousfield classes.

Background

The paper explains that, unlike homological Bousfield classes, cohomological Bousfield classes are not known in general to form a set. This is connected to foundational set-theoretic issues concerning well-generated triangulated categories and localizing subcategories. The question is stated explicitly as unknown by the authors.

References

In contrast, it is not known whether every cohomological Bousfield class ${A}$ is strictly localizing, nor is it known in general whether there is only a set of cohomological Bousfield classes.

— Local Bousfield classes via homological support  (2608.26876 - Barthel et al., 27 Aug 2026) in Remark following Example 2.1, Section 2, “Homological and cohomological Bousfield classes”