Distributivity of Compact Weight Scales

Definitions and mathematical statement

Here $\omega$ denotes the first infinite ordinal, and "smc $\infty$-category" is read as "symmetric monoidal $\infty$-category." For such a category $\mathcal V$, the source denotes its scale of $\omega$-compact weights by $\omega(\mathcal V)$.

The supplied conjecture is: for every symmetric monoidal $\infty$-category $\mathcal V$, \[ \mathcal V\text{ is }\omega\text{-presentable} \quad\Longrightarrow\quad \omega(\mathcal V)\text{ is distributive}. \] The expressions "$\omega$-presentable," "$\omega$-compact weight," "scale," and "distributive" retain their source-specific meanings. The supplied record does not define these notions, construct $\omega(\mathcal V)$, or state the distributivity law and its quantifiers. It also does not specify any required compatibility between the symmetric monoidal structure and presentability. Consequently, this is a schematic transcription of the conjecture, requiring the cited report's definitions and hypotheses before it becomes a self-contained mathematical statement. No particular lattice identity or colimit compatibility law is assumed here.

Short English statement

The conjecture asks whether every $\omega$-presentable symmetric monoidal $\infty$-category has a distributive scale of $\omega$-compact weights. The scale and the intended distributivity law require definitions absent from the supplied record.

Sources

• Local UnsolvedMath registry: permanent ID $30006036$, title "Distributivity of Compact Weight Scales," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2024; no individual proposer is identified.
• Supplied source citation: Homotopical Algebra and Higher Structures (2025), Oberwolfach Reports, pp. 2251--2308, DOI 10.4171/owr/2024/39. https://doi.org/10.4171/owr/2024/39. The record attributes the conjecture to this workshop report and describes its original-statement passage as retained verbatim; that description has not been independently checked.
• Supplied literature source: "Oberwolfach Report 39 (2024)," https://doi.org/10.4171/owr/2024/39. Literature checked at: 2026-08-22. This is supplied dated curation, not independently verified: it labels the problem open, describes distributivity of the scale of $\omega$-compact weights as an explicit conjecture, and reports that no resolution of the general statement was found. The background expressly calls for independent verification of time-sensitive status and cited claims.