Condensed-Set Equivalence of CH and Regular Model Structures

Definitions and mathematical statement

A condensed set is a sheaf of sets on the site of profinite sets with coverings given by finite jointly surjective families, subject to appropriate size conventions. The supplied record does not specify those conventions.

The source conjectures a comparison after replacing its ambient category $U$ by the category of condensed sets. Let $\mathcal D$ denote the common underlying category on which the resulting CH and regular model structures are intended to be defined, and write \[ W_{\mathrm{CH}},\;W_{\mathrm{reg}}\subseteq\operatorname{Mor}(\mathcal D) \] for their respective classes of weak equivalences. The requested assertion is \[ W_{\mathrm{CH}}=W_{\mathrm{reg}}, \] or, equivalently, \[ \forall X,Y\in\operatorname{Ob}(\mathcal D)\;\forall f\in\operatorname{Hom}_{\mathcal D}(X,Y),\qquad f\in W_{\mathrm{CH}}\iff f\in W_{\mathrm{reg}}. \]

Unresolved formulation qualifications. This is a schematic statement of the supplied conjecture: the record does not define $U$, specify $\mathcal D$, or give the constructions and hypotheses of either model structure after replacement by condensed sets. In particular, it does not determine whether the model structures are on condensed sets themselves or on a category of objects constructed from them. Their existence and common underlying category must be established from the source context before the displayed assertion becomes a fully specified conjecture. The clean statement expands CH as "compactly generated Hausdorff," whereas the supplied original excerpt uses only "CH-structure"; that expansion is retained as supplied terminology requiring contextual verification. The source additionally suggests that the conjecture should strengthen the analogue of its Theorem 1 and permit further results, but neither that theorem nor its proposed analogue is supplied.

Short English statement

The conjecture asks whether the CH and regular model structures have exactly the same weak equivalences after the ambient setting is replaced by condensed sets. The supplied excerpt leaves the underlying category and both model structures undefined, so the equality above remains schematic and requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005700$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No named proposer is supplied; the proposed year is recorded as 2023.
• Supplied source citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, pp. 3103--3138, DOI 10.4171/owr/2023/55. The original conjecture is located at p. 3129 in the supplied record. https://doi.org/10.4171/owr/2023/55
• Supplied literature source, labeled "Oberwolfach report": https://publications.mfo.de/bitstream/handle/mfo/4137/OWR_2023_55.pdf?isAllowed=y&sequence=4
• Supplied literature source, labeled "Corob Cook": https://doi.org/10.3390/axioms8030090. The record supplies no title or precise result connecting this reference to the conjecture.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the replacement by condensed sets as a conjecture, and reports that no later proof equating the weak equivalences was located. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. These are supplied verification claims; no source consultation or independent verification is asserted here.