The equivalence conjecture for c6-modules, B-pairs and vector bundles

From papers

Let RR be a ring and consider the corresponding c6c6-module functors, B{\square\square\square}B-pair functors, and vector bundle functors over FF{\square\square\square}FF functors on the indicated categories

sCommSimplicialIndSeminormedRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}\mathrm{Seminormed}^{\mathrm{formalseriescolimitcomp}}_R, sCommSimplicialIndmSeminormedRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}^m\mathrm{Seminormed}^{\mathrm{formalseriescolimitcomp}}_R, sCommSimplicialIndNormedRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}\mathrm{Normed}^{\mathrm{formalseriescolimitcomp}}_R, sCommSimplicialIndmNormedRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}^m\mathrm{Normed}^{\mathrm{formalseriescolimitcomp}}_R, sCommSimplicialIndBanachRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}\mathrm{Banach}^{\mathrm{formalseriescolimitcomp}}_R, sCommSimplicialIndmBanachRformalseriescolimitcomp,\mathrm{sComm}\mathrm{Simplicial}\mathrm{Ind}^m\mathrm{Banach}^{\mathrm{formalseriescolimitcomp}}_R,

or in AnalyticRingsRCS,formalcolimitclosure\mathrm{AnalyticRings}^{\mathrm{CS,formalcolimitclosure}}_R. The equivalence conjecture. The corresponding \infty-categories of φ\varphi-module functors, B{\square\square\square}B-pair functors, and vector bundle functors over FF{\square\square\square}FF functors are equivalent over these categories. This is presented as a direct conjectural consequence of the cited theorem, but the supplied material gives no resolution status.

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Sources & referencesView supporting material

Primary source

Xin Tong, “Topologization and Functional Analytification III”, arXiv:2405.14180 (2024).

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