Verify the Bousfield-localisation claim in Nardin and Shah’s construction

Verify that, in the construction of normed structures on presentable categories, the relevant category is a localisation of the indicated category at the relevant class of morphisms, and explain why saturation behaves well with respect to indexed products.

Background

The paper identifies a technical gap in the treatment of distributivity in Nardin and Shah’s work. Specifically, the cited construction asserts that a category is a localisation at a specified class of morphisms, but the paper’s authors point out that the categories were defined as classes of local objects, hence as Bousfield localisations. Passing from the original class of morphisms to its strong saturation is therefore necessary, and compatibility of this saturation process with indexed products must be established for the distributive argument to work.

References

We believe that there is a gap in Cor.~6.0.9 (proved in our seeting as \Cref{prop:distributivity_inverted_maps_presheaves}: in the proof they state that \enquote{by definition $g_\ast#1{\mathbb{P}{f\ast\mathcal{All}(f_\ast)$ is a localisation of $g_\ast #1{\mathbb{P}{\mathcal{T}{/V}(f\ast)$ at the relevant class of morphisms}. We could not verify that this is true by definition: the relevant categories were defined as the class of local objects of a class of morphism, i.e. as a Bousfield localisation.

— Normed idempotence of spectra and distributivity in equivariant higher algebra  (2610.22074 - Hilman et al., 15 Oct 2026) in Section 1, subsection “Relation to other work”