Monoidal Properties of Franke's Exotic Equivalence
Nikitas Nikandros, Constanze Roitzheim
TL;DR
This work addresses when Franke's exotic reconstruction $\mathcal{R}$ is compatible with monoidal structures, despite the underlying model categories not being Quillen equivalent. The authors develop a monoidal framework around crowned diagrams, homotopy Kan extensions, and the Franke $\mathcal{Q}$-functor, using a spectral sequence $E^2_{pq}=H_p(I;F_qX)$ to control homology data and a detailed cone analysis to compare tensor products with external smash products. The main contributions are Theorems A and B, establishing that under a hereditary abelian monoidal setting with enough projectives and the existence of $\mathcal{R}$, one has $\mathcal{R}(M_*\otimes^{\mathbb{L}}N_*) \cong \mathcal{R}(M_*) \wedge^{\mathbb{L}} \mathcal{R}(N_*)$, together with a compatible description of hocolims via crowned diagrams. These results provide a robust proof of monoidal compatibility for $\mathcal{R}$ in key situations (e.g., $KU$- and $E(1)$-type contexts) and clarify the multiplicative structure transfer for exotic equivalences, with potential applications to $K$-local spectra and related algebraic models.
Abstract
Franke's reconstruction functor R is known to provide examples of triangulated equivalences between homotopy categories of stable model categories, which are exotic in the sense that the underlying model categories are not Quillen equivalent. We show that, while not being a tensor-triangulated functor in general, R is compatible with monoidal products.