Symmetric monoidality of the trivial module functor for orthogonal epimorphisms

Let OEpi\mathsf{OEpi} be the category of orthogonal epimorphisms, let O()O(\ast) denote its endomorphism category of objects, and let RModO()\mathsf{RMod}_{O(\ast)} and RModOEpi\mathsf{RMod}_{\mathsf{OEpi}} be the corresponding categories of right modules. Write Triv\mathsf{Triv} for the trivial-module functor.

Symmetric monoidality conjecture. For OEpi\mathsf{OEpi},

Triv:RModO()RModOEpi\mathsf{Triv}: \mathsf{RMod}_{O(\ast)} \longrightarrow \mathsf{RMod}_{\mathsf{OEpi}}

is symmetric monoidal with respect to Day convolution.

This conjecture would imply a product rule for Weiss calculus, identifying the derivatives of a smash product with the Day convolution of the derivatives. The paper does not establish the required symmetric monoidality.

Sources & referencesView supporting material

Primary source

Connor Malin and Niall Taggart, “Koszul duality and a classification of stable Weiss towers”, arXiv:2409.01335 (2024).

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