Tambara modules as algebras

Tambara modules are the algebras of a monad. We start by noting that the hom profunctor is a monoid respect to Day convolution. This makes the following functor a monad on endoprofunctors, classically known as the Pastro-Street monad, $\Phi (P)=hom⊛P⊛hom;\text{mbox}where\Phi ﹕[C^{op}\times C,Set]\to [C^{op}\times C,Set].$ Theorem. The algebras of the Pastro-Street monad, the Φ-algebras, are precisely Tambara modules (Pastro and Street, 2008).

Corollary. The free Tambara module over a profunctor $H﹕C^{op}\times C\to \mathbf{Set}$ is $\Phi (H)$.

Example. Consider the profunctor ${よ}_B^A﹕{\mathbb{A}}^{op}\times \mathbb{A}\to \mathbf{Set}$ that produes a hole of types $A$ and $B$. That is, let ${よ}_B^A=\mathrm{hom}(•,A)\times \mathrm{hom}(B,•)$. The free Tambara module over it is the board with a hole of type $A$ and $B$, $\Phi ({よ}_B^A)={\int }^{M,N}\mathrm{hom}(•,M\otimes A\otimes N)\times \mathrm{hom}(M\otimes B\otimes N,•).$

Tags: Tambara module, Monad, duoidal category.