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, $$Φ(P) = hom \circledast P \circledast hom; \mbox{ where } Φ﹕[ℂ^{op}×ℂ,Set] → [ℂ^{op}×ℂ,Set].$$ 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 $H ﹕ ℂ^{op} × ℂ → \mathbf{Set}$ is $Φ(H)$.
Example. Consider the profunctor $よ^A_B ﹕ 𝔸^{op} × 𝔸 → \mathbf{Set}$ that produes a hole of types $A$ and $B$. That is, let $よ^A_B = \hom(•,A) × \hom(B,•)$. The free Tambara module over it is the board with a hole of type $A$ and $B$, $$Φ(よ^A_B) = ∫^{M,N} \hom(•,M⊗A⊗N) × \hom(M⊗B⊗N,•).$$
Tags: Tambara module, Monad, duoidal category.