Tambara module

Last updated Apr 23, 2024

Definition. Let $(𝔸, ⊗, I)$ be a monoidal category. A Tambara module is a profunctor $T ﹕ 𝔸^{op} × 𝔸 → \mathbf{Set}$ endowed with transformations $$t_l^M \colon T(X;Y) → T(M ⊗ X, M ⊗ Y),$$ $$t_r^M \colon T(X;Y) → T(X ⊗ M, Y ⊗ M),$$ that are natural in both $X$ and $Y$, but also dinatural on $M$. These must moreover satisfy the following axioms:

• $t_l^I = id$ and $t^I_r = id$, unitality;
• $t_l^M ⨾ t_l^N = t_l^{N ⊗ M}$ and $t_r^M ⨾ t_r^N = t_l^{M ⊗ N}$, multiplicativity;
• $t_l^M ⨾ t_r^N = t_r^N ⨾ t_l^M$, and compatibility.

References:

• Doubles for Monoidal Categories (Pastro and Street, 2008)