nLab Tambara module

Idea

A Tambara module is a profunctor endowed with additional structure that makes it interact nicely with the action of a monoidal category. In other words, Tambara modules generalize profunctors from categories to actegories. Tambara modules are used in the theory of optics (in computer science).

Definition

Definition

Let $(\mathbf M, i, \odot)$ be a monoidal category, let $\mathbf C$ and $\mathbf D$ be two left $\mathbf M$-actegories (in other terminology, left $\mathbf M$-modules). We denote $\mathbf M$ actions by $(-)\cdot (-)$.

A (left) Tambara module is a profunctor $P \,\colon\, \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ equipped with a family of morphisms called (left) strength

$$s_{a,b,m} \,\colon\, P(a,b) \longrightarrow P(m \cdot a, m \cdot b)$$

which is natural in $a$ and $b$ and dinatural in $m$, and satisfies two coherence laws:

1. $s_{a,b,i} = P(\rho_a, \rho_b^{-1})$, where $\rho$ comes with the module structures of $\mathbf C$ and $\mathbf D$.

2. $s_{a,b,m \odot n} = P(\mu^{-1}_{m,n,a}, \mu_{m,n,b}) \circ s_{a,b,m} \circ s_{a,b,n}$ where $\mu$ comes with the module structures of $\mathbf C$ and $\mathbf D$.

This is the same thing as a (left) strong profunctor.

A right Tambara module is defined in almost the exact same way except $\mathbf C$ and $\mathbf D$ are assumed to be right $\mathbf M$-actegories and everything is correspondigly ‘done on the other side’.

Definition

Suppose now $\mathbf C$ and $\mathbf D$ have both left and right $\mathbf M$-actegories structures. A Tambara bimodule is a profunctor $P : \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ equipped with compatible left and right Tambara module structures $s$ and $s'$, i.e. such that they satisfy

$$s'_{m \cdot a, m \cdot b, n} \circ s_{a,b,m} = s_{a \cdot n, b \cdot n, m} \circ s'_{a,b,n}$$

Definition

Let $P, Q: \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ be (left) Tambara modules. A morphism of Tambara modules $\alpha : P \to Q$ is a natural transformation $P \Rightarrow Q$ that commutes with $P$ and $Q$‘s strengths:

$$\alpha_{a,b} \circ s^P_{a,b,m} = s^Q_{a,b,m} \circ \alpha_{m \cdot a,m \cdot b}$$

Remark

Tambara modules are named after Daisuke Tambara who introduced them in Tamb06 to prove that for a $\mathbf V$-enriched category $\mathbf A$, $Z(\mathbf A^{op}, \mathbf V) \cong \mathbf{Tamb}(\mathbf A, \mathbf A)$.

Remark

Both Tamb06 and PS07 define Tambara module to mean Tambara bimodule. In the optical literature, however, left Tambara modules are called simply modules and they are the one used. This doesn’t pose any problem when $\mathbf M$ is symmetric monoidal since it’s evident how, in that case, any left module structure can be made into a right module structure, so that any left/right module is automatically a bimodule.

References

• Daisuke Tambara, Distributors on a tensor category, Hokkaido mathematical journal, 35(2):379–425, 2006.

• Bartosz Milewski, Profunctor optics: the categorical view (blog post)

• Mario Román, Profunctor optics and traversals, 2020 (arXiv)

• Craig Pastro?, Ross Street, Doubles for monoidal categories, 2007 (arXiv)