nLab strong profunctor

Idea

A strong profunctor or strong distributor is a profunctor endowed with additional structure that makes it interact nicely with the action of a monoidal category. In other words, strong profunctors generalize profunctors from categories to actegories. Strong profunctors are used in the theory of optics (in computer science), where they are often called Tambara modules (after Tambara 2006, although Tambara was not the first to study them).

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) strong profunctor 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 $\mathbf D$.

A right strong profunctor 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 correspondingly ‘done on the other side’.

Definition

Suppose now $\mathbf C$ and $\mathbf D$ have both left and right $\mathbf M$-actegories structures. A (bi)strong profunctor is a profunctor $P : \mathbf C^{op} \times \mathbf D \to \mathbf{Set}$ equipped with compatible left and right strong profunctor 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) profunctor. A morphism of profunctors $\alpha : P \to Q$ 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

The terminology “Tambara modules” are named after Daisuke Tambara who studied them in Tamb06 to prove that for a $\mathbf V$-enriched category $\mathbf A$, $Z(\mathbf A^{op}, \mathbf V) \cong \mathbf{Tamb}(\mathbf A, \mathbf A)$. However, they have been studied earlier, e.g. see Paré and Roman 1998.

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

Strong profunctors were defined in:

• Robert Paré and Leopoldo Román, Dinatural numbers (1998), JPAA

• 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)