Previously: Profunctors.

# Traversals

A traversal is a kind of optic that can focus on zero or more items at a time. Naively, we would expect to have a getter that returns a list of values, and a setter that replaces a list of values. Think of a tree with $N$ leaves: a traversal would return a list of leaves, and it would allow you to replace them with a new list. The problem is that the size of the list you pass to the setter cannot be arbitrary—it must match the number of leaves in the particular tree. This is why, in Haskell, the setter and the getter are usually combined in a single function:

s -> ([b] -> t, [a])


Still, Haskell is not able to force the sizes of both lists to be equal.

Since a list type can be represented as an infinite sum of tuples, I knew that the categorical version of this formula must involve a power series, or a polynomial functor:

$\mathbf{Set} \big(s, \sum_{n} \mathbf{Set}(b^n, t) \times a^n\big)$

but was unable to come up with an existential form for it.

Pickering, Gibbons, and Wu came up with a representation for traversals using profunctors that were cartesian, cocartesian, and monoidal at the same time, but the monoidal constraint didn’t fit neatly into the Tambara scheme:

class Profunctor p => Monoidal p where
par   :: p a b -> p c d -> p (a, c) (b, d)
empty :: p () ()


We’ve been struggling with this problem, when one of my students, Mario Román came up with the ingenious idea to make $n$ existential.

The idea is that a coend in the existential representation of optics acts like a sum (or like an integral—hence the notation). A sum over natural numbers is equivalent to the coend over the category of natural numbers.

At the root of all optics there is a monoidal action. For lenses, this action is given by “scaling”

$a \to a \times c$

For prisms, it’s the “translation”

$a \to a + c$

For grates it’s the exponentiation

$a \to a^c$

The composition of a prism and a lens is an affine transformation

$a \to c_0 + a \times c_1$

A traversal is similarly generated by a polynomial functor, or a power series functor:

$a \to \sum_n c_n \times a^n$

The key observation here is that there is a different object $c_n$ for every power of $a$, which can only be expressed using dependent types in programming. For every multiplicity of foci, the residue is of a different type.

In category theory, we can express the whole infinite sequence of residues as a functor from the monoidal category $\mathbb{N}$ of natural numbers to $\mathbf{Set}$. (The sum is really a coend over $\mathbb{N}$.)

The existential version of a traversal is thus given by:

$\int^{c \colon [\mathbb{N}, \mathbf{Set}]} \mathbf{Set}\big(s, \sum_n c_n \times a^n\big) \times \mathbf{Set}\big( \sum_m c_m \times b^m, t\big)$

We can now use the continuity of the hom-set to replace the mapping out of a sum with a product of mappings:

$\int^{c \colon [\mathbb{N}, \mathbf{Set}]} \mathbf{Set}\big(s, \sum_n c_n \times a^n\big) \times \prod_m \mathbf{Set}\big( c_m \times b^m, t\big)$

$\int^{c \colon [\mathbb{N}, \mathbf{Set}]} \mathbf{Set}\big(s, \sum_n c_n \times a^n\big) \times \prod_m \mathbf{Set}\big( c_m, \mathbf{Set}( b^m, t)\big)$

The product of hom-sets is really an end over $\mathbb{N}$, or a set of natural transformations in $[\mathbb{N}, \mathbf{Set}]$

$\int^{c \colon [\mathbb{N}, \mathbf{Set}]} \mathbf{Set}\big(s, \sum_n c_n \times a^n\big) \times [\mathbb{N}, \mathbf{Set}]\big( c_-, \mathbf{Set}( b^-, t)\big)$

and we can apply the Yoneda lemma to “integrate” over $c$ to get:

$\mathbf{Set}(s, \sum_n (\mathbf{Set}(b^n, t) \times a^n)\big)$

which is exactly the formula for traversals.

Once we understood the existential representation of traversals, the profunctor representation followed. The equivalent of Tambara modules for traversals is a category of profunctors equipped with the monoidal action parameterized by objects in $[\mathbb{N}, \mathbf{Set}]$:

$\alpha_{c, \langle a, b \rangle} \colon p \langle a, b \rangle \to p\langle \sum_n c_n \times a^n, \sum_m c_m \times b^m \rangle$

The double Yoneda trick works for these profunctors as well, proving the equivalence with the existential representation.

# Generalizations

As hinted in my blog post and formalized by Mitchell Riley, Tambara modules can be generalized to an arbitrary monoidal action. We have also realized that we can combine actions in two different categories. We could take an arbitrary monoidal category $\mathcal{M}$, define its action on two categories, $\mathcal{C}$ and $\mathcal{D}$ using strong monoidal functors:

$F \colon \mathcal{M} \to [\mathcal{C}, \mathcal{C}]$

$G \colon \mathcal{M} \to [\mathcal{D}, \mathcal{D}]$

These actions define the most general existential optic:

$\mathbf{Optic} \langle s, t \rangle \langle a, b \rangle = \int^{m \colon \mathcal{M}} \mathcal{C}(s, F_m a) \times \mathcal{D}(G_m b, t)$

Notice that the pairs of arguments are heterogenous—e.g., in $\langle a, b \rangle$, $a$ is from $\mathcal{C}$, and $b$ is from $\mathcal{D}$.

We have also generalized Tambara modules:

$\alpha_{m, \langle a, b \rangle} \colon p \langle a, b \rangle \to p \langle F_m a, G_m b\rangle$

and the Pastro Street derivation of the promonad. That lead us to a more general proof of isomorphism between the profunctor formulation and the existential formulation of optics. Just to be general enough, we did it for enriched categories, replacing $\mathbf{Set}$ with an arbitrary monoidal category.

Finally, we described some new interesting optics like algebraic and monadic lenses.

# The Physicist’s Explanation

The traversal result confirmed my initial intuition from general relativity that the most general optics are generated by the analog of diffeomorphisms. These are the smooth coordinate transformations under which Einstein’s theory is covariant.

Physicists have long been using symmetry groups to build theories. Laws of physics are symmetric with respect to translations, time shifts, rotations, etc.; leading to laws of conservation of momentum, energy, angular momentum, etc. There is an uncanny resemblance of these transformations to some of the monoidal actions in optics. The prism is related to translations, the lens to rotations or scaling, etc.

There are many global symmetries in physics, but the real power comes from local symmetries: gauge symmetries and diffeomorphisms. These give rise to the Standard Model and to Einstein’s theory of gravity.

A general monoidal action seen in optics is highly reminiscent of a diffeomorphism, and the symmetry behind a traversal looks like it’s generated by an analytical function.

In my opinion, these similarities are a reflection of a deeper principle of compositionality. There is only a limited set of ways we can decompose complex problems, and sooner or later they all end up in category theory.

The main difference between physics and category theory is that category theory is more interested in one-way mappings, whereas physics deals with invertible transformations. For instance, in category theory, monoids are more fundamental than groups.

Here’s how categorical optics might be seen by a physicist.

In physics we would start with a group of transformations. Its representations would be used, for instance, to classify elementary particles. In optics we start with a monoidal category $\mathcal{M}$ and define its action in the target category $\mathcal{C}$. (Notice the use of a monoid rather than a group.)

$F \colon \mathcal{M} \to [\mathcal{C}, \mathcal{C}]$

In physics we would represent the group using matrices, here we use endofunctors.

A profunctor is like a path that connects the initial state to the final state. It describes all the ways in which $a$ can evolve into $b$.

If we use mixed optics, final states come from a different category $\mathcal{D}$, but their transformations are parameterized by the same monoidal category:

$G \colon \mathcal{M} \to [\mathcal{D}, \mathcal{D}]$

A path may be arbitrarily extended, at both ends, by a pair of morphisms. Given a morphism in $\mathcal{C}$:

$f \colon a' \to a$

and another one in $\mathcal{D}$

$g \colon b \to b'$

the profunctor uses them to extend the path:

$p \langle a, b \rangle \to p \langle a', b' \rangle$

A (generalized) Tambara module is like the space of paths that can be extended by transforming their endpoints.

$\alpha_{m, \langle a, b \rangle} \colon p \langle a, b \rangle \to p \langle F_m a, G_m b\rangle$

If we have a path that can evolve $a$ into $b$, then the same path can be used to evolve $F_m a$ into $G_m b$. In physics, we would say that the paths are “invariant” under the transformation, but in category theory we are fine with a one-way mapping.

The profunctor representation is like a path integral:

$\int_{p \colon \mathbf{Tam}} \mathbf{Set}( p \langle a, b \rangle, p \langle s, t \rangle)$

We fix the end-states but we vary the paths. We integrate over all paths that have the “invariance” or extensibility property that defines the Tambara module.

For every such path, we have a mapping that takes the evolution from $a$ to $b$ and produces the evolution (along the same path) from $s$ to $t$.

The main theorem of profunctor optics states that if, for a given collection of states, $\langle a, b \rangle, \langle s, t \rangle$, such a mapping exists, then these states are related. There exists a transformation and a pair of morphisms that are secretly used in the path integral to extend the original path.

$\int^{m \colon \mathcal{M}} \mathcal{C}(s, F_m a) \times \mathcal{D}(G_m b, t)$

Again, the mappings are one-way rather than both ways. They let us get from $s$ to $F_m a$ and from $G_m b$ to $t$.

This pair of morphisms is enough to extend any path $p \langle a, b \rangle$ to $p \langle s, t \rangle$ by first applying $\alpha_m$ and then lifting the two morphisms. The converse is also true: if every path can be extended then such a pair of morphisms must exist.

What seems unique to optics is the interplay between transformations and decompositions: The way $m$ can be interpreted both as parameterizing a monoidal action and the residue left over after removing the focus.

# Conclusion

For all the details and a list of references you can look at our paper “Profunctor optics, a categorical update.” It’s the result of our work at the Adjoint School of Applied Category Theory in Oxford in 2019. It’s avaliable on arXiv.

I’d like to thank Mario Román for reading the draft and providing valuable feedback.