### Lens

Category theory extracts the essence of structure and composition. At its foundation it deals with the composition of arrows. Building on composition of arrows it then goes on describing the ways objects can be composed: we have products, coproducts and, at a higher level, tensor products. They all describe various modes of composing objects. In monoidal categories any two objects can be composed.

Unlike composition, which can be described uniformly, decomposition requires case-by-case treatment. It’s easy to decompose a cartesian product using projections. A coproduct (sum) can be decomposed using pattern matching. A generic tensor product, on the other hand, has no standard means of decompositon.

Optics is the essence of decomposition. It answers the question of what it means to decompose a composite.

We consider an object decomposable when:

• We can split it into the focus and the complement,
• We can replace the focus with something else, without changing the complement, to get a new composite object,
• We can zoom in; that is, if the focus is decomposable, we can compose the two decompositions,
• It’s possible for the whole object to be the focus.

Let’s translate these requirements into the language of category theory. We’ll start with the standard example: the lens, which is the optic for decomposing cartesian products.

The splitting means that there is a morphism from the composite object $s$ to the product $c \times a$, where $c$ is the complement and $a$ is the focus. This morphism is a member of the hom-set $\mathcal{C}(s, c \times a)$.

To replace the focus we need another morphism that takes the same complement $c$, combines it with the new focus $b$ to produce the new composite $t$. This morphism is a member of the hom-set $\mathcal{C}(c \times b, t)$

Here’s the important observation: We don’t care what the complement is. We are “focusing” on the focus. We carry the complement over to combine it with the new focus, but we don’t use it for anything else. It’s a featureless black box.

To erase the identity of the complement, we hide it inside a coend. A coend is a generalization of a sum, so it is written using the integral sign (see the Appendix for details). Programmers know it as an existential type, logicians call it an existential quantifier. We say that there exists a complement $c$, but we don’t care what it is. We “integrate” over all possible complements.

Here’s the existential definition of the lens:

$L(s, t; a, b) = \int^{c : \mathcal{C}} \mathcal{C}(s, c \times a) \times \mathcal{C}(c \times b, t)$

Just like we construct a coproduct using one of the injections, so the coend is constructed using one of (possibly infinite number of) injections. In our case we construct a lens $L(s, t; a, b)$ by injecting a pair of morphisms from the two hom-sets sharing the same $c$. But once the lens is constructed, there is no way to extract the original $c$ from it.

It’s not immediately obvious that this representation of the lens reproduces the standard setter/getter form. However, in a cartesian closed category, we can use the currying adjunction to transform the second hom-set:

$\mathcal{C}(c \times b, t) \cong \mathcal{C}(c, [b, t])$

Here, $[b, t]$ is the internal hom, or the function object representing morphisms from $b$ to $t$. We can then use the co-Yoneda lemma to reduce the coend:

$\int^{c : \mathcal{C}} \mathcal{C}(s, c \times a) \times \mathcal{C}(c, [b, t]) \cong \mathcal{C}(s, [b, t] \times a) \cong \mathcal{C}(s \times b, t) \times \mathcal{C}(s, a)$

The first part of this product is the setter: it takes the source object $s$ and the new focus $b$ to produce the new target $t$. The second part is the getter that extracts the focus $a$.

Even though all optics have similar form, each of them reduces differently.

Here’s another example: the prism. We just replace the product with the coproduct (sum).

$P(s, t; a, b) = \int^{c : \mathcal{C}} \mathcal{C}(s, c + a) \times \mathcal{C}(c + b, t)$

This time the reduction goes through the universal property of the coproduct: a mapping out of a sum is a product of mappings:

$\mathcal{C}(c + b, t) \cong\mathcal{C}(c, t) \times\mathcal{C}(b, t)$

Again, we use the co-Yoneda to reduce the coend:

$\int^{c : \mathcal{C}} \mathcal{C}(s, c + a) \times\mathcal{C}(c, t) \times\mathcal{C}(b, t) \cong\mathcal{C}(s, t + a) \times\mathcal{C}(b, t)$

The first one extracts the focus $a$, if possible, otherwise it constructs a $t$ (by secretly injecting a $c$). The second constructs a $t$ by injecting a $b$.

We can easily generalize existential optics to an arbitrary tensor product in a monoidal category:

$O(s, t; a, b) = \int^{c : \mathcal{C}} \mathcal{C}(s, c \otimes a) \times \mathcal{C}(c \otimes b, t)$

In general, though, this form cannot be further reduced using the co-Yoneda trick.

But what about the third requirement: the zooming-in property of optics? In the case of the lens and the prism it works because of associativity of the product and the sum. In fact it works for any tensor product. If you can decompose $s$ into $c \otimes a$, and further decompose $a$ into $c' \otimes a'$, then you can decompose $s$ into $(c \otimes c') \otimes a'$. Zooming-in is made possible by the associativity of the tensor product.

Focusing on the whole object plays the role of the unit of zooming.

These two properties are used in the definition of the category of optics. The objects in this category are pairs of object in $\mathcal{C}$. A morphism from a pair $\langle s, t \rangle$ to $\langle a, b \rangle$ is the optic $O(s, t; a, b)$. Zooming-in is the composition of morphisms.

But this is still not the most general setting. The useful insight is that the multiplication (product) in a lens, and addition (coproduct) in a prism, look like examples of linear transformations, with the residue $c$ playing the role of a parameter. In fact, a composition of a lens with a prism produces a 2-parameter affine transformation, which also behaves like an optic. We can therefore generalize optics to work with an arbitrary monoidal action (first hinted in the discussion at the end of this blog post). Categories with such actions are known as actegories.

The idea is that you define a family of endofunctors $A_m$ in $\mathcal{C}$ that is parameterized by objects from a monoidal category $\mathcal{M}$. So far we’ve only discussed examples where the parameters were taken from the same category $\mathcal{C}$ and the action was either multiplication or addition. But there are many examples in which $\mathcal{M}$ is not the same as $\mathcal{C}$.

The zooming principles are satisfied if the action respects the tensor product in $\mathcal{M}$:

$A_{m \otimes n} \cong A_m \circ A_n$

$A_1 \cong \mathit{Id}$

(Here, $1$ is the unit object with respect to the tensor product $\otimes$ in $\mathcal{M}$, and $\mathit{Id}$ is the identity endofunctor.)

The actegorical version of the optic doesn’t deal directly with the residue. It tells us that the “unimportant” part of the composite object can be parameterized by some $m \colon \mathcal{M}$.

This additional abstraction allows us to transport the residue between categories. It’s enough that we have one action $L_m$ in $\mathcal{C}$ and another $R_m$ in $\mathcal{D}$ to create this mixed optics (first introduced by Mitchell Riley):

$O(s, t; a, b) = \int^{m : \mathcal{M}} \mathcal{C}(s, L_m a) \times \mathcal{D}(R_m b, t)$

The separation of the focus from the complement using monoidal actions is reminiscent of what physicists call the distinction between “physical”  and “gauge” degrees of freedom.

An in-depth presentation of optics, including their profunctor representation, is available in this paper.

### Appendix: Coends and the Co-Yoneda Lemma

A coend is defined for a profunctor, that is a functor of two variables, one contravariant and one covariant, $p \colon \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}$. It’s a cross between a coproduct and a trace, as it’s constructed using injections of diagonal elements (with some identifications):

$\iota_{a} \colon p \langle a, a \rangle \to \int^{c : \mathcal{C}} p \langle c, c \rangle$

Co-Yoneda lemma is the identity that works for any covariant functor (copresheaf) $F \colon \mathcal{C} \to \mathbf{Set}$:

$\int^{c \colon \mathcal{C}} F(c) \times \mathcal{C}(c, x) \cong F(x)$

Dependent types, in programming, are families of types indexed by elements of an indexing type. For instance, counted vectors are families of tuples indexed by natural numbers—the lengths of the vectors.

In category theory we model dependent types as fibrations. We start with the total space $E$, the base space $B$, and a projection, or a display map, $p \colon E \to B$. The fibers of $p$ correspond to members of the type family. For instance, the total space, or the bundle, of counted vectors is the list type $\mathit{List} (A)$ (a free monoid generated by $A$) with the projection $\mathit{len} \colon \mathit{List} (A) \to \mathbb{N}$ that returns the length of a list.

Another way of looking at dependent types is as objects in the slice category $\mathcal{C}/B$. Counted vectors, for instance, are represented as objects in $\mathcal{C}/\mathbb{N}$ given by pairs $\langle \mathit{List} (A), \mathit{len} \rangle$. Morphisms in the slice category correspond to fibre-wise mappings between bundles.

We often require that $\mathcal{C}$ be a locally cartesian closed category, that is a category whose slice categories are cartesian closed. In such categories, the base-change functor $f^*$ has both the left adjoint, the dependent sum $\Sigma_f$; and the right adjoint, the dependent product $\Pi_f$. The base-change functor is defined as a pullback:

This pullback defines a cartesian product in the slice category $\mathcal{C}/B$ between two objects: $\langle B', f \rangle$ and $\langle E, p \rangle$. In a locally cartesian closed category, this product has the right adjoint, the internal hom in $\mathcal{C}/B$.

# Dependent optics

The most general optic is given by two monoidal actions $L_m$ and $R_m$ in two categories $\mathcal{C}$ and $\mathcal{D}$. It can be written as the following coend of the product of two hom-sets:

$O(A, A'; S, S') = \int^{m \colon \mathcal{M}} \mathcal{C}( S, L_m A) \times \mathcal{D}(R_m A', S')$

Monoidal actions are parameterized by objects in a monoidal category $(\mathcal{M}, \otimes, 1)$.

Dependent optics are a special case of general optics, where one or both categories in question are slice categories. When the monoidal action is defined in the slice category, the transformations must respect fibrations. For instance, the action in the bundle $\langle E, p \rangle$ over $B$ must commute with the projection:

$p \circ L_m = p$

This is reminiscent of gauge transformations in physics, which act on fibers in bundles over spacetime. The action must respect the monoidal structure of $\mathcal{M}$ so, for instance,

$L_{m \otimes n} \cong L_m \circ L_n$

$L_1 \cong \mathit{Id}$

We can define a dependent (mixed) optic as:

$\int^{m : \mathcal{M}} (\mathcal{C}/B)( S, L_m A) \times (\mathcal{D}/B')(R_m A', S')$

Just like regular optics, dependent optics can be represented using Tambara modules, which are profunctors with the additional structure given by transformations:

$\alpha_{m, \langle A, A' \rangle} \colon P \langle A, A' \rangle \to P\langle L_m A, R_m A' \rangle$

where $A$ and $A'$ are objects in the appropriate slice categories.
The optic is then given by the following end in the Tambara category:

$O(A, A'; S, S') = \int_{p : \mathbf{Tam}} \mathbf{Set}(P \langle A, A' \rangle, P \langle S, S' \rangle)$

# Dependent lens

The primordial optic, the lens, is defined by the monoidal action of a product. By analogy, we define a dependent lens by the action of the product in a slice category. The action parameterized by an object $\langle C, q \rangle$ on another object $\langle A, p \rangle$ is given by the pullback:

$M_C A = C \times_B A$

Since a pullback is the product in the slice category $\mathcal{C}/B$, it is automatically associative and unital, so it can be used to define a dependent lens:

$\mathit{DLens}(A, A'; S, S') = \int^{\langle C, p \rangle : \mathcal{C}/B} (\mathcal{C}/B)( S, C \times_B A) \times (\mathcal{C}/B)(C \times_B A', S')$

Since $\mathcal{C}$ is locally cartesian closed, there is an adjunction between the product and the exponential. We can use it to get:

$\cong \int^{\langle C, p \rangle : \mathcal{C}/B} (\mathcal{C}/B)( S, C \times_B A) \times (\mathcal{C}/B)(C , [A', S']_B)$

We can then apply the Yoneda lemma to get the setter/getter form:

$(\mathcal{C}/B)( S, [A', S']_B \times_B A)$

The internal hom $[A', S']_B$ in a locally cartesian closed category can be expressed using a dependent product:

$\left [\left \langle A' \atop p \right \rangle, \left \langle S' \atop q \right \rangle \right ] \cong \Pi_p \left(p^* \left \langle S' \atop q \right \rangle \right)$

where $p \colon A' \to B$ is the fibration of $A'$, $\Pi_p$ is the right adjoint to the base change functor, and $p^*$ is the base-change functor along $p$.

The dependent lens can be written as:

$(\mathcal{C} / B) \left( \left \langle S \atop r \right \rangle, \Pi_p \left(p^* \left \langle S' \atop q \right \rangle \right) \times \left \langle A \atop r' \right \rangle \right)$

In particular, if $B$ is $\mathbb{N}$, this is equal to an infinite tuple of functions:

$O(A, B; S, T) \cong \prod_n \left( s_n \to \left((b_n \to t_n) \times a_n \right) \right)$

or fiber-wise pairs of setter/getter $\langle s_n \to b_n \to t_n, s_n \to a_n \rangle$ indexed by $n$.

# Traversals

Traversals are optics whose monoidal action is generated by polynomial functors of the form:

$M_{c} a = \sum_{n \colon \mathbb{N}} c_n \times a^n$

The coefficients $c_n$ can be expressed as a fibration $\langle C, p \colon C \to \mathbb{N} \rangle$, with $C = \sum_n c_n$, the sum of the fibers. The set of powers of $a$ can be similarly written as $\langle L(a), \mathit{len} \rangle$, with $L(a)$ the type of list of $a$ (a free monoid generated by $a$), and $\mathit{len}$ the function that assigns the length to a list. The monoidal action can then be written using a product (pullback) in the slice category $\mathbf{Set}/\mathbb{N}$:

$\left \langle {C \atop p} \right \rangle \times \left \langle {L(a) \atop \mathit{len}} \right \rangle$

There is an obvious forgetful functor $U \colon \mathbf{Set}/\mathbb{N} \to \mathbf{Set}$, which can be used to express the polynomial action:

$M_c a = U\left( \left \langle {C \atop p} \right \rangle \times \left \langle {L(a) \atop \mathit{len}} \right \rangle \right)$

The traversal is the optic:

$\int^{\langle C, p \rangle : \mathbf{Set}/\mathbb{N}} \mathbf{Set} \left(s, M_c a \right) \times \mathbf{Set}(M_c b, t)$

Eqivalently, the second factor can be rewritten as:

$\mathbf{Set}\left( \sum_{n \colon \mathbb{N}} c_n \times b^n, t\right) \cong \prod_{n \colon \mathbb{N}} \mathbf{Set}(c_n \times b^n, t)$

This, in turn, is equivalent to a single hom-set in the slice category:

$\cong (\mathbf{Set}/\mathbb{N})\left(\left \langle {C \atop p} \right \rangle \times \left \langle {L(b) \atop \mathit{len}} \right \rangle, \left \langle {\mathbb{N} \times t \atop \pi_1} \right \rangle \right)$

where $\pi_1$ is the projection from the cartesian product.

The traversal is therefore a mixed optic:

$\int^{\langle C, p \rangle : \mathbf{Set}/\mathbb{N}} \mathbf{Set} \left(s, M_c a \right) \times (\mathbf{Set}/\mathbb{N})\left( \left \langle {C \atop p} \right \rangle \times \left \langle {L(b) \atop \mathit{len}} \right \rangle, \left \langle {\mathbb{N} \times t \atop \pi_1} \right \rangle \right)$

The second factor can be transformed using the internal hom adjunction:

$(\mathbf{Set}/\mathbb{N})\left(\left \langle {C \atop p} \right \rangle, \left[ \left \langle {L(b) \atop \mathit{len}} \right \rangle, \left \langle {\mathbb{N} \times t \atop \pi_1} \right \rangle \right] \right)$

We can then use the ninja Yoneda lemma on the optic to “integrate” over $\langle C, p \rangle$ and get:

$O(a, b; s, t) \cong \mathbf{Set} \left( s, U\left( \left[ \left \langle {L(b) \atop \mathit{len}} \right \rangle, \left \langle {\mathbb{N} \times t \atop \pi_1} \right \rangle \right] \times \left \langle {L(a) \atop \mathit{len}} \right \rangle \right) \right)$

$s \to \sum_n \left( (b^n \to t) \times a^n \right)$

Abstract: I present a uniform derivation of profunctor optics: isos, lenses, prisms, and grates based on the Yoneda lemma in the (enriched) profunctor category. In particular, lenses and prisms correspond to Tambara modules with the cartesian and cocartesian tensor product.

This blog post is the result of a collaboration between many people. The categorical profunctor picture solidified after long discussions with Edward Kmett. A lot of the theory was developed in exchanges on the Lens IRC channel between Russell O’Connor, Edward Kmett and James Deikun. They came up with the idea to use the Pastro functor to freely generate Tambara modules, which was the missing piece that completed the picture.

My interest in lenses started long time ago when I first made the connection between the universal quantification over functors in the van Laarhoven representation of lenses and the Yoneda lemma. Since I was still learning the basics of category theory, it took me a long time to find the right language to make the formal derivation. Unbeknownst to me Mauro Jaskellioff and Russell O’Connor independently had the same idea and they published a paper about it soon after I published my blog. But even though this solved the problem of lenses, prisms still seemed out of reach of the Yoneda lemma. Prisms require a more general formulation using universal quantification over profunctors. I was able to put a dent in it by deriving Isos from profunctor Yoneda, but then I was stuck again. I shared my ideas with Russell, who reached for help on the IRC channel, and a Haskell proof of concept was quickly established. Two years later, after a brainstorm with Edward, I was finally able to gather all these ideas in one place and give them a little categorical polish.

## Yoneda Lemma

The starting point is the Yoneda lemma, which states that the set of natural transformations between the hom-functor C(a, -) in the category C and an arbitrary functor f from C to Set is (naturally) isomorphic with the set f a:

[C, Set](C(a, -), f) ≅ f a

Here, f is a member of the functor category [C, Set], where natural transformation form hom-sets.

The set of natural transformations may be represented as an end, leading to the following formulation of the Yoneda lemma:

∫x Set(C(a, x), f x) ≅ f a

This notation makes the object x explicit, which is often very convenient. It can be easily translated to Haskell, by replacing the end with the universal quantifier. We get:

forall x. (a -> x) -> f x ≅ f a

A special case of the Yoneda lemma replaces the functor f with a hom-functor in C:

f x = C(b, x)

and we get:

∫x Set(C(a, x), C(b, x)) ≅ C(b, a)

This form of the Yoneda lemma is useful in showing the Yoneda embedding, which states that any category C can be fully and faithfully embedded in the functor category [C, Set]. The embedding is a functor, and the above formula defines its action on morphisms.

We will be interested in using the Yoneda lemma in the functor category. We simply replace C with [C, Set] in the previous formula, and do some renaming of variables:

∫f Set([C, Set](g, f), [C, Set](h, f)) ≅ [C, Set](h, g)

The hom-sets in the functor category are sets of natural transformations, which can be rewritten using ends:

∫f Set(∫x Set(g x, f x), ∫x Set(h x, f x))
≅ ∫x Set(h x, g x)

This is a short recap of adjunctions. We start with two functors going between two categories C and D:

L :: C -> D
R :: D -> C

We say that L is left adjoint to R iff there is a natural isomorphism between hom-sets:

D(L x, y) ≅ C(x, R y)

In particular, we can define an adjunction in a functor category [C, Set]. We start with two higher order (endo-) functors:

L :: [C, Set] -> [C, Set]
R :: [C, Set] -> [C, Set]

We say that L is left adjoint to R iff there is a natural isomorphism between two sets of natural transformations:

[C, Set](L f, g) ≅ [C, Set](f, R g)

where f and g are functors from C to Set. We can rewrite natural transformations using ends:

∫x Set((L f) x, g x) ≅ ∫x Set(f x, (R g) x)

In Haskell, you may think of f and g as type constructors (with the corresponding Functor instances), in which case L and R are types that are parameterized by these type constructors (similar to how the monad or functor classes are).

Here’s a little trick. Since the fixed objects in the formula for Yoneda embedding are arbitrary, we can pick them to be images of other objects under some functor L that we know is left adjoint to another functor R:

∫x Set(D(L a, x), D(L b, x)) ≅ D(L b, L a)

Using the adjunction, this is isomorphic to:

∫x Set(C(a, R x), C(b, R x)) ≅ C(b, (R ∘ L) a)

Notice that the composition R ∘ L of adjoint functors is a monad in C. Let’s write this monad as Φ.

The interesting case is the adjunction between a forgetful functor U and a free functor F. We get:

∫x Set(C(a, U x), C(b, U x)) ≅ C(b, Φ a)

The end is taken over x in a category D that has some additional structure (we’ll see examples of that later); but the hom-sets are in the underlying simpler category C, which is the target of the forgetful functor U.

The Yoneda-with-adjunction formula generalizes to the category of functors:

∫f Set(∫x Set((L g) x, f x), ∫x Set((L h) x, f x))
≅ ∫x Set((L h) x, (L g) x)

∫f Set(∫x Set((g x, (R f) x), ∫x Set(h x, (R f) x))
≅ ∫x Set(h x, (Φ g) x)

Here, Φ is the monad R ∘ L in the category of functors.

An interesting special case is when we substitute hom-functors for g and h:

g x = C(a, x)
h x = C(s, x)

We get:

∫f Set(∫x Set((C(a, x), (R f) x), ∫x Set(C(s, x), (R f) x))
≅ ∫x Set(C(s, x), (Φ C(a, -)) x)

We can then use the regular Yoneda lemma to “integrate over x” and reduce it down to:

∫f Set((R f) a, (R f) s)) ≅ (Φ C(a, -)) s

Again, we are particularly interested in the forgetful/free adjunction:

∫f Set((U f) a, (U f) s)) ≅ (Φ C(a, -)) s

Φ = U ∘ F

The simplest application of this identity is when the functors in question are identity functors. We get:

∫f Set(f a, f s)) ≅ C(a, s)

forall f. Functor f => f a -> f s  ≅ a -> s

You may think of this formula as defining the trivial kind of optic that simply turns a to s.

## Profunctors

Profunctors are just functors from a product category Cop×D to Set. All the results from the last section can be directly applied to the profunctor category [Cop×D, Set]. Keep in mind that morphisms in this category are natural transformations between profunctors. Here’s the key formula:

∫p Set((U p)<a, b>, (U p)<s, t>)) ≅ (Φ (Cop×D)(<a, b>, -)) <s, t>

I have replaced a with a pair <a, b> and s with a pair <s, t>. The end is taken over all profunctors that exhibit some structure that U forgets, and F freely creates. Φ is the monad U ∘ F. It’s a monad that acts on profunctors to produce other profunctors.

Notice that a hom-set in the category Cop×D is a set of pairs of morphisms:

<f, g> :: (Cop×D)(<a, b>, <s, t>)
f :: s -> a
g :: b -> t

the first one going in the opposite direction.

The simplest application of this identity is when we don’t impose any constraints on the profunctors, in which case Φ is the identity monad. We get:

∫p Set(p <a, b>, p <s, t>) ≅ (Cop×D)(<a, b>, <s, t>)

Haskell translation of this formula gives the well-known representation of Iso:

forall p. Profunctor p => p a b -> p s t ≅ Iso s t a b

where:

data Iso s t a b = Iso (s -> a) (b -> t)

Interesting things happen when we impose more structure on our profunctors.

## Enriched Categories

First, let’s generalize profunctors to work on enriched categories. We start with some monoidal category V whose objects serve as hom-objects in an enriched category A. The category V will essentially replace Set in our constructions. For instance, we’ll work with profunctors that are enriched functors from the (enriched) product category to V:

p :: Aop ⊗ A -> V

Notice that we use a tensor product of categories. The objects in such a category are pairs of objects, and the hom-objects are tensor products of individual hom-objects. The definition of composition in a product category requires that the tensor product in V be symmetric (up to isomorphism).

For such profunctors, there is a suitable generalization of the end:

∫x p x x

It’s an object in V together with a V-natural family of projections:

pry :: ∫x p x x -> p y y

We can formulate the Yoneda lemma in an enriched setting by considering enriched functors from A to V. We get the following generalization:

∫x [A(a, x), f x] ≅ f a

Notice that A(a, x) is now an object of V — the hom-object from a to x. The notation [v, w] generalizes the internal hom. It is defined as the right adjoint to the tensor product in V:

V(x ⊗ v, w) ≅ V(x, [v, w])

We are assuming that V is closed, so the internal hom is defined for every pair of objects.

Enriched functors, or V-functors, between two enriched categories C and D form a functor category [C, D] that is itself enriched over V. The hom-object between two functors f and g is given by the end:

[C, D](f, g) = ∫x D(f x, g x)

We can therefore use the Yoneda lemma in a category of enriched functors, or in the category of enriched profunctors. Therefore the result of the previous section holds in the enriched setting as well:

∫p [(U p)<a, b>, (U p)<s, t>] ≅ (Φ (Aop⊗A)(<a, b>, -)) <s, t>

with the understanding that:

(Aop⊗A)(<a, b>, -))

is an enriched hom functor mapping pairs of objects in A to objects in V, plus the appropriate action on hom-objects. This hom-functor is the profunctor on which Φ acts.

## Tambara Modules

An enriched category A may have a monoidal structure of its own. We’ll use the same tensor product notation for its structure as we did for the underlying monoidal category V. There is also a tensorial unit object i in A.

A Tambara module is a V-functor p from Aop⊗A to V, which transforms under the tensor action of A according to a family of morphisms, natural in all three arguments:

α a x y :: p x y -> p (a ⊗ x) (a ⊗ y)

Notice that these are morphisms in the underlying category V, which is also the target of the profunctor.

We impose the usual unit law:

α i x y = id

and associativity:

α a⊗b x y = α a b⊗x b⊗y ∘ α b x y

Strictly speaking one can separately define left and right action but, for simplicity, we’ll assume that the product is symmetric (up to isomorphism).

The intuition behind Tambara modules is that some of the profunctor values are not independent of others. Once we have calculated p x y, we can obtain the value of p at any of the points on the path <a⊗x, a⊗y> by applying α.

Tambara modules form a category that’s enriched over V. The construction of this enrichment is non-trivial. The hom-object between two profunctors p and q in a category of profunctors is given by the end:

[Aop⊗A, V](p, q) = ∫<x y> V(p x y, q x y)

This object generalizes the set of natural transformations. Conceptually, not all natural transformation preserve the Tambara structure, so we have to define a subobject of this hom-object that does. The intuition is that the end is a generalized product of its components. It comes equipped with projections. For instance, the projection pr<x,y> picks the component:

V(p x y, q x y)

But there is also a projection pr<a⊗x, a⊗y> that picks:

V(p a⊗x a⊗y, q a⊗x a⊗y)

from the same end. These two objects are not completely independent, because they can both be transformed into the same object. We have:

V(id, αa) :: V(p x y, q x y) -> V(p x y, q a⊗x a⊗y)
V(αa, id) :: V(a⊗x a⊗y, q a⊗x a⊗y) -> V(p x y, q a⊗x a⊗y)

We are using the fact that the mapping:

<v, w> -> V(v, w)

is itself a profunctor Vop×V -> V, so it can be used to lift pairs of morphisms in V.

Now, given any triple a, x, and y, we want the two paths to be equivalent, which means finding the equalizer between each pair of morphisms:

V(id, αa) ∘ pr<x, y>
V(αa, id) ∘ pr<a⊗x, a⊗y>

Since we want our hom-object to satisfy the above condition for any triple, we have to construct it as an intersection of all those equalizers. Here, an intersection means an object of V together with a family of monomorphisms, each embedding it into a particular equalizer.

It’s possible to construct a forgetful functor from the Tambara category to the category of profunctors [Aop⊗A, V]. It forgets the existence of α and it maps hom-objects between the two categories. Composition in the Tambara category is defined is such a way as to be compatible with this forgetful functor.

The fact that Tambara modules form a category is important, because we want to be able to use the Yoneda lemma in that category.

## Tambara Optics

The key observation is that the forgetful functor from the Tambara category has a left adjoint, and that their composition forms a monad in the category of profunctors. We’ll plug this monad into our general formula.

The construction of this monad starts with a comonad that is given by the following end:

(Θ p) s t = ∫c p (c⊗s) (c⊗t)

For a given profunctor p, this comonad builds a new profunctor that is essentially a gigantic product of all values of this profunctor “shifted” by tensoring its arguments with all possible objects c.

The monad we are interested in is the left adjoint to this comonad (calculated using a Kan extension):

(Φ p) s t = ∫ c x y A(s, c⊗x) ⊗ A(c⊗y, t) ⊗ p x y

Notice that we have two separate tensor products in this formula: one in V, between the hom-objects and the profunctor, and one in A, under the hom-objects. This monad takes an arbitrary profunctor p and produces a new profunctor Φ p.

We can now use our earlier formula:

∫p [(U p)<a, b>, (U p)<s, t>)] ≅ (Φ (Aop⊗A)(<a, b>, -)) <s, t>

inside the Tambara category. To calculate the right hand side, let’s evaluate the action of Φ on the hom-profunctor:

(Φ (Aop⊗A)(<a, b>, -)) <s, t>
= ∫ c x y A(s, c⊗x) ⊗ A(c⊗y, t) ⊗ (Aop⊗A)(<a, b>, <x, y>)

We can “integrate over” x and y using the Yoneda lemma to get:

∫ c A(s, c⊗a) ⊗ A(c⊗b, t)

We get the following result:

∫p [(U p)<a, b>, (U p)<s, t>)] ≅ ∫ c A(s, c⊗a) ⊗ A(c⊗b, t)

where the end on the left is taken over all Tambara modules, and U is the forgetful functor from the Tambara category to the category of profunctors.

If the category in question is closed, we can use the adjunction:

A(c⊗b, t) ≅ A(c, [b, t])

and “perform the integration” over c to arrive at the set/get formulation:

∫ c A(s, c⊗a) ⊗ A(c, [b, t]) ≅ A(s, [b, t]⊗a)

It corresponds to the familiar Haskell lens type:

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

(This final trick doesn’t work for prisms, because there is no right adjoint to Either.)

A Tambara module is parameterized by the choice of the tensor product ten. We can write a general definition:

class (Profunctor p) => TamModule (ten :: * -> * -> *) p where
leftAction  :: p a b -> p (c ten a) (c ten b)
rightAction :: p a b -> p (a ten c) (b ten c)

This can be further specialized for two obvious monoidal structures: product and sum:

type TamProd p = TamModule (,) p
type TamSum p = TamModule Either p

The former is equivalent to what it called a Strong (or Cartesian) profunctor in Haskell, the latter is equivalent to a Choice (or Cocartesian) profunctor.

Replacing ends and coends with universal and existential quantifiers in Haskell, our main formula becomes (pseudocode):

forall p. TamModule ten p => p a b -> p s t
≅ exists c. (s -> c ten a, c ten b -> t)

The two sides of the isomorphism can be defined as the following data structures:

type TamOptic ten s t a b
= forall p. TamModule ten p => p a b -> p s t
data Optic ten s t a b
= forall c. Optic (s -> c ten a) (c ten b -> t)

Chosing product for the tensor, we recover two equivalent definitions of a lens:

type Lens s t a b = forall p. Strong p => p a b -> p s t
data Lens s t a b = forall c. Lens (s -> (c, a)) ((c, b) -> t)

Chosing the coproduct, we get:

type Prism s t a b = forall p. Choice p => p a b -> p s t
data Prism s t a b = forall c. Prism (s -> Either c a) (Either c b -> t)

These are the well-known existential representations of lenses and prisms.

The monad Φ (or, equivalently, the free functor that generates Tambara modules), is known in Haskell under the name Pastro for product, and Copastro for coproduct:

data Pastro p a b where
Pastro :: ((y, z) -> b) -> p x y -> (a -> (x, z))
-> Pastro p a b
data Copastro p a b where
Copastro :: (Either y z -> b) -> p x y -> (a -> Either x z)
-> Copastro p a b

They are the left adjoints of Tambara and Cotambara, respectively:

newtype Tambara p a b = Tambara forall c. p (a, c) (b, c)
newtype Cotambara p a b = Cotambara forall c. p (Either a c) (Either b c)

which are special cases of the comonad Θ.

## Discussion

It’s interesting that the work on Tambara modules has relevance to Haskell optics. It is, however, just one example of an even larger pattern.

The pattern is that we have a family of transformations in some category A. These transformations can be used to select a class of profunctors that have simple transformation laws. Using a tensor product in a monoidal category to transform objects, in essence “multiplying” them, is just one example of such symmetry. A more general pattern involves a family of transformations f that is closed under composition and includes a unit. We specify a transformation law for profunctors:

class Profunctor p => Related p where
α f a b :: forall f. Trans f => p a b -> p (f a) (f b)

This requirement picks a class of profunctors that we call Related.

Why are profunctors relevant as carriers of symmetry? It’s because they generalize a relationship between objects. The profunctor transformation law essentially says that if two objects a and b are related through p then so are the transformed objects; and that there is a function α that relates the proofs of this relationship. This is in the spirit of profunctors as proof-relevant relations.

As an analogy, imagine that we are comparing people, and the transformation we’re interested in is aging. We notice that family relationships remain invariant under aging: if a is a sibling of b, they will remain siblings as they age. This is not true about other relationships, for instance being a boss of another person. But family bonds are not the only ones that survive the test of time. Another such relation is being older or younger than the other person.

Now imagine that you pick four people at random points in time and you find out that any time-invariant relation between two of them, a and b, also holds between s and t. You have to conclude that there is some connection between s and age-adjusted a, and between age-adjusted b and t. In other words there exists a time shift that transforms one pair to another.

Considering all possible relations from the class Related corresponds to taking the end over all profunctors from this class:

type Optic p s t a b = forall p. Related p =>
p a b -> p s t

The end is a generalization of a product, so it’s enough that one of the components is empty for the whole end to be empty. It means that, for a particular choice of the four types a, b, s, and t, we have to be able to construct a whole family of morphisms, one for every p. We have seen that this end exists only if the four types are connected in a very peculiar way — for instance, if a and b are somehow embedded in s and t.

In the simplest case, we may choose the four types to be related by the transformation:

s = f a
t = f b

For these types, we know that the end exists:

forall p. Related p =>
p a b -> p s t

because there is a family of appropriate morphisms: our αf a b. In general, though, we can get away with weaker connection.

Let’s look at an example of a family of transformations generated by pairing with arbitrary type c:

fc a = (c, a)

Profunctors that respect these transformations are Tambara modules over a cartesian product (or, in lens parlance, Strong profunctors). For the choice:

s = (c, a)
t = (c, b)

the end in question trivially exists. As we’ve seen, we can weaken these conditions. It’s enough that one way (lax) transformations exist:

s -> (c, a)
t <- (c, b)

These morphisms assert that s can be split into a pair, and that t can be constructed from a pair (but not the other way around).

## Other Optics

With the understanding that optics may be defined using a family of transformations, we can analyze another optic called the Grate. It’s based on the following family:

type Reader e a = e -> a

Notice that, unlike the case of Tambara modules, this family is parameterized by a contravariant parameter e.

We are interested in profunctors that transform under these transformations:

class Profunctor p => Closed p where
closed :: p a b -> p (x -> a) (x -> b)

They let us form the optic:

type Grate s t a b = forall p. Closed p => p a b -> p s t

It turns out that there is a profunctor functor that freely generates Closed profunctors. We have the obvious comonad:

newtype Closure p a b = Closure forall x. p (x -> a) (x -> b)

data Environment p u v where
Environment :: ((c -> y) -> v) -> p x y -> (u -> (c -> x))
-> Environment p a b

or, in categorical notation:

(Φ p) u v = ∫ c x y A([c, y], v) ⊗ p x y ⊗ A(u, [c, x])

Using our construction, we apply this monad to the hom-profunctor:

(Φ (Aop⊗A)(<a, b>, -)) <s, t>
= ∫ c x y A([c, y], t) ⊗ (Aop⊗A)(<a, b>, <x, y>) ⊗ A(s, [c, x])
≅ ∫ c A([c, b], t) ⊗ A(s, [c, a])

Translating it back to Haskell, we get a representation of Grate as an existential type:

Grate s t a b = forall c. Grate ((c -> b) -> t) (s -> (c -> a))

This is very similar to the existential representation of a lens or a prism. It has the intuitive interpretation that s can be thought of as a container of a‘s indexed by some hidden type c.

We can also “perform the integration” using the Yoneda lemma, internal-hom-adjunction, and the symmetry of the product:

∫ c A([c, b], t) ⊗ A(s, [c, a])
≅ ∫ c A([c, b], t) ⊗ A(s ⊗ c, a)
≅ ∫ c A([c, b], t) ⊗ A(c, [s, a])
≅ A([[s, a], b], t)

to get the more familiar form:

Grate s t a b ≅ ((s -> a) -> b) -> t

## Conclusion

I find it fascinating that constructions that were first discovered in Haskell to make Haskell’s optics composable have their categorical counterparts. This was not at all obvious, if only because some of them use parametricity arguments. Parametricity is the property of the language, not easily translatable to category theory. Now we know that the profunctor formulation of isos, lenses, prisms, and grates follows from the Yoneda lemma. The work is not complete yet. I haven’t been able to derive the same formulation for traversals, which combine two different tensor products plus some monoidal constraints.

## Bibliography

1. Haskell lens library, Edward Kmett
2. Distributors on a tensor category, D. Tambara
3. Doubles for monoidal categories, Craig Pastro, Ross Street
4. Profunctor optics, Modular data accessors,
Matthew Pickering, Jeremy Gibbons, and Nicolas Wu
5. CPS based functional references, Twan van Laarhoven
6. Isomorphism lenses, Twan van Laarhoven
7. Theorem for Second-Order Functionals, Mauro Jaskellioff and Russell O’Connor

In the previous post I explored the application of the Yoneda lemma in the functor category to derive some results from the Haskell lens library. In particular I derived the profunctor representation of isos. There is one more trick that is used in the lens library: combining the Yoneda lemma with adjunctions. Jaskelioff and O’Connor used this trick in the context of free/forgetful adjunctions, but it can be easily generalized to any pair of adjoint higher order functors.

An adjunction between two functors, L and R (left and right functor) is a natural isomorphism between hom-sets:

C(L d, c) ≅ D(d, R c)

The left functor L goes from the category D to C, and the right functor R goes in the opposite direction. Formally, having an adjunction allows us to shift the action of the functor from one end of the hom-set to the other. The shortcut notation for an adjunction is L ⊣ R.

Since adjunctions can be defined for arbitrary categories, they will also work between functor categories. In that case objects are functors and hom-sets are sets of natural transformations. For instance, Let’s consider an adjunction between two higher order functors:

ρ :: [C, C'] -> [D, D']
λ :: [D, D'] -> [C, C']

Here, [C, C'] is a category of functors between two categories C and C’, [D, D'] is a category of functors between D and D’, and ρ maps functors (and natural transformations) between these two categories. λ goes in the opposite direction. The adjunction λ ⊣ ρ is expressed as a natural isomorphism between sets of natural transformations:

[C, C'](λ g, h)  ≅  [D, D'](g, ρ h)

The two objects in functor categories are themselves functors:

h :: C -> C'
g :: D -> D'

Here’s the same adjunction written using ends:

∫x∈C C'((λ g) x, h x)  ≅  ∫y∈D D'(g y, (ρ h) y)

The end notation is easily translatable to Haskell. The end corresponds to a universal quantifier forall, and hom-sets become function types:

forall x. (lambda g) x -> h x ≅ forall y. g y -> (rho h) y

Since lambda and rho act on functors, they have kinds (*->*)->(*->*).

Let’s recall the formula for the Yoneda embedding of the functor category:

∫f Set(∫x D(g x, f x), ∫y D(h y, f y))
≅ ∫z D(h z, g z)

Here, g, h, and f, are functors — objects in the functor category [C, D]. The ends represent natural transformations — morphisms in the functor category. The end over f is a higher order natural transformation.

Since g and h are arbitrary, let’s replace them with the results of the action of some higher order functors, λ g and λ' h. The idea is that λ and λ' are left halves of some higher order adjunctions.

∫f Set(∫x D'((λ g) x, f x), ∫y D'((λ' h) y, f y))
≅ ∫z D'((λ' h) z, (λ g) z)

The right halves of these adjunctions are, respectively, ρ and ρ'.

λ  ⊣ ρ
λ' ⊣ ρ'

Let’s apply these adjunctions inside the hom-sets:

∫f Set(∫x D(g x, (ρ f) x), ∫y D(h y, (ρ' f) y))
≅ ∫z D(h z, (ρ' (λ g)) z)

Let’s focus our attention on the category of sets. If we replace D with Set, we can pick g and h to be hom-functors (which are the simplest representable functors) parameterized by some arbitrary objects b and t:

g = C(b, -)
h = C(t, -)

We get:

∫f Set(∫x Set(C(b, x), (ρ f) x), ∫y Set(C(t, y), (ρ' f) y)
≅ ∫z Set(C(t, z), (ρ' (λ C(b, -))) z)

Remember, hom-functors behave like Dirac delta functions under the integration sign. That is to say, we can use the Yoneda lemma to “integrate” over x, y, and z:

∫f Set((ρ f) b, (ρ' f) t)
≅ (ρ' (λ C(b, -))) t

We are now free to pick a pair of adjoint higher order functors to suit our goal. Here’s one such choice for ρ: the functor that maps a functor f (an endofunctor in C) to a set of morphisms from some fixed object a to f acting on another object. This is an operation that lifts a functor to a profunctor. In Haskell it’s defined as UpStar. This higher-order functor is parameterized by the choice of the object a in C:

κa f = C(a, f -)

It can also be written in terms of the exponential object:

κa f = (f -)a

This functor has an obvious left adjoint:

λa g = a × g -

This follows from the standard adjunction between the product and the exponential.

Our pick for ρ' is the same functor but taken at a different carrier, s:

ρ' = κs

With those choices, the left side of the identity

∫f Set((ρ f) b, (ρ' f) t)
≅ (ρ' (λ C(b, -))) t

becomes:

∫f Set(C(a, f b), C(s, f t))

This is the categorical version of the van Laarhoven lens.

Let’s now evaluate the right hand side. First we apply λa to the hom-functor C(b, -) to get:

λa C(b, -) = a × C(b, -)

The action of ρ' produces the result:

C(s, (a × C(b, t)))

This, in turn, is the categorical version of the getter/setter representation of the lens.

## Translation

In Haskell, our formula derived from the higher-order Yoneda lemma with the adjoint pair:

∫f Set((ρ f) b, (ρ' f) t)
≅ (ρ' (λ C(b, -))) t

takes the form:

forall f. Functor f => (rho f) b -> (rho' f) t
≅ (rho' (lambda ((->)b))) t

With our choice for ρ as the up-star functor:

rho  f = a -> f -
rho' f = s -> f -

type Rho  a f b = a -> f b
type Rho' s f t = s -> f t

we get:

forall f. Functor f => (a -> f b) -> (s -> f t)
≅ (rho' (lambda ((->)b))) t

To get the λ, we plug our ρ into the adjunction formula. We get:

forall x. (lambda g) x -> h x ≅ forall x. g x -> a -> h x

which has the obvious solution:

lambda g = (a, g -)

type Lambda a g x = (a, g x)

Indeed, with the currying and flipping of arguments, we get the adjunction:

forall x. (a, g x) -> h x ≅ forall x. g x -> a -> h x

Now let’s evaluate the right hand side:

(rho' (lambda ((->) b))) t

lambda (b -> -) = (a, b -> -)

The action of rho' gives us:

rho' (a, b -> -) = s -> (a, b -> -)

Altogether:

(rho' (lambda ((->) b))) t = s -> (a, b -> t)

So the right hand side is just the getter/setter pair:

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

The final result is the well known van Laarhoven representation of the lens:

forall f. Functor f => (a -> f b) -> (s -> f t)
≅ (s -> a, s -> b -> t)

This is not a new result, but I like the elegance of this derivation — especially the role played by the exponential adjunction and the lifting of a functor to a profunctor. This formulation has the additional advantage of being generalizable towards the profunctor formulation of lenses.

The connection between the Haskell lens library and category theory is a constant source of amazement to me. The most interesting part is that lenses are formulated in terms of higher order functions that are polymorphic in functors (or, more generally, profunctors). Consider, for instance, this definition:

type Lens s t a b = forall f. Functor f => (a -> f b) -> (s -> f t)

In Haskell, saying that a function is polymorphic in functors, which form a class parameterized by type constructors of the kind *->* (or *->*->*, in the case of profunctors) and supporting a special method called fmap (or dimap, respectively) is rather mind-boggling.

In category theory, on the other hand, functors are standard fare. You can form categories of functors. The properties of such categories are described by pretty much the same machinery as those of any other category.

In particular, one of the most important theorems of category theory, the Yoneda lemma, works in the category of functors out of the box. I have previously shown how to employ the Yoneda lemma to derive the representation for Haskell lenses (see my original blog post and, independently, this paper by Jaskelioff and O’Connor — or a more recent expanded post). Continuing with this program, I’m going to show how to use the Yoneda lemma with profunctors. But let’s start with the basics.

By the way, if you feel intimidated by mathematical notation, don’t worry, I have provided a translation to Haskell. However, math notation is often more succinct and almost always more general. I guess, the same ideas could be expressed using C++ templates, but it would look like an incomprehensible mess.

## Functor Categories

Functors between any two given categories C and D can themselves be organized into a category, which is often called [C, D] or DC. The objects in that category are functors, and the morphisms are natural transformations. Given two functors f and g, the hom-set between them can be either called

Nat(f, g)

or

[C, D](f, g)

depending how much information you want to expose. (For simplicity, I’ll assume that the categories are small, so that the “sets” or natural transformations are sets indeed.)

What’s interesting is that, since functor categories are just categories, we can have functors going between them. We sometimes call them higher order functors. We can also have higher order functors going from a functor category to a regular category, in particular to the category of sets, Set. An example of such a functor is a hom-functor in a functor category. You construct this functor (also called a representable functor) when you fix one end of the hom-set and vary the other. In any category, the mapping:

x -> C(a, x)

is a functor from C to Set. We often use a shorthand notation for this functor:

C(a, -)

If we replace C by a functor category then, for a fixed functor g, the mapping:

f -> [C, D](g, f)

is a higher order functor. It maps f to a set of natural transformations — itself an object in Set.

Representable functors play an important role in the Yoneda lemma. Take the set of natural transformations from a representable functor in C to any functor f that goes from C to Set. This set is in one-to-one correspondence with the set of values of this functor at the object a:

[C, Set](C(a, -), f) ≅ f a

This correspondence is an isomorphism, which is natural both in a and f.

The set of natural transformations between two functors f and g can also be expressed as an end:

[C, D](f, g) = ∫x∈C D(f x, g x)

The end notation is sometimes more convenient because it makes the object x (the “integration variable”) explicit. The Yoneda lemma, in this notation, becomes:

∫x∈C Set(C(a, x), f x) ≅ f a

If you’re familiar with distributions, this formula will immediately resonate with you — it looks like the definition of the Dirac delta function:

∫ dx δ(a - x) f(x) ≅ f(a)

We can apply the Yoneda lemma to a functor category to get:

Nat([C, D](g, -), φ) ≅ φ g

or, in the end notation,

∫f Set(∫x D(g x, f x), φ f) ≅ φ g

Here, the “integration variable” f is itself a functor from C to D, and so is g; φ, however, is a higher order functor. It maps functors from [C, D] to sets from Set. The natural transformations in this formula are higher order natural transformations between higher order functors.

Furthermore, if we substitute for φ another instance of the representable functor, [C, D](h, -), we get the formula for the higher order Yoneda embedding:

Nat([C, D](g, -), [C, D](h, -)) ≅ [C, D](h, g)

which reduces higher order natural transformations to lower order natural transformations. Notice the inversion of g and h on the right hand side.

Using the end notation, this becomes:

∫f Set(∫x D(g x, f x), ∫y D(h y, f y))
≅ ∫z D(h z, g z)

We can further specialize this formula by replacing D with Set. We can then choose both functors to be hom-functors (for some fixed a and b):

g = C(a, -)
h = C(b, -)

We get:

∫f Set(∫x Set(C(a, x), f x), ∫y Set(C(b, y), f y))
≅ ∫z Set(C(b, z), C(a, z))

This can be simplified by applying the Yoneda lemma to the internal ends (“integrating” over x, y, and z) to get:

∫f Set(f a, f b) ≅ C(a, b)

This simple formula has some interesting possibilities that I will explore later.

## Translation

All this might be easier to digest for programmers when translated to Haskell. Natural transformations are polymorphic functions:

forall x. f x -> g x

Here, f and g are arbitrary Haskell Functors. It’s a straightforward translation of the end formula:

∫x∈Set Set(f x, g x)

where the end is replaced by the universal quantifier, and the hom-set in Set by a function type. I have deliberately used Set rather than Hask as the category of Haskell types, because I’m not going to pretend that I care about non-termination.

A higher order functor of the kind we are interested in is a mapping from functors to types, which could be defined as follows:

class HFunctor (phi :: (* -> *) -> *) where
hfmap :: (forall a. f a -> g a) -> (phi f -> phi g)

The higher order hom-functor is defined as:

newtype HHom f g = HHom (forall a. f a -> g a)

Indeed, it’s easy to define hfmap for it:

instance HFunctor (HHom f) where
hfmap nat (HHom nat') = HHom (nat . nat')

The types give it away:

nat    :: forall a. g a -> h a
nat'   :: forall a. f a -> g a
result :: HHom (forall a. f a -> h a)

Higher order natural transformations between such functors will have the signature:

type HNat (phi :: (* -> *) -> *) (psi :: (* -> *) -> *) =
forall f. Functor f => phi f -> psi f

The standard Yoneda lemma establishes the isomorphism between f a and the following higher order polymorphic function:

forall x. (a -> x) -> f x  ≅  f a

The Yoneda lemma for higher order functors is the equivalence between φ g and:

forall f. Functor f => forall x. (g x -> f x) -> φ f  ≅  φ g

Compare this again with:

∫f Set(∫x Set(g x, f x), φ f) ≅ φ g

The higher order Yoneda embedding takes the form of the equivalence between:

forall f. Functor f => forall x. (g x -> f x) -> forall y. (h y -> f y)

and

forall z. h z -> g z

The earlier result of the double application of the Yoneda lemma:

∫f Set(f a, f b) ≅ C(a, b)

translates to:

forall f. Functor f => f a -> f b ≅ a -> b

One direction of this equivalence simply reiterates the definition of a functor: a function a->b can be lifted to any functor. The other direction is a little more interesting. Given two types, a and b, if there is a function from f a to f b for any functor f, than there is a direct function from a to b. In Set, where there are functions between any two types, with the exception of a->Void, this is not a big surprise.

But there are other categories embedded in Set, and the same categorical formula will lead to more interesting translations. In particular, think of categories where the hom-set is not equivalent to a simple function type with trivial composition. A good example is the basic formulation of lens as the getter/setter pair, or a function of type:

type Lens s t a b = s -> (a, b -> t)

Such functions don’t compose naturally, but their functor-polymorphic representations do.

## Profunctors

You’ve seen the reusability of categorical constructs in action. We can have functors operate on functors, and natural transformations that work between higher order functors. The same Yoneda lemma works as well in the category of types and functions, as in the category of functors and natural transformations. From that perspective, a profunctor is just a special case of a functor. Given two categories C and D, a profunctor is a functor:

Cop × D -> Set

It’s a map from a product category to Set. Because the first component of the product is the opposite category (all morphisms reversed), this functor is contravariant in the first argument.

Let’s translate this definition to Haskell. We substitute all three categories with the same category of types and functions, which is essentially Set (remember, we ignore the bottom values). So a profunctor is a functor from Setop×Set to Set. It’s a mapping of types — a two-argument type constructor p  — and a mapping of morphisms. A morphism in Setop×Set is a pair of functions going between pairs (a, b) and (s, t). Because of contravariance, the first function goes in the opposite direction:

(s -> a, b -> t)

A profunctor lifts this pair to a single function:

p a b -> p s t

The lifting is done by the function dimap, which is usually written in the curried form:

class Profunctor p where
dimap :: (s -> a) -> (b -> t) -> p a b -> p s t

All said and done, a profunctor is still a functor, so we can reuse all the machinery of functor calculus, including all versions of the Yoneda lemma.

[Cop×D, Set]((Cop×D)(<c, d>, -), p) ≅ p <c, d>

or, in the end notation:

∫<x, y>∈Cop×D Set((Cop×D)(<c, d>, <x, y>), p <x, y>) ≅ p <c, d>

Here, p is the profunctor operating on pairs of objects, such as <c, d>. A hom-set in the product category Cop×D goes between two such pairs:

(Cop×D)(<c, d>, <x, y>)

Here’s the straightforward translation to Haskell:

forall x y. (x -> c) -> (d -> y) -> p x y ≅ p c d

Notice the customary currying and the reversal of source with target in the first function argument due to contravariance.

Since profunctors are just functors, they form a functor category:

[Cop×D, Set]

(not to be confused with Prof, the profunctor category, where profunctors serve as morphisms rather than objects):

We can easily rewrite the higher-order Yoneda lemma replacing functors with profunctors:

∫p Set(∫<x, y> Set(q <x, y>, p <x, y>), π p) ≅ π q

And this is what it looks like in Haskell:

forall p. Profunctor p => (forall x y. q x y -> p x y) -> pi p ≅ pi q

Here, π is a higher order functor acting on profunctors, with values in Set. In Haskell it’s defined by a type class:

class HFunProf (pi :: (* -> * -> *) -> *) where
fhpmap :: (forall a b. p a b -> q a b) -> (pi p -> pi q)

Natural transformations between such functors have the type:

type HNatProf (pi :: (* -> * -> *) -> *) (rho :: (* -> * -> *) -> *) =
forall p. Profunctor p => pi p -> rho p

Notice that we are now defining functions that are polymorphic in profunctors. This is getting us closer to the profunctor formulation of the lens library, in particular to prisms and isos.

## Understanding Isos

An iso is a perfect example of a data structure straddling the gap between lenses and prisms. Its first order definition is simple:

type Iso s t a b = (s -> a, b -> t)

The name derives from isomorphism, which is a special case of an iso (I think a cuter name for an iso would be Mirror). The crucial observation is that this is nothing but the type corresponding to a hom-set in the product category Setop×Set:

(Setop×Set)(<a b>, <s t>)

We know how to compose such morphisms:

compIso :: Iso s t a b -> Iso a b u v -> Iso s t u v
(f1, g1) compIso (f2, g2) = (f2 . f1, g1 . g2)

but it’s not as straightforward as function composition. Fortunately, there is a higher order representation of isos, which composes using simple function composition. The trick is to make it profunctor-polymorphic:

type Iso s t a b = forall p. Profunctor p => p a b -> p s t

Why are the two definitions isomorphic? There is a standard argument based on parametricity, which I will skip, because there is a better explanation.

Recall the earlier result of applying the Yoneda lemma to the functor category:

forall f. Functor f => f a -> f b ≅ a -> b

The similarity is striking, isn’t it? That’s because, the categorical formula for both identities is the same:

∫f Set(f a, f b) ≅ C(a, b)

All we need is to replace C with Cop×D and rewrite it in terms of pairs of objects:

∫p Set(p <a b>, p <s t>) ≅ (Cop×D)(<a b>, <s t>)

But that’s exactly what we need:

forall p. Profunctor p => p a b -> p s t  ≅ (s -> a, b -> t)

The immediate advantage of the profunctor-polymorphic representation is that you can compose two isos using straightforward function composition. Instead of using compIso, we can use the dot:

p :: Iso s t a b
q :: Iso a b u v
r :: Iso s t u v
r = p . q

Of course, the full power of lenses is in the ability to compose (and type-check) combinations of different elements of the library.

Note: The definition of Iso in the lens library involves a functor f:

type Iso s t a b = forall p f. (Profunctor p, Functor f) =>
p a (f b) -> p s (f t)

This functor can be absorbed into the definition of the profunctor p without any loss of generality.

Next: Combining adjunctions with the Yoneda lemma.

## Acknowledgments

I’m grateful to Gershom Bazerman and Gabor Greif for useful comments and to André van Meulebrouck for checking the grammar and spelling.

Lenses are a fascinating subject. Edward Kmett’s lens library is an indispensable tool in every Haskell programmer’s toolbox. I set out to write this blog post with the goal of describing some new insights into their categorical interpretation, but then I started reviewing all the different formulations of lenses and their relations to each other. So this post turned into a little summary of the theoretical underpinning of lenses.

If you’re already familiar with lenses, you may skip directly to the last section, which describes some new results.

# The Data-Centric Picture

Lenses start as a very simple idea: an accessor/mutator or getter/setter pair — something familiar to every C++, Java, or C# programmer. In Haskell they can be described as two functions:

get :: s -> a
set :: s -> a -> s

Given an object of type s, the first function produces the value of the object’s sub-component of type a — which is the focus of the particular lens. The second function takes the object together with a new value for the sub-component, and produces a modified object.

A simple example is a lens that focuses on the first component of a pair:

get :: (a, a') -> a
get (x, y) = x
set :: (a, a') -> a -> (a, a')
set (x, y) x' = (x', y)

In Haskell we can go even further and define a lens for a polymorphic object, in which the setter changes not only the value of the sub-component, but also its type. This will, of course, also change the type of the resulting object. So, in general we have:

get :: s -> a
set :: s -> b -> t

Our pair example doesn’t change much on the surface:

get :: (a, a') -> a
get (x, y) = x
set :: (a, a') -> b -> (b, a')
set (x, y) x' = (x', y)

The difference is that the type of x' can now be different from the type of x. The first component of the pair is of type a before the update, and of type b (the same as that of x') after it. This way we can turn a pair of, say, (Int, Bool) to a pair of (String, Bool).

We can always go back to the monomorphic version of the lens by choosing b equal to a and t equal to s.

Not every pair of functions like these constitutes a lens. A lens has to obey a few laws (first formulated by Pierce in the database context). In particular, if you get a component after you set it, you should get back the value you just put in:

get . set s = id

If you call set with the value you obtained from get, you should get back the unchanged object:

set s (get s) = s

Finally, a set should overwrite the result of a previous set:

set (set s a) b = set s b

So this is what I would call a “classical” lens. It’s formulated in easy to understand programming terms.

# The Algebraic Picture

It was Russell O’Connor who noticed that when you refactor the common element from the getter/setter pair, you get an interesting algebraic structure. Instead of writing the lens as two functions, we can write it as one function returning a pair:

s -> (a, b -> t)

Think of this as factoring out the “this” pointer in OO and returning the interface. Of course, the difference is that, in functional programming, the setter does not mutate the original — it returns a new version of the object instead.

Let’s for a moment concentrate on the monomorphic version of the lens — one in which a is the same as b, and s is the same as t. We can define a data structure:

data Store a s = Store a (a -> s)

and rewrite the lens as:

s -> Store a s

For instance, our first-component-of-a-pair lens will take the form:

fstl :: (a, b) -> Store a (a, b)
fstl (x, y) = Store x (\x' -> (x', y))

The first observation is that, for any a, Store a is a functor:

instance Functor (Store a) where
fmap f (Store x h) = Store x (f . h)

It’s a well-known fact that you can define algebras for a functor, so-called F-algebras. An algebra for a functor f consists of a type s called the carrier type and a function called the action:

alg :: f s -> s

This is almost like a lens (with f replaced by Store a), except that the arrow goes the wrong way. Not to worry: there is a dual notion called a coalgebra. It consists of a carrier type s and a function:

coalg :: s -> f s

Substitute Store a for f and you see that a lens is nothing but a coalgebra for this functor. This is not saying much — we have just given a mathematical name to a programming construct, no big deal. Except that Store a is more than a functor — it’s also a comonad.

You know that you can define a monad in Haskell using return and join. Reverse the arrows on those two, and you get extract and duplicate, the two functions that define a comonad.

class (Functor w) => Comonad w where
extract :: w a -> a
duplicate :: w a -> w (w a)

Here, w is a type constructor that is also a functor.

This is how you can implement those two functions for Store a:

instance Comonad (Store a) where
-- extract :: Store a s -> s
extract (Store x h) = h x
-- duplicate :: Store a s -> Store a (Store a s)
duplicate (Store x h) = Store x (\y -> Store y h)

So now we have two structures: a comonad w and a coalgebra:

type Coalgebra w s = s -> w s

A lens is a special case of this coalgebra where w is Store a.

Every time you have two structures, you may legitimately ask the question: Are they compatible? Just by looking at types, you may figure out some obvious compatibility conditions. In particular, since they go the opposite way, it would make sense for extract to undo the action of coalg:

coalg :: Coalgebra w s
extract . coalg = id

Also, duplicating the result of coalg should be the same as applying coalg twice (the second time lifted by fmap):

fmap coalg . coalg = duplicate . coalg

If these two conditions are satisfied, we call coalg a comonad coalgebra.

And here’s the clencher:

These two conditions when applied to the Store a comonad are equivalent to our earlier lens laws.

Let’s see how it works. First we’ll express the result of our lens coalgebra acting on some object s in terms of get and set (curried set s is a function a->s):

coalg s = Store (get s) (set s)

The first condition extract . coalg = id immediately gives us the law:

set s (get s) = s

When we act with duplicate on coalg s, we get:

Store (get s) (\y -> Store y (set s))

On the other hand, when we fmap our coalg over coalg s, we get:

Store (get s) ((\s' -> Store (get s') (set s')) . set s)

Two functions — the second components of the Store objects in those equations — must be equal when acting on any a. The first function produces:

Store a (set s)

In the second one, we first apply set s to a to get set s a, which we then pass to the lambda to get:

Store (get (set s a)) (set (set s a))

This reproduces the other two (monomorphic) lens laws:

get (set s a) = a

and

set (set s a) = set s

This algebraic construction can be extended to type-changing lenses by replacing Store with its indexed version:

data IStore a b t = IStore a (b -> t)

and the comonad with its indexed counterpart. The indexed store is also called Context in the lens parlance, and an indexed comonad is also called a parametrized comonad.

So what’s an indexed comonad? Let’s start with an indexed functor. It’s a type constructor that takes three types, a, b, and s, and is a functor in the third argument:

class IxFunctor f where
imap :: (s -> t) -> f a b s -> f a b t

IStore is obviously an indexed functor:

instance IxFunctor IStore where
-- imap :: (s -> t) -> IStore a b s -> IStore a b t
imap f (IStore x h) = IStore x (f . h)

An indexed comonad has the indexed versions of extract and duplicate:

class IxComonad w where
iextract :: w a a t -> t
iduplicate :: w a b t -> w a j (w j b t)

Notice that iextract is “diagonal” in the index types, whereas the double application of w shares one index, j, between the two applications. This plays very well with the unit and multiplication interpretation of a monad — here it looks just like matrix multiplication (although we are dealing with a comonad rather than a monad).

It’s easy to see that the instantiation of the indexed comonad for IStore works the same way as the instantiation of the comonad for Store. The types just work out that way.

instance IxComonad IStore where
-- iextract :: IStore a a t -> t
iextract (IStore a h) = h a
-- iduplicate :: IStore a b t -> IStore a c (IStore c b t)
iduplicate (IStore a h) = IStore a (\c -> IStore c h)

There is also an indexed version of a comonad coalgebra, where the coalgebra is replaced by a family of mappings from some carrier type s to w a b t; with the type t determined by s together with the choice of of the indexes a and b:

type ICoalg w s t a b = s -> w a b t

The compatibility conditions that make it an (indexed) comonad coalgebra are almost identical to the standard compatibility conditions, except that we have to be careful about the index types. Here’s the first condition:

icoalg_aa :: ICoalg w s t a a
iextract . icoalg_aa = id

Let’s analyze the types. The inner part has the type:

icoalg_aa :: s -> w a a t

We apply iextract to it, which has the type:

iextract :: w a a t -> t

and get:

iextract . icoalg_aa :: s -> t

The right hand side of the condition has the type:

id :: s -> s

It follows that, for the diagonal components of ICoalg w s t, t must be equal to s. The diagonal part of ICoalg w s t is therefore a family of regular coalgebras.

As we have done with the monomorphic lens, we can express (ICoalg IStore s t a b), when acting on s, in terms of get and set:

icoalg_ab s = IStore (get s) (set s)

But now get s is of type a, while set s if of the type b -> t. We can still apply extract to the diagonal term IStore a a t as required by the first compatibility condition. When equating the result to id, we recover the lens law:

set s (get s) = s

Similarly, it’s straightforward to see that the second compatibility condition:

icoalg_bc :: ICoalg w s t b c
icoalg_ab :: ICoalg w s t a b
icoalg_ac :: ICoalg w s t a c
imap icoalg_bc . icoalg_ab = iduplicate . icoalg_ac

is equivalent to the other two lens laws.

# The Parametric Picture

Despite being theoretically attractive, standard lenses were awkward to use and, in particular, to compose. The breakthrough came when Twan van Laarhoven realized that there is a higher-order representation for them that has very nice compositional properties. Composing lenses to focus on sub-objects of sub-objects turned into simple function composition.

Here’s Twan’s representation (generalized by Russell for the polymorphic case):

type Lens s t a b = forall f. Functor f => (a -> f b) -> (s -> f t)

So a lens is a polymorphic higher order function with a twist. The twist is that it’s polymorphic with respect to a functor rather than a type.

You can think of it this way: the caller provides a function to modify a particular field of s, turning it from type a to f b. What the caller gets back is a function that transforms the whole of s to f t. The idea is that the lens knows how to reconstruct the object, while putting it under a functor f — if you tell it how to modify a field, also under this functor.

For instance, continuing with our example, here’s the van Laarhoven lens that focuses on the first component of a pair:

vL :: Lens (a, c) (b, c) a b
vL h (x, x') = fmap (\y -> (y, x')) (h x)

Here, s is (a, c) and t is (b, c).

To see that the van Laarhoven representation is equivalent to the get/set one, let’s first change the order of arguments and pull s outside of the forall quantifier:

Lens s t a b = s -> (forall f. Functor f => (a -> f b) -> f t)

Here’s how you can read this definition: For a given s, if you give me a function from a to f b, I will produce a value of type f t. And I don’t care what functor you use!

What does it mean not to care about the functor? It means that the lens must be parametrically polymorphic in f. It can’t do case analysis on a functor. It must be implemented using the same formula for f being the list functor, or the Maybe functor, or the Const functor, etc. There’s only one thing all these functors have in common, and that’s the fmap function; so that’s what we are allowed to use in the implementation of the lens.

Now let’s think what we can do with a function a -> f b that we were given. There’s only one thing: apply it to some value of type a. So we must have access to a value of type a. The result of this application is some value of type f b, but we need to produce a value of the type f t. The only way to do it is to have a function of type b->t and sneak it under the functor using fmap (so here’s where the generic functor comes in). We conclude that the implementation of the function:

forall f. Functor f => (a -> f b) -> f t

must be hiding a value of type a and a function b->t. But that’s exactly the contents of IStore a b t. Parametricity tells us that there is an IStore hiding inside the van Laarhoven lens. The lens is equivalent to:

s -> IStore a b t

In fact, with a clever choice of functors we can recover both get and set from the van Laarhoven representation.

First we select our functor to be Const a. Note that the parameter a is not the one over which the functor is defined. Const a takes a second parameter b over which it is functorial. And, even though it takes b as a type parameter, it doesn’t use it at all. Instead, like a magician, it palms an a, and then reveals it at the end of the trick.

newtype Const a b = Const { getConst :: a }

instance Functor (Const a) where
-- fmap :: (s -> t) -> Const a s -> Const a t
fmap _ (Const a) = (Const a)

The constructor Const happens to be a function

Const :: a -> Const a b

which has the required form to be the first argument to the lens:

a -> f b

When we apply the lens, let’s call it vL, to the function Const, we get another function:

vL Const :: s -> Const a t

We can apply this function to s, and then, in the final reveal, retrieve the value of a that was smuggled inside Const a:

get vL s = getConst $vL Const s Similarly, we can recover set from the van Laarhoven lens using the Identity functor: newtype Identity a = Identity { runIdentity :: a } We define: set vL s x = runIdentity$ vL (Identity . const x) s

The beauty of the van Laarhoven representation is that it composes lenses using simple function composition. A lens takes a function and returns a function. This function can, in turn, be passed as the argument to another lens, and so on.

There’s an interesting twist to this kind of composition — the function composition operator in Haskell is the dot, just like the field accessor in OO languages like Java or C++; and the composition follows the same order as the composition of accessors in those languages. This was first observed by Conal Elliott in the context of semantic editor combinators.

Consider a lens that focuses on the a field inside some object s. It’s type is:

Lens s t a b

When given a function:

h :: a -> f b

it returns a function:

h' :: s -> f t

Now consider another lens that focuses on the s field inside some even bigger object u. It’s type is:

Lens u w s t

It expects a function of the type:

g :: s -> f t

We can pass the result of the first lens directly to the second lens to form a composite:

Lens u w s t . Lens s t a b

We get a lens that focuses on the a field of the object s that is the sub-object of the big object u. It works just like in Java, where you apply a dot to the result of a getter or a setter, to dig deeper into a subobject.

Not only do lenses compose using regular function composition, but we can also use the identity function as the identity lens. So lenses form a category. It’s time to have a serious look at category theory. Warning: Heavy math ahead!

# The Categorical Picture

I used parametricity arguments to justify the choice of the van Laarhoven representation for the lens. The lens function is supposed to have the same form for all functors f. Parametricity arguments have an operational feel to them, which is okay, but I feel like a solid categorical justification is more valuable than any symbol-shuffling argument. So I worked on it, and eventually came up with a derivation of the van Laarhoven representation using the Yoneda lemma. Apparently Russell O’Connor and Mauro Jaskelioff had similar feelings because they came up with the same result independently. We used the same approach, going through the Store functor and applying the Yoneda lemma twice, once in the functor category, and once in the Set category (see the Bibliography).

I would like to present the same result in a more general setting of the Yoneda embedding. It’s a direct consequence of the Yoneda lemma, and it states that any category can be embedded (fully and faithfully) in the category of functors from that category to Set.

Here’s how it works: Let’s fix some object a in some category C. For any object x in that category there is a hom-set C(a, x) of morphisms from a to x. A hom-set is a set — an object in the category Set of sets. So we have defined a mapping from C to Set that takes an x and maps it to the set C(a, x). This mapping is called C(a, _), with the underscore serving as a placeholder for the argument.

It’s easy to convince yourself that this mapping is in fact a functor from C to Set. Indeed, take any morphism f from x to y. We want to map this morphism to a function (a morphism in Set) that goes between C(a, x) and C(a, y). Let’s define this lifted function component-wise: given any element h from C(a, x) we can map it to f . h. It’s just a composition of two morphisms from C. The resulting morphism is a member of C(a, y). We have lifted a morphism f from C to Set thus establishing that C(a, _) is a functor.

Now consider two such functors, C(a, _) and C(b, _). The Yoneda embedding theorem tells us that there is a one-to-one correspondence between the set of natural transformations between these two functors and the hom-set C(b, a).

Nat(C(a, _), C(b, _)) ≅ C(b, a)

Notice the reversed order of a and b on the right-hand side.

Let’s rephrase what we have just seen. For every a in C, we can define a functor C(a, _) from C to Set. Such a functor is a member of the functor category Fun(C, Set). So we have a mapping from C to the functor category Fun(C, Set). Is this mapping a functor?

We have just seen that there is a mapping between morphisms in C and natural transformations in Fun(C, Set) — that’s the gist of the Yoneda embedding. But natural transformations are morphisms in the functor category. So we do have a functor from C to the functor category Fun(C, Set). It maps objects to objects and morphisms to morphisms. It’s a contravariant functor, because of the reversal of a and b. Moreover, it maps the hom-sets in the two categories one-to-one, so it’s a fully faithful functor, and therefore it defines an embedding of categories. Every category C can be embedded in the functor category Fun(C, Set). That’s called the Yoneda embedding.

There’s an interesting consequence of the Yoneda embedding: Every functor category can be embedded in its own functor category — just replace C with a functor category in the Yoneda embedding. Recall that functors between any two categories form a category. It’s a category in which objects are functors and morphisms are natural transformations. Yoneda embedding works for that category too, which means that a functor category can be embedded in a category of functors from that functor category to Set.

Let’s see what that means. We can fix one functor, say R and consider the hom-set from R to some arbitrary functor f. Since we are in a functor category, this hom-set is a set of natural transformations between the two functors, Nat(R, f).

Now let’s pick another functor S. It also defines a set of natural transformations Nat(S, f). We can keep picking functors and mapping them to sets (sets of natural transformations). In fact we know from the previous argument that this mapping is itself a functor. This time it’s a functor from a functor category to Set.

What does the Yoneda embedding tell us about any two such functors? That the set of natural transformations between them is isomorphic to the (reversed) hom-set. But this time hom-sets are sets of natural transformations. So we have:

Nat(Nat(R, _), Nat(S, _)) ≅ Nat(S, R)

All natural transformations in this formula are regular natural transformation except for the outer one, which is more interesting. You may recall that a natural transformation is a family of morphisms parameterized by objects. But in this case objects are functors, and morphisms are themselves natural transformations. So it’s a family of natural transformations parameterized by functors. Keep this in mind as we proceed.

To get a better feel of what’s happening, let’s translate this to Haskell. In Haskell we represent natural transformations as polymorphic functions. This makes sense, since a natural transformation is a family of morphisms (here functions) parameterized by objects (here types). So a member of Nat(R, f) can be represented as:

forall x. R x -> f x

Similarly, the second natural transformation in our formula turns into:

forall y. S y -> f y

As I said, the outer natural transformation in the Yoneda embedding is a family of natural transformations parameterized by a functor, so we get:

forall f. Functor f => (forall x. R x -> f x) -> (forall y. S y -> f y)

You can already see one element of the van Laarhoven representation: the quantification over a functor.

The right hand side of the Yoneda embedding is a natural transformation:

forall z. S z -> R z

The next step is to pick the appropriate functors for R and S. We’ll take R to be IStore a b and S to be IStore s t.

Let’s work on the first part:

forall x. IStore a b x -> f x

A function from IStore a b x is equivalent to a function of two arguments, one of them of type a and another of type b->x:

forall x. a -> (b -> x) -> f x

We can pull a out of forall to get:

a -> (forall x. (b -> x) -> f x)

If you squint a little, you recognize that the thing in parentheses is a natural transformation between the functor C(b, _) and f, where C is the category of Haskell types. We can now apply the Yoneda lemma, which says that this set of natural transformations is isomorphic to the set f b:

forall x. (b -> x) -> f x ≅ f b

We can apply the same transformation to the second part of our identity:

forall y. (IStore s t y -> f y) ≅ s -> f t

Taking it all together, we get:

forall f. Functor f => (a -> f b) -> (s -> f t)
≅ forall z. IStore s t z -> IStore a b z

Let’s now work on the right hand side:

forall z. IStore s t z -> IStore a b z
≅ forall z. s -> (t -> z) -> IStore a b z

Again, pulling s out of forall and applying the Yoneda lemma, we get:

s -> IStore a b t

But that’s just the standard representation of the lens:

s -> IStore a b t ≅ (s -> a, s -> b -> t) = (get, set)

Thus the Yoneda embedding of the functor category leads to the van Laarhoven representation of the lens:

forall f. Functor f => (a -> f b) -> (s -> ft)
≅ (s -> a, s -> b -> t)

This is all very satisfying, but you may wonder what’s so special about the IStore functor? The crucial step in the derivation of the van Laarhoven representation was the application of the Yoneda lemma to get this identity:

forall x. IStore a b x -> f x ≅ a -> f b

Let’s rewrite it in the more categorical language:

Nat(IStore a b, f) ≅ C(a, f b)

The set of natural transformations from the functor IStore a b to the functor f is isomorphic to the hom-set between a and f b. Any time you see an isomorphism of hom-sets (and remember that Nat is the hom-set in the functor category), you should be on the lookout for an adjunction. And indeed, we have an adjunction between two functors. One functor is defined as:

a -> IStore a b

It takes an object a in C and maps it to a functor IStore a b parameterized by some other object b. The other functor is:

f -> f b

It maps a functor, an object in the functor category, to an object in C. This functor is also parameterized by the same b. Since this is a flipped application, I’ll call it Flapp:

newtype Flapp b f = Flapp (f b)

So, for any b, the functor-valued functor IStore _ b is left adjoint to Flapp b. This is what makes IStore special.

As a side note: IStore a b is a covariant functor in a and a contravariant functor in b. However, Store a is not functorial in a, because a appears in both positive and negative position in its definition. So the adjunction trick doesn’t work for a simple (monomorphic) lens.

We can now turn the tables and use the adjunction to define the functor IStore in an arbitrary category (notice that the Yoneda lemma worked only for Set-valued functors). We just define a functor-valued functor IStore to be the left adjoint to Flapp, provided it exists.

Nat(IStore a b, f) ≅ C(a, f b)

Here, Nat is a set of natural transformations between endofunctors in C.

We can substitute the so defined functor into the Yoneda embedding formula we used earlier:

Nat((Nat(IStore a b, f), Nat(IStore s t, f))
≅ Nat(IStore s t, IStore a b)

We can now use the adjunction, rather than the Yoneda lemma, to eliminate some of the occurrences IStore:

Nat(C(a, f b), C(s, f t))
≅ C(s, IStore a b t)

This is slightly more general than the original van Laarhoven equivalence.

We can go even farther and reproduce the Jaskelioff and O’Connor trick of constraining the generic functor in the definition of the van Laarhoven lens to a pointed or applicative functor. This results in a multi-focus lens. In particular, if we use pointed functors, we get lenses with zero or one targets, so called affine lenses. Restricting the functors further to applicative leads to lenses with any number of targets, or traversals.

The trick is that any pointed or applicative functor can be stripped of the additional functionality and treated just like any other functor. This act of “forgetting” about pure and <*> may itself be considered a functor in the functor category. It’s called, appropriately, a forgetful functor. The left adjoint to a forgetful functor (if it exists) is called a free functor. It takes an arbitrary functor and creates a pointed functor by generating an artificial pure; or it creates an applicative functor by adding <*>. This adjunction is described by a natural isomorphism of hom-sets — in this case sets of natural transformations:

Nat(S, U f) ≅ Nat(S*, f)

Here, U is the forgetful functor, and S* is the free applicative/pointed version of the functor S. The functor f ranges across applicative (respectively, pointed) functors.

Now we can try to substitute the free version of IStore in the Yoneda embedding formula:

Nat((Nat(IStore* a b, f), Nat(IStore* s t, f))
≅ Nat(IStore* s t, IStore* a b)

The formula holds for any applicative (pointed) functor f and a set of natural transformations over such functors.

The first step is to use the forgetful/free adjunction:

Nat((Nat(IStore a b, U f), Nat(IStore s t, U f))
≅ Nat(IStore s t, U IStore* a b)

Then we can use our defining adjunction for IStore to get:

Nat(C(a, U f b), C(s, U f t))
≅ C(s, U IStore* a b t)

forall f. Applicative f => (a -> f b) -> (s -> f t)
≅ s -> IAppStore a b t

(the action of U is implicit).

The free applicative version of IStore is defined as:

data IAppStore a b t =
Unit t
| IAppStore a (IAppStore a b (b -> t))

These are all known results, but the use of the Yoneda embedding and the adjunction to define the IStore functor makes the derivation more compact and slightly more general.

# Acknowledgments

I’m grateful to Mauro Jaskelioff, Gershom Bazerman, and Joseph Abrahamson for reading the draft and providing helpful comments and to André van Meulebrouck for editing help.

# Bibliography

1. Edward Kmett, The Haskell Lens Library
2. Simon Peyton Jones, Lenses: Compositional Data Access and Manipulation. A Skills Matter video presentation.
3. Joseph Abrahamson, A Little Lens Starter Tutorial
4. Joseph Abrahamson, Lenses from Scratch
5. Artyom, lens over tea tutorial
6. Twan van Laarhoven, CPS based functional references
7. Bartosz Milewski, Lenses, Stores, and Yoneda
8. Mauro Jaskelioff, Russell O’Connor, A Representation Theorem for Second-Order Functionals
9. Bartosz Milewski, Understanding Yoneda

Edward Kmett’s lens library made lenses talk of the town. This is, however, not a lens tutorial (let’s wait for the upcoming Simon Peyton Jones’s intro to lenses (edit: here it is)). I’m going to concentrate on one aspect of lenses that I’ve found intriguing — the van Laarhoven representation.

Quick introduction: A lens is a data structure with a getter and a setter.

get :: b -> a
set :: b -> a -> b

Given object of type b, the getter returns the value of type a of the substructure on which the lens is focused (say, a field of a record, or an element of a list). The setter takes the object and a new value and returns a modified object, with the focus substructure replaced by the new value.

A lens can also be represented as a single function that returns a Store structure. Given the object of type b the lens returns a pair of: old value at the focus, and a function to modify it. This pair represents the Store comonad (I followed the naming convention in Russell O’Connor’s paper, see bibliography):

data Store a b = Store
{ pos  :: a
, peek :: a -> b
}

The two elements of Store are just like getter and setter except that the initial b -> was factored out.

Here’s the unexpected turn of events: Twan van Laarhoven came up with a totally different representation for a lens. Applied to the Store comonad it looks like this:

forall g . Functor g => (a -> g a) -> g b

The Store with pos and peek is equivalent to this single polymorphic function. Notice that the polymorphism is in terms of the functor g, which means that the function may only use the functor-specific fmap, nothing else. Recall the signature of fmap — for a given g:

fmap :: (a -> b) -> g a -> g b

The only way the van Laarhoven function could produce the result g b is if it has access to a function a -> b and a value of type a (which is exactly the content of Store). It can apply the function a -> g a to this value and fmap the function a -> b over it. Russell O’Connor showed the isomorphism between the two representations of the Store comonad.

This is all hunky-dory, but what’s the theory behind this equivalence? When is a data structure equivalent to a polymorphic function? I’ve seen this pattern in the Yoneda lemma (for a refresher, see my tutorial: Understanding Yoneda). In Haskell we usually see a slightly narrower application of Yoneda. It says that a certain set of polymorphic functions (natural transformations) is equivalent to a certain set of values in the image of the functor g:

forall t . (a -> t) -> g t
≈ g a

Here, (a -> t) is a function, which is mapped to g t — some functor g acting on type t. This mapping is defined once for all possible types t — it’s a natural transformation — hence forall t. All such natural transformations are equivalent to (parameterized by, isomomophic to) a type (set of values) g a. I use the symbol ≈ for type isomorphism.

There definitely is a pattern, but we have to look at the application of Yoneda not to Haskell types (the Hask category) but to Haskell functors.

# Yoneda on Functors

We are in luck because functors form a category. Objects in the Functor Category are functors between some underlying categories, and morphisms are natural transformations between those functors.

Here’s the Yoneda construction in the Functor Category. Let’s fix one functor f and consider all natural transformations of f to an arbitrary functor g. These transformations form a set, which is an object in the Set Category. For each choice of g we get a different set. Let’s call this mapping from functors to sets a representation, rep. This mapping, the canonical Yoneda embedding, is actually a functor. I’ll call this a second order functor, since it takes a functor as its argument. Being a functor, its action on morphisms in the Functor Category is also well-defined — morphisms being natural transformations in this case.

Now let’s consider an arbitrary mapping from functors to sets, let’s call it eta and let’s assume that it’s also a 2nd order functor. We have now two such functors, rep and eta. The Yoneda lemma considers mappings between these 2nd order functors, but not just any mappings — natural transformations. Since those transformations map 2nd order functors, I’ll also call them 2nd order natural transformations (thick yellow arrow in the picture).

What does it mean for a transformation to be natural? Technically, it means that a certain diagram commutes (see my Yoneda tutorial), but the Haskell intuition is the following: A natural transformation must be polymorphic in its argument. It treats this argument generically, without considering its peculiarities. In our case the 2nd order natural transformation will be polymorphic in its functor argument g.

The Yoneda lemma tells us that all such 2nd order natural transformations from rep to eta are in one-to-one correspondence with the elements of eta acting on f. I didn’t want to overcrowd the picture, but imagine another red arrow going from f to some set. That’s the set that parameterizes these natural transformations.

Let’s get down to earth. We’ll specialize the general Functor Category to the category of endofunctors in Hask and replace the Set Category with Hask (a category of Haskell types, which are treated as sets of values).

Elements of the Functor Category will be represented in Haskell through the Functor class. If f and g are two functors, a natural transformation between them can be defined pointwise (e.g., acting on actual types) as a polymorphic function.

forall t . f t -> g t

Another way of looking at this formula is that it describes a mapping from an arbitrary functor g to the Haskell type forall t . f t -> g t, where f is some fixed functor. This mapping is the Yoneda embedding we were talking about in the previous section. We’ll call it RepF:

{-# LANGUAGE Rank2Types #-}

type RepF f g = (Functor f, Functor g)
=> forall t . f t -> g t

The second ingredient we need is the mapping Eta that, just like Rep maps functors to types. Since the kind of the functor g is * -> *, the kind of Eta is (* -> *) -> *.

type family Eta :: (* -> *) -> *

Putting all this together, the Yoneda lemma tells us that the following types are equivalent:

{-# LANGUAGE Rank2Types, TypeFamilies #-}

type family Eta :: (* -> *) -> *

type NatF f = Functor f
=> forall g . Functor g
=> forall t. (f t -> g t) -> Eta g

-- equivalent to

type NatF' = Functor f => Eta f

There are many mappings that we can substitute for Eta, but I’d like to concentrate on the simplest nontrivial one, parameterized by some type b. The action of this EtaB on a functor g is defined by the application of g to b.

{-# LANGUAGE Rank2Types #-}

type EtaB b g = Functor g => g b

Now let’s consider 2nd order natural transformations between RepF and EtaB or, more precisely, between RepF f and EtaB b for some fixed f and b. These transformations must be polymorphic in g:

type NatFB f b = Functor f
=> forall g . Functor g
=> forall t. (f t -> g t) -> EtaB b g

This can be further simplified by applying EtaB to its arguments:

type NatFB f b = Functor f
=> forall g . Functor g
=> forall t. (f t -> g t) -> g b

The final step is to apply the Yoneda lemma which, in this case, tells us that the above type is equivalent to the type obtained by acting with EtaB b on f. This type is simply f b.

Do you see how close we are to the van Laarhoven equivalence?

forall g . Functor g => (a -> g a) -> g b
≈ (a, a -> b)

We just need to find the right f. But before we do that, one little exercise to get some familiarity with the Yoneda lemma for functors.

# Undoing fmap

Here’s an interesting choice for the functor f — function application. For a given type a the application (->) a is a functor. Indeed, it’s easy to implement fmap for it:

instance Functor ((->) a) where
-- fmap :: (t -> u) -> (a -> t) -> (a -> u)
fmap f g = f . g

Let’s plug this functor in the place of f in our version of Yoneda:

type NatApA a b =
forall g . Functor g
=> forall t. ((a -> t) -> g t) -> g b

NatApA a b ≈ a -> b

Here, f t was replaced by a -> t and f b by a -> b. Now let’s dig into this part of the formula:

forall t. ((a -> t) -> g t)

Isn’t this just the left hand side of the regular Yoneda? It’s a natural transformation between the Yoneda embedding functor (->) a and some functor g. The lemma tells us that this is equivalent to the type g a. So let’s make the substitution:

type NatApA a b =
forall g . Functor g
=> g a -> g b

NatApA a b ≈ a -> b

On the one hand we have a function g a -> g b which maps types lifted by the functor g. If this function is polymorphic in the functor g than it is equivalent to a function a -> b. This equivalence shows that fmap can go both ways. In fact it’s easy to show the isomorphism of the two types directly. Of course, given a function a -> b and any functor g, we can construct a function g a -> g b by applying fmap. Conversely, if we have a function g a -> g b that works for any functor g then we can use it with the trivial identity functor Identity and recover a -> b.

So this is not a big deal, but the surprise for me was that it followed from the Yoneda lemma.

# The Store Comonad and Yoneda

After this warmup exercise, I’m ready to unveil the functor f that, when plugged into the Yoneda lemma, will generate the van Laarhoven equivalence. This functor is:

Product (Const a) ((->) a)

When acting on any type b, it produces:

(Const a b, a -> b)

The Const functor ignores its second argument and is equivalent to its first argument:

newtype Const a b = Const { getConst :: a }

So the right hand side of the Yoneda lemma is equivalent to the Store comonad (a, a -> b).

Let’s look at the left hand side of the Yoneda lemma:

type NatFB f b = Functor f
=> forall g . Functor g
=> forall t. (f t -> g t) -> g b

and do the substitution:

forall g . Functor g
=> forall t. ((a, a -> t) -> g t) -> g b

Here’s the curried version of the function in parentheses:

forall t. (a -> (a -> t) -> g t)

Since the first argument a doesn’t depend on t, we can move it in front of forall:

a -> forall t . (a -> t) -> g t

We are now free to apply the 1st order Yoneda

forall t . (a -> t) -> g t ≈ g a

Which gives us:

a -> forall t . (a -> t) -> g t ≈ a -> g a

Substituting it back to our formula, we get:

forall g . Functor g => (a -> g a) -> g b
≈ (a, a -> b)

Which is exactly the van Laarhoven equivalence.

# Conclusion

I have shown that the equivalence of the two formulations of the Store comonad and, consequently, the Lens, follows from the Yoneda lemma applied to the Functor Category. This equivalence is a special case of the more general formula for functor-polymorphic representations:

type family Eta :: (* -> *) -> *

Functor f
=> forall g . Functor g
=> forall t. (f t -> g t) -> Eta g
≈ Functor f => Eta f

This formula is parameterized by two entities: a functor f and a 2nd order functor Eta.

In this article I restricted myself to one particular 2nd order functor, EtaB b, but it would be interesting to see if more complex Etas lead to more interesting results.

In the choice of the functor f I also restricted myself to just a few simple examples. It would be interesting to try, for instance, functors that generate recursive data structures and try to reproduce some of the biplate and multiplate results of Russell O’Connor’s.

# Appendix: Another Exercise

Just for the fun of it, let’s try substituting Const a for f:

The naive substitution would give us this:

forall g . Functor g
=> (forall t . Const a t -> g t) -> g b
≈ Const a b

But the natural transformation on the left:

forall t . Const a t -> g t

cannot be defined for all gs. Const a t doesn’t depend on t, so the co-domain of the natural transformation can only include functors that don’t depend on their argument — constant functors. So we are really only looking for gs that are Const c for any choice of c. All the allowed variation in g can be parameterized by type c:

forall c . (Const a t -> Const c t) -> Const c b
≈ Const a b

If you remove the Const noise, the conclusion turns out to be pretty trivial:

forall c . (a -> c) -> c ≈ a

It says that a polymorphic function that takes a function a -> c and returns a c must have some fixed a inside. This is pretty obvious, but it’s nice to know that it can be derived from the Yoneda lemma.

# Acknowledgment

Special thanks go to the folks on the Lens IRC channel for correcting my Haskell errors and for helpful advice.

# Update

After I finished writing this blog, I contacted Russel O’Connor asking him for comments. It turned out that he and Mauro Jaskelioff had been working on a paper in which they independently came up with almost exactly the same derivation of the van Laarhoven representation for the lens starting from the Yoneda lemma. They later published the paper, A Representation Theorem for Second-Order Functionals, including the link to this blog. It contains a much more rigorous proof of the equivalence.

# Bibliography

1. Twan van Laarhoven, [CPS based functional references](http://twanvl.nl/blog/haskell/cps-functional-references)
2. Russell O’Connor, [Functor is to Lens as Applicative is to Biplate](http://arxiv.org/pdf/1103.2841v2.pdf)