This post is based on the talk I gave at Functional Conf 2022. There is a video recording of this talk.

Disclaimers

Data types may contain secret information. Some of it can be extracted, some is hidden forever. We’re going to get to the bottom of this conspiracy.

No data types were harmed while extracting their secrets.

No coercion was used to make them talk.

We’re talking, of course, about unsafeCoerce, which should never be used.

Implementation hiding

The implementation of a function, even if it’s available for inspection by a programmer, is hidden from the program itself.

What is this function, with the suggestive name double, hiding inside?

x double x
2 4
3 6
-1 -2

Best guess: It’s hiding 2. It’s probably implemented as:

double x = 2 * x


How would we go about extracting this hidden value? We can just call it with the unit of multiplication:

double 1
> 2


Is it possible that it’s implemented differently (assuming that we’ve already checked it for all values of the argument)? Of course! Maybe it’s adding one, multiplying by two, and then subtracting two. But whatever the actual implementation is, it must be equivalent to multiplication by two. We say that the implementaion is isomorphic to multiplying by two.

Functors

Functor is a data type that hides things of type a. Being a functor means that it’s possible to modify its contents using a function. That is, if we’re given a function a->b and a functorful of a‘s, we can create a functorful of b‘s. In Haskell we define the Functor class as a type constructor equipped with the method fmap:

class Functor f where
fmap :: (a -> b) -> f a -> f b


A standard example of a functor is a list of a‘s. The implementation of fmap applies a function g to all its elements:

instance Functor [] where
fmap g [] = []
fmap g (a : as) = (g a) : fmap g as


Saying that something is a functor doesn’t guarantee that it actually “contains” values of type a. But most data structures that are functors will have some means of getting at their contents. When they do, you can verify that they change their contents after applying fmap. But there are some sneaky functors.

For instance Maybe a tells us: Maybe I have an a, maybe I don’t. But if I have it, fmap will change it to a b.

instance Functor Maybe where
fmap g Empty = Empty
fmap g (Just a) = Just (g a)


A function that produces values of type a is also a functor. A function e->a tells us: I’ll produce a value of type a if you ask nicely (that is call me with a value of type e). Given a producer of a‘s, you can change it to a producer of b‘s by post-composing it with a function g :: a -> b:

instance Functor ((->) e) where
fmap g f = g . f


Then there is the trickiest of them all, the IO functor. IO a tells us: Trust me, I have an a, but there’s no way I could tell you what it is. (Unless, that is, you peek at the screen or open the file to which the output is redirected.)

Continuations

A continuation is telling us: Don’t call us, we’ll call you. Instead of providing the value of type a directly, it asks you to give it a handler, a function that consumes an a and returns the result of the type of your choice:

type Cont a = forall r. (a -> r) -> r


You’d suspect that a continuation either hides a value of type a or has the means to produce it on demand. You can actually extract this value by calling the continuation with an identity function:

runCont :: Cont a -> a
runCont k = k id


In fact Cont a is for all intents and purposes equivalent to a–it’s isomorphic to it. Indeed, given a value of type a you can produce a continuation as a closure:

mkCont :: a -> Cont a
mkCont a = \k -> k a


The two functions, runCont and mkCont are the inverse of each other thus establishing the isomorphism Cont a ~ a.

The Yoneda Lemma

Here’s a variation on the theme of continuations. Just like a continuation, this function takes a handler of a‘s, but instead of producing an x, it produces a whole functorful of x‘s:

type Yo f a = forall x. (a -> x) -> f x


Just like a continuation was secretly hiding a value of the type a, this data type is hiding a whole functorful of a‘s. We can easily retrieve this functorful by using the identity function as the handler:

runYo :: Functor f => Yo f a -> f a
runYo g = g id


Conversely, given a functorful of a‘s we can reconstruct Yo f a by defining a closure that fmap‘s the handler over it:

mkYo :: Functor f => f a -> Yo f a
mkYo fa = \g -> fmap g fa


Again, the two functions, runYo and mkYo are the inverse of each other thus establishing a very important isomorphism called the Yoneda lemma:

Yo f a ~ f a

Both continuations and the Yoneda lemma are defined as polymorphic functions. The forall x in their definition means that they use the same formula for all types (this is called parametric polymorphism). A function that works for any type cannot make any assumptions about the properties of that type. All it can do is to look at how this type is packaged: Is it passed inside a list, a function, or something else. In other words, it can use the information about the form in which the polymorphic argument is passed.

Existential Types

One cannot speak of existential types without mentioning Jean-Paul Sartre.

An existential data type says: There exists a type, but I’m not telling you what it is. Actually, the type has been known at the time of construction, but then all its traces have been erased. This is only possible if the data constructor is itself polymorphic. It accepts any type and then immediately forgets what it was.

Here’s an extreme example: an existential black hole. Whatever falls into it (through the constructor BH) can never escape.

data BlackHole = forall a. BH a


Even a photon can’t escape a black hole:

bh :: BlackHole
bh = BH "Photon"


We are familiar with data types whose constructors can be undone–for instance using pattern matching. In type theory we define types by providing introduction and elimination rules. These rules tell us how to construct and how to deconstruct types.

But existential types erase the type of the argument that was passed to the (polymorphic) constructor so they cannot be deconstructed. However, not all is lost. In physics, we have Hawking radiation escaping a black hole. In programming, even if we can’t peek at the existential type, we can extract some information about the structure surrounding it.

Here’s an example: We know we have a list, but of what?

data SomeList = forall a. SomeL [a]


It turns out that to undo a polymorphic constructor we can use a polymorphic function. We have at our disposal functions that act on lists of arbitrary type, for instance length:

length :: forall a. [a] -> Int


The use of a polymorphic function to “undo” a polymorphic constructor doesn’t expose the existential type:

len :: SomeList -> Int
len (SomeL as) = length as


Indeed, this works:

someL :: SomeList
someL = SomeL [1..10]
> len someL
> 10


Extracting the tail of a list is also a polymorphic function. We can use it on SomeList without exposing the type a:

trim :: SomeList -> SomeList
trim (SomeL []) = SomeL []
trim (SomeL (a: as)) = SomeL as


Here, the tail of the (non-empty) list is immediately stashed inside SomeList, thus hiding the type a.

But this will not compile, because it would expose a:

bad :: SomeList -> a
bad (SomeL as) = head as


Producer/Consumer

Existential types are often defined using producer/consumer pairs. The producer is able to produce values of the hidden type, and the consumer can consume them. The role of the client of the existential type is to activate the producer (e.g., by providing some input) and passing the result (without looking at it) directly to the consumer.

Here’s a simple example. The producer is just a value of the hidden type a, and the consumer is a function consuming this type:

data Hide b = forall a. Hide a (a -> b)


All the client can do is to match the consumer with the producer:

unHide :: Hide b -> b
unHide (Hide a f) = f a


This is how you can use this existential type. Here, Int is the visible type, and Char is hidden:

secret :: Hide Int
secret = Hide 'a' (ord)


The function ord is the consumer that turns the character into its ASCII code:

> unHide secret
> 97


Co-Yoneda Lemma

There is a duality between polymorphic types and existential types. It’s rooted in the duality between universal quantifiers (for all, $\forall$) and existential quantifiers (there exists, $\exists$).

The Yoneda lemma is a statement about polymorphic functions. Its dual, the co-Yoneda lemma, is a statement about existential types. Consider the following type that combines the producer of x‘s (a functorful of x‘s) with the consumer (a function that transforms x‘s to a‘s):

data CoYo f a = forall x. CoYo (f x) (x -> a)


What does this data type secretly encode? The only thing the client of CoYo can do is to apply the consumer to the producer. Since the producer has the form of a functor, the application proceeds through fmap:

unCoYo :: Functor f => CoYo f a -> f a
unCoYo (CoYo fx g) = fmap g fx


The result is a functorful of a‘s. Conversely, given a functorful of a‘s, we can form a CoYo by matching it with the identity function:

mkCoYo :: Functor f => f a -> CoYo f a
mkCoYo fa = CoYo fa id


This pair of unCoYo and mkCoYo, one the inverse of the other, witness the isomorphism

CoYo f a ~ f a

In other words, CoYo f a is secretly hiding a functorful of a‘s.

Contravariant Consumers

The informal terms producer and consumer, can be given more rigorous meaning. A producer is a data type that behaves like a functor. A functor is equipped with fmap, which lets you turn a producer of a‘s to a producer of b‘s using a function a->b.

Conversely, to turn a consumer of a‘s to a consumer of b‘s you need a function that goes in the opposite direction, b->a. This idea is encoded in the definition of a contravariant functor:

class Contravariant f where
contramap :: (b -> a) -> f a -> f b


There is also a contravariant version of the co-Yoneda lemma, which reverses the roles of a producer and a consumer:

data CoYo' f a = forall x. CoYo' (f x) (a -> x)


Here, f is a contravariant functor, so f x is a consumer of x‘s. It is matched with the producer of x‘s, a function a->x.

As before, we can establish an isomorphism

CoYo' f a ~ f a

by defining a pair of functions:

unCoYo' :: Contravariant f => CoYo' f a -> f a
unCoYo' (CoYo' fx g) = contramap g fx

mkCoYo' :: Contravariant f => f a -> CoYo' f a
mkCoYo' fa = CoYo' fa id


Existential Lens

A lens abstracts a device for focusing on a part of a larger data structure. In functional programming we deal with immutable data, so in order to modify something, we have to decompose the larger structure into the focus (the part we’re modifying) and the residue (the rest). We can then recreate a modified data structure by combining the new focus with the old residue. The important observation is that we don’t care what the exact type of the residue is. This description translates directly into the following definition:

data Lens' s a =
forall c. Lens' (s -> (c, a)) ((c, a) -> s)


Here, s is the type of the larger data structure, a is the type of the focus, and the existentially hidden c is the type of the residue. A lens is constructed from a pair of functions, the first decomposing s and the second recomposing it.

Given a lens, we can construct two functions that don’t expose the type of the residue. The first is called get. It extracts the focus:

toGet :: Lens' s a -> (s -> a)
toGet (Lens' frm to) = snd . frm


The second, called set replaces the focus with the new value:

toSet :: Lens' s a -> (s -> a -> s)
toSet (Lens' frm to) = \s a -> to (fst (frm s), a)


Notice that access to residue not possible. The following will not compile:

bad :: Lens' s a -> (s -> c)
bad (Lens' frm to) = fst . frm


But how do we know that a pair of a getter and a setter is exactly what’s hidden in the existential definition of a lens? To show this we have to use the co-Yoneda lemma. First, we have to identify the producer and the consumer of c in our existential definition. To do that, notice that a function returning a pair (c, a) is equivalent to a pair of functions, one returning c and another returning a. We can thus rewrite the definition of a lens as a triple of functions:

data Lens' s a =
forall c. Lens' (s -> c) (s -> a) ((c, a) -> s)


The first function is the producer of c‘s, so the rest will define a consumer. Recall the contravariant version of the co-Yoneda lemma:

data CoYo' f s = forall c. CoYo' (f c) (s -> c)


We can define the contravariant functor that is the consumer of c‘s and use it in our definition of a lens. This functor is parameterized by two additional types s and a:

data F s a c = F (s -> a) ((c, a) -> s)


This lets us rewrite the lens using the co-Yoneda representation, with f replaced by (partially applied) F s a:

type Lens' s a = CoYo' (F s a) s


We can now use the isomorphism CoYo' f s ~ f s. Plugging in the definition of F, we get:

Lens' s a ~ CoYo' (F s a) s
CoYo' (F s a) s ~ F s a s
F s a s ~ (s -> a) ((s, a) -> s)


We recognize the two functions as the getter and the setter. Thus the existential representation of the lens is indeed isomorphic to the getter/setter pair.

Type-Changing Lens

The simple lens we’ve seen so far lets us replace the focus with a new value of the same type. But in general the new focus could be of a different type. In that case the type of the whole thing will change as well. A type-changing lens thus has the same decomposition function, but a different recomposition function:

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


As before, this lens is isomorphic to a get/set pair, where get extracts an a:

toGet :: Lens s t a b -> (s -> a)
toGet (Lens frm to) = snd . frm


and set replaces the focus with a new value of type b to produce a t:

toSet :: Lens s t a b -> (s -> b -> t)
toSet (Lens frm to) = \s b -> to (fst (frm s), b)


Other Optics

The advantage of the existential representation of lenses is that it easily generalizes to other optics. The idea is that a lens decomposes a data structure into a pair of types (c, a); and a pair is a product type, symbolically $c \times a$

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


A prism does the same for the sum data type. A sum $c + a$ is written as Either c a in Haskell. We have:

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


We can also combine sum and product in what is called an affine type $c_1 + c_2 \times a$. The resulting optic has two possible residues, c1 and c2:

data Affine s t a b =
forall c1 c2. Affine (s -> Either c1 (c2, a))
(Either c1 (c2, b) -> t)


The list of optics goes on and on.

Profunctors

A producer can be combined with a consumer in a single data structure called a profunctor. A profunctor is parameterized by two types; that is p a b is a consumer of a‘s and a producer of b‘s. We can turn a consumer of a‘s and a producer of b‘s to a consumer of s‘s and a producer of t‘s using a pair of functions, the first of which goes in the opposite direction:

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


The standard example of a profunctor is the function type p a b = a -> b. Indeed, we can define dimap for it by precomposing it with one function and postcomposing it with another:

instance Profunctor (->) where
dimap in out pab = out . pab . in


Profunctor Optics

We’ve seen functions that were polymorphic in types. But polymorphism is not restricted to types. Here’s a definition of a function that is polymorphic in profunctors:

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


This function says: Give me any producer of b‘s that consumes a‘s and I’ll turn it into a producer of t‘s that consumes s‘s. Since it doesn’t know anything else about its argument, the only thing this function can do is to apply dimap to it. But dimap requires a pair of functions, so this profunctor-polymorphic function must be hiding such a pair:

s -> a
b -> t


Indeed, given such a pair, we can reconstruct it’s implementation:

mkIso :: (s -> a) -> (b -> t) -> Iso s t a b
mkIso g h = \p -> dimap g h p


All other optics have their corresponding implementation as profunctor-polymorphic functions. The main advantage of these representations is that they can be composed using simple function composition.

Main Takeaways

• Producers and consumers correspond to covariant and contravariant functors
• Existential types are dual to polymorphic types
• Existential optics combine producers with consumers in one package
• In such optics, producers decompose, and consumers recompose data
• Functions can be polymorphic with respect to types, functors, or profunctors

A PDF version of this post is available on github.

Abstract

Co-presheaf optic is a new kind of optic that generalizes the polynomial lens. Its distinguishing feature is that it’s not based on the action of a monoidal category. Instead the action is parameterized by functors between different co-presheaves. The composition of these actions corresponds to composition of functors rather than the more traditional tensor product. These functors and their composition have a representation in terms of profunctors.

Motivation

A lot of optics can be defined using the existential, or coend, representation:

$\mathcal{O}\langle a, b\rangle \langle s, t \rangle = \int^{m \colon \mathcal M} \mathcal C (s, m \bullet a) \times \mathcal D ( m \bullet b, t)$

Here $\mathcal M$ is a monoidal category with an action on objects of two categories $\mathcal C$ and $\mathcal D$ (I’ll use the same notation for both actions). The actions are composed using the tensor product in $\mathcal M$:

$n \bullet (m \bullet a) = (n \otimes m) \bullet a$

The idea of this optic is that we have a pair of morphisms, one decomposing the source $s$ into the action of some $m$ on $a$, and the other recomposing the target $t$ from the action of the same $m$ on $b$. In most applications we pick $\mathcal D$ to be the same category as $\mathcal C$.

Recently, there has been renewed interest in polynomial functors. Morphisms between polynomial functors form a new kind of optic that doesn’t neatly fit this mold. They do, however, admit an existential representation or the form:

$\int^{c_{k i}} \prod_{k \in K} \mathbf{Set} \left(s_k, \sum_{n \in N} a_n \times c_{n k} \right) \times \prod_{i \in K} \mathbf{Set} \left(\sum_{m \in N} b_m \times c_{m i}, t_i \right)$

Here the sets $s_k$ and $t_i$ can be treated as fibers over the set $K$, while the sets $a_n$ and $b_m$ are fibers over a different set $N$.

Alternatively, we can treat these fibrations as functors from discrete categories to $\mathbf{Set}$, that is co-presheaves. For instance $a_n$ is the result of a co-presheaf $a$ acting on an object $n$ of a discrete category $\mathcal N$. The products over $K$ can be interpreted as ends that define natural transformations between co-presheaves. The interesting part is that the matrices $c_{n k}$ are fibrated over two different sets. I have previously interpreted them as profunctors:

$c \colon \mathcal N^{op} \times \mathcal K \to \mathbf{Set}$

In this post I will elaborate on this interpretation.

Co-presheaves

A co-presheaf category $[\mathcal C, Set ]$ behaves, in many respects, like a vector space. For instance, it has a “basis” consisting of representable functors $\mathcal C (r, -)$; in the sense that any co-presheaf is as a colimit of representables. Moreover, colimit-preserving functors between co-presheaf categories are very similar to linear transformations between vector spaces. Of particular interest are functors that are left adjoint to some other functors, since left adjoints preserve colimits.

The polynomial lens formula has a form suggestive of vector-space interpretation. We have one vector space with vectors $\vec{s}$ and $\vec{t}$ and another with $\vec{a}$ and $\vec{b}$. Rectangular matrices $c_{n k}$ can be seen as components of a linear transformation between these two vector spaces. We can, for instance, write:

$\sum_{n \in N} a_n \times c_{n k} = c^T a$

where $c^T$ is the transposed matrix. Transposition here serves as an analog of adjunction.

We can now re-cast the polynomial lens formula in terms of co-presheaves. We no longer intepret $\mathcal N$ and $\mathcal K$ as discrete categories. We have:

$a, b \colon [\mathcal N, \mathbf{Set}]$

$s, t \colon [\mathcal K, \mathbf{Set}]$

In this interpretation $c$ is a functor between categories of co-presheaves:

$c \colon [\mathcal N, \mathbf{Set}] \to [\mathcal K, \mathbf{Set}]$

We’ll write the action of this functor on a presheaf $a$ as $c \bullet a$.

We assume that this functor has a right adjoint and therefore preserves colimits.

$[\mathcal K, \mathbf{Set}] (c \bullet a, t) \cong [\mathcal N, \mathbf{Set}] (a, c^{\dagger} \bullet t)$

where:

$c^{\dagger} \colon [\mathcal K, \mathbf{Set}] \to [\mathcal N, \mathbf{Set}]$

We can now generalize the polynomial optic formula to:

$\mathcal{O}\langle a, b\rangle \langle s, t \rangle = \int^{c} [\mathcal K, \mathbf{Set}] \left(s, c \bullet a \right) \times [\mathcal K, \mathbf{Set}] \left(c \bullet b, t \right)$

The coend is taken over all functors that have a right adjoint. Fortunately there is a better representation for such functors. It turns out that colimit preserving functors:

$c \colon [\mathcal N, \mathbf{Set}] \to [\mathcal K, \mathbf{Set}]$

are equivalent to profunctors (see the Appendix for the proof). Such a profunctor:

$p \colon \mathcal N^{op} \times \mathcal K \to \mathbf{Set}$

is given by the formula:

$p \langle n, k \rangle = c ( \mathcal N(n, -)) k$

where $\mathcal N(n, -)$ is a representable co-presheaf.

The action of $c$ can be expressed as a coend:

$(c \bullet a) k = \int^{n} a(n) \times p \langle n, k \rangle$

The co-presheaf optic is then a coend over all profunctors $p \colon \mathcal N^{op} \times \mathcal K \to \mathbf{Set}$:

$\int^{p} [\mathcal K, \mathbf{Set}] \left(s, \int^{n} a(n) \times p \langle n, - \rangle \right) \times [\mathcal K, \mathbf{Set}] \left(\int^{n'} b(n') \times p \langle n', - \rangle, t \right)$

Composition

We have defined the action $c \bullet a$ as the action of a functor on a co-presheaf. Given two composable functors:

$c \colon [\mathcal N, \mathbf{Set}] \to [\mathcal K, \mathbf{Set}]$

and:

$c' \colon [\mathcal K, \mathbf{Set}] \to [\mathcal M, \mathbf{Set}]$

we automatically get the associativity law:

$c' \bullet (c \bullet a) = (c' \circ c) a$

The composition of functors between co-presheaves translates directly to profunctor composition. Indeed, the profunctor $p' \diamond p$ corresponding to $c' \circ c$ is given by:

$(p' \diamond p) \langle n, m \rangle = (c' \circ c) ( \mathcal N(n, -)) m$

and can be evaluated to:

$(c' ( c ( \mathcal N(n, -))) m \cong \int^{k} c ( \mathcal N(n, -)) k \times p' \langle k, m \rangle$

$\cong \int^{k} p \langle n, k \rangle \times p' \langle k, m \rangle$

which is the standard definition of profunctor composition.

Consider two composable co-presheaf optics, $\mathcal{O}\langle a, b\rangle \langle s, t \rangle$ and $\mathcal{O}\langle a', b' \rangle \langle a, b \rangle$. The first one tells us that there exists a $c$ and a pair of natural transformations:

$l_c (s, a ) = [\mathcal K, \mathbf{Set}] \left(s, c \bullet a \right)$

$r_c (b, t) = [\mathcal K, \mathbf{Set}] \left(c \bullet b, t \right)$

The second tells us that there exists a $c'$ and a pair:

$l'_{c'} (a, a' ) = [\mathcal K, \mathbf{Set}] \left(a, c' \bullet a' \right)$

$r'_{c'} (b', b) = [\mathcal K, \mathbf{Set}] \left(c' \bullet b', b \right)$

The composition of the two should be an optic of the type $\mathcal{O}\langle a', b'\rangle \langle s, t \rangle$. Indeed, we can construct such an optic using the composition $c' \circ c$ and a pair of natural transformations:

$s \xrightarrow{l_c (s, a )} c \bullet a \xrightarrow{c \,\circ \, l'_{c'} (a, a')} c \bullet (c' \bullet a') \xrightarrow{assoc} (c \circ c') \bullet a'$

$(c \circ c') \bullet b' \xrightarrow{assoc^{-1}} c \bullet (c' \bullet b') \xrightarrow{c \, \circ \, r'_{c'} (b', b)} c \bullet b \xrightarrow{r_c (b, t)} t$

Generalizations

By duality, there is a corresponding optic based on presheaves. Also, (co-) presheaves can be naturally generalized to enriched categories, where the correspondence between left adjoint functors and enriched profunctors holds as well.

Appendix

I will show that a functor between two co-presheaves that has a right adjoint and therefore preserves colimits:

$c \colon [\mathcal N, \mathbf{Set}] \to [\mathcal K, \mathbf{Set}]$

is equivalent to a profunctor:

$p \colon \mathcal N^{op} \times \mathcal K \to \mathbf{Set}$

The profunctor is given by:

$p \langle n, k \rangle = c ( \mathcal N(n, -)) k$

and the functor $c$ can be recovered using the formula:

$c (a) k = \int^{n'} a (n') \times p \langle n', k \rangle$

where:

$a \colon [\mathcal N, \mathbf{Set}]$

I’ll show that these formulas are the inverse of each other. First, inserting the formula for $c$ into the definition of $p$ should gives us $p$ back:

$\int^{n'} \mathcal N(n, -) (n') \times p\langle n', k \rangle \cong p \langle n, k \rangle$

which follows from the co-Yoneda lemma.

Second, inserting the formula for $p$ into the definition of $c$ should give us $c$ back:

$\int^{n'} a n' \times c(\mathcal N(n', -)) k \cong c (a) k$

Since $c$ preserves all colimits, and any co-presheaf is a colimit of representables, it’s enough that we prove this identity for a representable:

$a (n) = \mathcal N (r, n)$

We have to show that:

$\int^{n'} \mathcal N (r, n') \times c(\mathcal N(n', -)) k \cong c ( \mathcal N (r, -) ) k$

and this follows from the co-Yoneda lemma.

From the outside it might seem like physics and mathematics are a match made in heaven. In practice, it feels more like physicists are given a very short blanket made of math, and when they stretch it to cover their heads, their feet are freezing, and vice versa.

Physicists turn reality into numbers. They process these numbers using mathematics, and turn them into predictions about other numbers. The mapping between physical reality and mathematical models is not at all straightforward. It involves a lot of arbitrary choices. When we perform an experiment, we take the readings of our instruments and create one particular parameterization of nature. There usually are many equivalent parameterizations of the same process and this is one of the sources of redundancy in our description of nature. The Universe doesn’t care about our choice of units or coordinate systems.

This indifference, after we plug the numbers into our models, is reflected in symmetries of our models. A change in the parameters of our measuring apparatus must be compensated by a transformation of our model, so that the results of calculations still match the outcome of the experiment.

But there is an even deeper source of symmetries in physics. The model itself may introduce additional redundancy in order to simplify the calculations or, sometimes, make them possible. It is often necessary to use parameter spaces that allow the description of non-physical states–states that could never occur in reality.

Computer programmers are familiar with such situations. For instance, we often use integers to access arrays. But an integer can be negative, or it can be larger than the size of the array. We could say that an integer can describe “non-physical” states of the array. We also have freedom of parameterization of our input data: we can encode true as one, and false as zero; or the other way around. If we change our parameterization, we must modify the code that deals with it. As programmers we are very well aware of the arbitrariness of the choice of representation, but it’s even more true in physics. In physics, these reparameterizations are much more extensive and they have their mathematical description as groups of transformations.

But what we see in physics is very strange: the non-physical degrees of freedom introduced through redundant parameterizations turn out to have some measurable consequences.

Symmetries

If you ask physicists what the foundations of physics are, they will probably say: symmetry. Depending on their area of research, they will start talking about various symmetry groups, like SU(3), U(1), SO(3,1), general diffeomorphisms, etc. The foundations of physics are built upon fields and their symmetries. For physicists this is such an obvious observation that they assume that the goal of physics is to discover the symmetries of nature. But are symmetries the property of nature, or are they the artifact of our tools? This is a difficult question, because the only way we can study nature is through the prism or mathematics. Mathematical models of reality definitely exhibit lots of symmetries, and it’s easy to confuse this with the statement that nature itself is symmetric.

But why would models exhibit symmetry? One explanation is that symmetries are the effect of redundant descriptions.

I’ll use the example of electromagnetism because of its relative simplicity (some of the notation is explained in the Appendix), but the redundant degrees of freedom and the symmetries they generate show up everywhere in physics. The Standard Model is one big gauge theory, and Einstein’s General Relativity is built on the principle of invariance with respect to local coordinate transformations.

Electromagnetic field

Maxwell’s equations are a mess, until you rewrite them using 4-dimensional spacetime. The two vector fields, the electric field and the magnetic field are combined into one 4-dimensional antisymmetric tensor $F^{\mu \nu}$:

$F^{\mu\nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{bmatrix}$

Because of antisymmetry, $F^{\mu \nu}$ has only six independent components. The components of $F^{\mu \nu}$ are physical fields that can be measured using test charges and magnetic needles.

The derivatives of these fields satisfy two sets of Maxwell’s equations. The first set of four describes the dependence of fields on sources—electric charges and currents:

$\partial_{\mu} F^{\mu \nu} = j^{\nu}$

The second set of four equations describe constraints imposed on these fields:

$\partial_{[\rho} F_{\mu \nu ]} = 0$

For a particular set of sources and an initial configuration, we could try to solve these equations numerically. A brute force approach would be to divide space into little cubes, distribute our charges and currents between them, replace differential equations with difference equations, and turn on the crank.

First, we would check if the initial field configuration satisfied the constraints. Then we would calculate time derivatives of the fields. We would turn time derivatives into time differences by multiplying them by a small time period, get the next configuration, and so on. With the size of the cubes and the quantum of time small enough, we could get a reasonable approximation of reality. A program to perform these calculations isn’t much harder to write than a lot of modern 3-d computer games.

Notice that this procedure has an important property. To calculate the value of a field in a particular cube, it’s enough to know the values at its nearest neighbors and its value at the previous moment of time. The nearest-neighbor property is called locality and the dependence on the past, as opposed to the future, is called causality. The famous Conway Game of Life is local and causal, and so are cellular automata.

We were very lucky to be able to formulate a model that pretty well approximates reality and has these properties. Without such models, it would be extremely hard to calculate anything. Essentially all classical physics is written in the language of differential equations, which means it’s local, and its time dependence is carefully crafted to be causal. But it should be stressed that locality and causality are properties of particular models. And locality, in particular, cannot be taken for granted.

Electromagnetic Potential

The second set of Maxwell’s equations can be solved by introducing a new field, a 4-vector $A_{\mu}$ called the vector potential. The field tensor can be expressed as its anti-symmetrized derivative

$F_{\mu \nu} = \partial_{[ \mu} A_{\nu ]}$

Indeed, if we take its partial derivative and antisymmetrize the three indices, we get:

$\partial_{[\rho} F_{\mu \nu ]} = \partial_{[\rho} \partial_{ \mu} A_{\nu ]} = 0$

which vanishes because derivatives are symmetric, $\partial_{\mu} \partial_{\nu} = \partial_{\nu} \partial_{\mu}$.

Note for mathematicians: Think of $A_{\mu}$ as a connection in the U(1) fiber bundle, and $F_{\mu \nu}$ as its curvature. The second Maxwell equation is the Bianchi identity for this connection.

This field $A_{\mu}$ is not physical. We cannot measure it. We can measure its derivatives in the form of $F_{\mu \nu}$, but not the field itself. In fact we cannot distinguish between $A_{\mu}$ and the transformed field:

$A'_{\mu} = A_{\mu} + \partial_{\mu} \Lambda$

Here, $\Lambda(x)$ is a completely arbitrary, time dependent scalar field. This is, again, because of the symmetry of partial derivatives:

$F_{\mu \nu}' = \partial_{[ \mu} A'_{\nu ]} = \partial_{[ \mu} A_{\nu ]} + \partial_{[ \mu} \partial_{\nu ]} \Lambda = \partial_{[ \mu} A_{\nu ]} = F_{\mu \nu}$

Adding a derivative of $\Lambda$ is called a gauge transformation, and we can formulated a new law: Physics in invariant under gauge transformations. There is a beautiful symmetry we have discovered in nature.

But wait a moment: didn’t we just introduce this symmetry to simplify the math?

Well, it’s a bit more complicated. To explain that, we have to dive even deeper into technicalities.

The Action Principle

You cannot change the past and your cannot immediately influence far away events. These are the reasons why differential equations are so useful in physics. But there are some types of phenomena that are easier to explain by global rather than local reasoning. For instance, if you span an elastic rubber band between two points in space, it will trace a straight line. In this case, instead of diligently solving differential equations that describe the movements of the rubber band, we can guess its final state by calculating the shortest path between two points.

Surprisingly, just like the shape of the rubber band can be calculated by minimizing the length of the curve it spans, so the evolution of all classical systems can be calculated by minimizing (or, more precisely, finding a stationary point of) a quantity called the action. For mechanical systems the action is the integral of the Lagrangian along the trajectory, and the Lagrangian is given by the difference between  kinetic and potential energy.

Consider the simple example of an object thrown into the air and falling down due to gravity. Instead of solving the differential equations that relate acceleration to force, we can reformulate the problem in terms of minimizing the action. There is a tradeoff: we want to minimize the kinetic energy while maximizing the potential energy. Potential energy is larger at higher altitudes, so the object wants to get as high as possible in the shortest time, stay there as long as possible, before returning to earth. But the faster it tries to get there, the higher its kinetic energy. So it performs a balancing act resulting is a perfect parabola (at least if we ignore air resistance).

The same principle can be applied to fields, except that the action is now given by a 4-dimensional integral over spacetime of something called the Lagrangian density which, at every point, depends only of fields and their derivatives. This is the classical Lagrangian density that describes the electromagnetic field:

$L = - \frac{1}{4} F^{\mu \nu} F_{\mu \nu} = \frac{1}{2}(\vec{E}^2 - \vec{B}^2)$

and the action is:

$S = \int L(x)\, d^4 x$

However, if you want to derive Maxwell’s equations using the action principle, you have to express it in terms of the potential $A_{\mu}$ and its derivatives.

Noether’s Theorem

The first of the Maxwell’s equations describes the relationship between electromagnetic fields and the rest of the world:

$\partial_{\mu} F^{\mu \nu} = j^{\nu}$

Here “the rest of the world” is summarized in a 4-dimensional current density $j^{\nu}$. This is all the information about matter that the fields need to know. In fact, this equation imposes additional constraints on the matter. If you differentiate it once more, you get:

$\partial_{\nu}\partial_{\mu} F^{\mu \nu} = \partial_{\nu} j^{\nu} = 0$

Again, this follows from the antisymmetry of $F^{\mu \nu}$ and the symmetry of partial derivatives.

The equation:

$\partial_{\nu} j^{\nu} = 0$

is called the conservation of electric charge. In terms of 3-d components it reads:

$\dot{\rho} = \vec{\nabla} \vec{J}$

or, in words, the change in charge density is equal to the divergence of the electric current. Globally, it means that charge cannot appear or disappear. If your isolated system starts with a certain charge, it will end up with the same charge.

Why would the presence of electromagnetic fields impose conditions on the behavior of matter? Surprisingly, this too follows from gauge invariance. Electromagnetic fields must interact with matter in a way that makes it impossible to detect the non-physical vector potentials. In other words, the interaction must be gauge invariant. Which makes the whole action, which combines the pure-field Lagrangian and the interaction Lagrangian, gauge invariant.

It turns out that any time you have such an invariance of the action, you automatically get a conserved quantity. This is called the Noether’s theorem and, in the case of electromagnetic theory, it justifies the conservation of charge. So, even though the potentials are not physical, their symmetry has a very physical consequence: the conservation of charge.

Quantum Electrodynamics

The original idea of quantum field theory (QFT) was that it should extend the classical theory. It should be able to explain all the classical behavior plus quantum deviations from it.

This is no longer true. We don’t insist on extending classical behavior any more. We use QFT to, for instance, describe quarks, for which there is no classical theory.

The starting point of any QFT is still the good old Lagrangian density. But in quantum theory, instead of minimizing the action, we also consider quantum fluctuations around the stationary points. In fact, we consider all possible paths. It just so happens that the contributions from those paths that are far away from the classical solutions tend to cancel each other. This is the reason why classical physics works so well: classical trajectories are the most probable ones.

In quantum theory, we calculate probabilities of transitions from the initial state to the final state. These probabilities are given by summing up complex amplitudes for every possible path and then taking the absolute value of the result. The amplitudes are given by the exponential of the action:

$e^{i S / \hbar }$

Far away from the stationary point of the action, the amplitudes corresponding to adjacent paths vary very quickly in phase and they cancel each other. The summation effectively acts like a low-pass filter for these amplitudes. We are observing the Universe through a low-pass filter.

In quantum electrodynamics things are a little tricky. We would like to consider all possible paths in terms of the vector potential $A_{\mu}(x)$. The problem is that two such paths that differ only by a gauge transformation result in exactly the same action, since the Lagrangian is written in terms of gauge invariant fields $F^{\mu \nu}$. The action is therefore constant along gauge transformations and the sum over all such paths would result in infinity. Once again, the non-physical nature of the potential raises its ugly head.

Another way of describing the same problem is that we expect the quantization of electromagnetic field to describe the quanta of such field, namely photons. But a photon has only two degrees of freedom corresponding to two polarizations, whereas a vector potential has four components. Besides the two physical ones, it also introduces longitudinal and time-like polarizations, which are not present in the real world.

To eliminate the non-physical degrees of freedom, physicists came up with lots of clever tricks. These tricks are relatively mild in the case of QED, but when it comes to non-Abelian gauge fields, the details are quite gory and involve the introduction of even more non-physical fields called ghosts.

Still, there is no way of getting away from vector potentials. Moreover, the interaction of the electromagnetic field with charged particles can only be described using potentials. For instance, the Lagrangian for the electron field $\psi$ in the electromagnetic field is:

$\bar{\psi}(i \gamma^{\mu}D_{\mu} - m) \psi$

The potential $A_{\mu}$ is hidden inside the covariant derivative

$D_{\mu} = \partial_{\mu} - i e A_{\mu}$

where $e$ is the electron charge.

Note for mathematicians: The covariant derivative locally describes parallel transport in the U(1) bundle.

The electron is described by a complex Dirac spinor field $\psi$. Just as the electromagnetic potential is non-physical, so are the components of the electron field. You can conceptualize it as a “square root” of a physical field. Square roots of numbers come in pairs, positive and negative—Dirac field describes both negative electrons and positive positrons. In general, square roots are complex, and so are Dirac fields. Even the field equation they satisfy behaves like a square root of the conventional Klein-Gordon equation. Most importantly, Dirac field is only defined up to a complex phase. You can multiply it by a complex number of modulus one, $e^{i e \Lambda}$ (the $e$ in the exponent is the charge of the electron). Because the Lagrangian pairs the field $\psi$ with its complex conjugate $\bar{\psi}$, the phases cancel, which shows that the Lagrangian does not depend on the choice of the phase.

In fact, the phase can vary from point to point (and time to time) as long as the phase change is compensated by the the corresponding gauge transformation of the electromagnetic potential. The whole Lagrangian is invariant under the following simultaneous gauge transformations of all fields:

$\psi' = e^{i e \Lambda} \psi$

$\bar{\psi}' = \bar{\psi} e^{-i e \Lambda}$

$A_{\mu}' = A_{\mu} + \partial_{\mu} \Lambda$

The important part is the cancellation between the derivative of the transformed field and the gauge transformation of the potential:

$(\partial_{\mu} - i e A'_{\mu}) \psi' = e^{i e \Lambda}( \partial_{\mu} + i e \partial_{\mu} \Lambda - i e A_{\mu} - i e \partial_{\mu} \Lambda) \psi = e^{i e \Lambda} D_{\mu} \psi$

Note for mathematicians: Dirac field forms a representation of the U(1) group.

Since the electron filed is coupled to the potential, does it mean that an electron can be used to detect the potential? But the potential is non-physical: it’s only defined up to a gauge transformation.

The answer is really strange. Locally, the potential is not measurable, but it may have some very interesting global effects. This is one of these situations where quantum mechanics defies locality. We may have a region of space where the electromagnetic field is zero but the potential is not. Such potential must, at least locally, be of the form: $A_{\mu} = \partial_{\mu} \phi$. Such potential is called pure gauge, because it can be “gauged away” using $\Lambda = -\phi$.

But in a topologically nontrivial space, it may be possible to define a pure-gauge potential that cannot be gauged away by a continuous function. For instance, if we remove a narrow infinite cylinder from a 3-d space, the rest has a non-trivial topology (there are loops that cannot be shrunk to a point). We could define a 3-d vector potential that circulates around the cylinder. For any fixed radius around the cylinder, the field would consist of constant-length vectors that are tangent to the circle. A constant function is a derivative of a linear function, so this potential could be gauged away using a function $\Lambda$ that linearly increases with the angle around the cylinder, like a spiral staircase. But once we make a full circle, we end up on a different floor. There is no continuous $\Lambda$ that would eliminate this potential.

This is not just a theoretical possibility. The field around a very long thin solenoid has this property. It’s all concentrated inside the solenoid and (almost) zero outside, yet its vector potential cannot be eliminated using a continuous gauge transformation.

Classically, there is no way to detect this kind of potential. But if you look at it from the perspective of an electron trying to pass by, the potential is higher on one side of the solenoid and lower on the other, and that means the phase of the electron field will be different, depending whether it passes on the left, or on the right of it. The phase itself is not measurable but, in quantum theory, the same electron can take both paths simultaneously and interfere with itself. The phase difference is translated into the shift in the interference pattern. This is called the Aharonov-Bohm effect and it has been confirmed experimentally.

Note for mathematicians: Here, the base space of the fiber bundle has non-trivial homotopy. There may be non-trivial connections that have zero curvature.

Aharonov-Bohm experiment

Space Pasta

I went into some detail to describe the role redundant degrees of freedom and their associated symmetries play in the theory of electromagnetic fields.

We know that the vector potentials are not physical: we have no way of measuring them directly. We know that in quantum mechanics they describe non-existent particles like longitudinal and time-like photons. Since we use redundant parameterization of fields, we introduce seemingly artificial symmetries.

And yet, these “bogus symmetries” have some physical consequences: they explain the conservation of charge; and the “bogus degrees of freedom” explain the results of the Aharonov-Bohm experiment. There are some parts of reality that they capture. What are these parts?

One possible answer is that we introduce redundant parametrizations in order to describe, locally, the phenomena of global or topological nature. This is pretty obvious in the case of the Aharonov-Bohm experiment where we create a topologically nontrivial space in which some paths are not shrinkable. The charge conservation case is subtler.

Consider the path a charged particle carves in space-time. If you remove this path, you get a topologically non-trivial space. Charge conservation makes this path unbreakable, so you can view it as defining a topological invariant of the surrounding space. I would even argue that charge quantization (all charges are multiples of 1/3 of the charge or the electron) can be explained this way. We know that topological invariants, like the Euler characteristic that describes the genus of a manifold, take whole-number values.

We’d like physics to describe the whole Universe but we know that current theories fail in some areas. For instance, they cannot tell us what happens at the center of a black hole or at the Big Bang singularity. These places are far away, either in space or in time, so we don’t worry about them too much. There’s still a lot of Universe left for physicist to explore.

Except that there are some unexplorable places right under our noses. Every elementary particle is surrounded by a very tiny bubble that’s unavailable to physics. When we try to extrapolate our current theories to smaller and smaller distances, we eventually hit the wall. Our calculations result in infinities. Some of these infinities can be swept under the rug using clever tricks like renormalization. But when we get close to Planck’s distance, the effects of gravity take over, and renormalization breaks down.

So if we wanted to define “physical space” as the place where physics is applicable, we’d have to exclude all the tiny volumes around the paths of elementary particles. Removing the spaghetti of all such paths leaves us with a topological mess. This is the mess on which we define all our theories. The redundant descriptions and symmetries are our way of probing the excluded spaces.

Appendix

A point in Minkowski spacetime is characterized by four coordinates $x^{\mu}$ $\mu = 0, 1, 2, 3$, where $x^0$ is the time coordinate, and the rest are space coordinates. We use the system of units in which the speed of light $c$ is one.

Repeated indices are, by Einstein convention, summed over (contracted). Indices between square brackets are anisymmetrized (that is summed over all permutations, with the minus sign for odd permutations). For instance

$F_{0 1} = \partial_{[0} A_{1]} = \partial_{0} A_{1} - \partial_{1} A_{0} = \partial_{t} A_{x} - \partial_{x} A_{t}$

Indexes are raised and lowered by contracting them with the Minkowski metric tensor:
$\eta_{\mu\nu} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}$

Partial derivatives with respect to these coordinates are written as:

$\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}$

4-dimensional antisymmetric tensor $F^{\mu \nu}$ is a $4 \times 4$ matrix, but because of antisymmetry, it reduces to just 6 independent entries, which can be rearranged into two 3-d vector fields. The vector $\vec E$ is the electric field, and the vector $\vec B$ is the magnetic field.

$F^{\mu\nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{bmatrix}$

The sources of these fields are described by a 4-dimensional vector $j^{\mu}$. Its zeroth component describes the distribution of electric charges, and the rest describes electric current density.

The second set of Maxwell’s equations can also be written using the completely antisymmetric Levi-Civita tensor with entries equal to 1 or -1 depending on the parity of the permutation of the indices:

$\epsilon^{\mu \nu \rho \sigma} \partial_{\nu} F_{\rho \sigma} = 0$

A PDF of this post is available on github.

Motivation

In this post I’ll be looking at a subcategory of $\mathbf{Poly}$ that consists of polynomial functors in which the fibration is done over one fixed set $N$:

$P(y) = \sum_{n \in N} s_n \times \mathbf{Set}(t_n, y)$

The reason for this restriction is that morphisms between such functors, which are called polynomial lenses, can be understood in terms of monoidal actions. Optics that have this property automatically have profunctor representation. Profunctor representation has the advantage that it lets us compose optics using regular function composition.

Previously I’ve explored the representations of polynomial lenses as optics in terms on functors and profunctors on discrete categories. With just a few modifications, we can make these categories non-discrete. The trick is to replace sums with coends and products with ends; and, when appropriate, interpret ends as natural transformations.

Monoidal Action

Here’s the existential representation of a lens between polynomials in which all fibrations are over the same set $N$:

$\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle \cong$

$\int^{c_{k i}} \prod_{k \in N} \mathbf{Set} \left(s_k, \sum_{n \in N} a_n \times c_{n k} \right) \times \prod_{i \in N} \mathbf{Set} \left(\sum_{m \in N} b_m \times c_{m i}, t_i \right)$

This makes the matrices $c_{n k}$ “square.” Such matrices can be multiplied using a version of matrix multiplication.

Interestingly, this idea generalizes naturally to a setting in which $N$ is replaced by a non-discrete category $\mathcal{N}$. In this setting, we’ll write the residues $c_{m n}$ as profunctors:

$c \langle m, n \rangle \colon \mathcal{N}^{op} \times \mathcal{N} \to \mathbf{Set}$

They are objects in the monoidal category in which the tensor product is given by profunctor composition:

$(c \diamond c') \langle m, n \rangle = \int^{k \colon \mathcal{N}} c \langle m, k \rangle \times c' \langle k, n \rangle$

and the unit is the hom-functor $\mathcal{N}(m, n)$. (Incidentally, a monoid in this category is called a promonad.)

In the case of $\mathcal{N}$ a discrete category, these definitions decay to standard matrix multiplication:

$\sum_k c_{m k} \times c'_{k n}$

and the Kronecker delta $\delta_{m n}$.

We define the monoidal action of the profunctor $c$ acting on a co-presheaf $a$ as:

$(c \bullet a) (m) = \int^{n \colon \mathcal{N}} a(n) \times c \langle n, m \rangle$

This is reminiscent of a vector being multiplied by a matrix. Such an action of a monoidal category equips the co-presheaf category with the structure of an actegory.

A product of hom-sets in the definition of the existential optic turns into a set of natural transformations in the functor category $[\mathcal{N}, \mathbf{Set}]$.

$\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle \cong \int^{c \colon [\mathcal{N}^{op} \times \mathcal{N}, Set]} [\mathcal{N}, \mathbf{Set}] \left(s, c \bullet a\right) \times [\mathcal{N}, \mathbf{Set}] \left(c \bullet b, t\right)$

Or, using the end notation for natural transformations:

$\int^{c} \left( \int_m \mathbf{Set}\left(s(m), (c \bullet a)(m)\right) \times \int_n \mathbf{Set} \left((c \bullet b)(n), t(n)\right) \right)$

As before, we can eliminate the coend if we can isolate $c$ in the second hom-set using a series of isomorphisms:

$\int_n \mathbf{Set} \left(\int^k b(k) \times c\langle k, n \rangle , t(n) \right)$

$\cong \int_n \int_k \mathbf{Set}\left( b(k) \times c\langle k, n \rangle , t (n)\right)$

$\cong \int_{n, k} \mathbf{Set}\left(c\langle k, n \rangle , [b(k), t (n)]\right)$

I used the fact that a mapping out of a coend is an end. The result, after applying the Yoneda lemma to eliminate the end over $k$, is:

$\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle \cong \int_m \mathbf{Set}\left(s(m), \int^j a(j) \times [b(j), t(m)] \right)$

or, with some abuse of notation:

$[\mathcal{N}, \mathbf{Set}] ( s, [b, t] \bullet a)$

When $\mathcal{N}$ is discrete, this formula decays to the one for polynomial lens.

Profunctor Representation

Since this poly-lens is a special case of a general optic, it automatically has a profunctor representation. The trick is to define a generalized Tambara module, that is a category $\mathcal{T}$ of profunctors of the type:

$P \colon [\mathcal{N}, \mathbf{Set}]^{op} \times [\mathcal{N}, \mathbf{Set}] \to \mathbf{Set}$

with additional structure given by the following family of transformations, in components:

$\alpha_{c, s, t} \colon P\langle s, t \rangle \to P \left \langle c \bullet s, c \bullet t \right \rangle$

The profunctor representation of the polynomial lens is then given by an end over all profunctors in this Tambara category:

$\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle \cong \int_{P \colon \mathcal{T}} \mathbf{Set}\left ( (U P)\langle a, b \rangle, (U P) \langle s, t \rangle \right)$

Where $U$ is the obvious forgetful functor from $\mathcal{T}$ to the underlying profunctor category\$.

Lenses and, more general, optics are an example of hard-core category theory that has immediate application in programming. While working on polynomial lenses, I had a vague idea how they could be implemented in a programming language. I thought up an example of a polynomial lens that would focus on all the leaves of a tree at once. It could retrieve or modify them in a single operation. There already is a Haskell optic called traversal that could do it. It can safely retrieve a list of leaves from a tree. But there is a slight problem when it comes to replacing them: the size of the input list has to match the number of leaves in the tree. If it doesn’t, the traversal doesn’t work.

A polynomial lens adds an additional layer of safety by keeping track of the sizes of both the trees and the lists. The problem is that its implementation requires dependent types. Haskell has some support for dependent types, so I tried to work with it, but I quickly got bogged down. So I decided to bite the bullet and quickly learn Idris. This was actually easier than I expected and this post is the result.

Counted Vectors and Trees

I started with the “Hello World!” of dependent types: counted vectors. Notice that, in Idris, type signatures use a single colon rather than the Haskell’s double colon. You can quickly get used to it after the compiler slaps you a few times.

data Vect : Type -> Nat -> Type where
VNil : Vect a Z
VCons : (x: a) -> (xs : Vect a n) -> Vect a (S n)

If you know Haskell GADTs, you can easily read this definition. In Haskell, we usually think of Nat as a “kind”, but in Idris types and values live in the same space. Nat is just an implementation of Peano artithmetics, with Z standing for zero, and (S n) for the successor of n. Here, VNil is the constructor of an empty vector of size Z, and VCons prepends a value of type a to the tail of size n resulting in a new vector of size (S n). Notice that Idris is much more explicit about types than Haskell.

The power of dependent types is in very strict type checking of both the implementation and of usage of functions. For instance, when mapping a function over a vector, we can make sure that the result is the same size as the argument:

mapV : (a -> b) -> Vect a n -> Vect b n
mapV f VNil = VNil
mapV f (VCons a v) = VCons (f a) (mapV f v)

When concatenating two vectors, the length of the result must be the sum of the two lengths, (plus m n):

concatV : Vect a m -> Vect a n -> Vect a (plus m n)
concatV VNil v = v
concatV (VCons a w) v = VCons a (concatV w v)

Similarly, when splitting a vector in two, the lengths must match, too:

splitV : (n : Nat) -> Vect a (plus n m) -> (Vect a n, Vect a m)
splitV Z v = (VNil, v)
splitV (S k) (VCons a v') = let (v1, v2) = splitV k v'
in (VCons a v1, v2)

Here’s a more complex piece of code that implements insertion sort:

sortV : Ord a => Vect a n -> Vect a n
sortV VNil = VNil
sortV (VCons x xs) = let xsrt = sortV xs
in (ins x xsrt)
where
ins : Ord a => (x : a) -> (xsrt : Vect a n) -> Vect a (S n)
ins x VNil = VCons x VNil
ins x (VCons y xs) = if x < y then VCons x (VCons y xs)
else VCons y (ins x xs)

In preparation for the polynomial lens example, let’s implement a node-counted binary tree. Notice that we are counting nodes, not leaves. That’s why the node count for Node is the sum of the node counts of the children plus one:

data Tree : Type -> Nat -> Type where
Empty : Tree a Z
Leaf  : a -> Tree a (S Z)
Node  : Tree a n -> Tree a m -> Tree a (S (plus m  n))

All this is not much different from what you’d see in a Haskell library.

Existential Types

So far we’ve been dealing with function that return vectors whose lengths can be easily calculated from the inputs and verified at compile time. This is not always possible, though. In particular, we are interested in retrieving a vector of leaves from a tree that’s parameterized by the number of nodes. We don’t know up front how many leaves a given tree might have. Enter existential types.

An existential type hides part of its implementation. An existential vector, for instance, hides its size. The receiver of an existential vector knows that the size “exists”, but its value is inaccessible. You might wonder then: What can be done with such a mystery vector? The only way for the client to deal with it is to provide a function that is insensitive to the size of the hidden vector. A function that is polymorphic in the size of its argument. Our sortV is an example of such a function.

Here’s the definition of an existential vector:

data SomeVect : Type -> Type where
HideV : {n : Nat} -> Vect a n -> SomeVect a

SomeVect is a type constructor that depends on the type a—the payload of the vector. The data constructor HideV takes two arguments, but the first one is surrounded by a pair of braces. This is called an implicit argument. The compiler will figure out its value from the type of the second argument, which is Vect a n. Here’s how you construct an existential:

secretV : SomeVect Int
secretV = HideV (VCons 42 VNil)

In this case, the compiler will deduce n to be equal to one, but the recipient of secretV will have no way of figuring this out.

Since we’ll be using types parameterized by Nat a lot, let’s define a type synonym:

Nt : Type
Nt = Nat -> Type

Both Vect a and Tree a are examples of this type.

We can also define a generic existential for stashing such types:

data Some : Nt -> Type where
Hide : {n : Nat} -> nt n -> Some nt

and some handy type synonyms:

SomeVect : Type -> Type
SomeVect a = Some (Vect a)
SomeTree : Type -> Type
SomeTree a = Some (Tree a)

Polynomial Lens

We want to translate the following categorical definition of a polynomial lens:

$\mathbf{PolyLens}\langle s, t\rangle \langle a, b\rangle = \prod_{k} \mathbf{Set}\left(s_k, \sum_{n} a_n \times [b_n, t_k] \right)$

We’ll do it step by step. First of all, we’ll assume, for simplicity, that the indices $k$ and $n$ are natural numbers. Therefore the four arguments to PolyLens are types parameterized by Nat, for which we have a type alias:

PolyLens : Nt -> Nt -> Nt -> Nt -> Type

The definition starts with a big product over all $k$‘s. Such a product corresponds, in programming, to a polymorphic function. In Haskell we would write it as forall k. In Idris, we’ll accomplish the same using an implicit argument {k : Nat}.

The hom-set notation $\mathbf{Set}(a, b)$ stands for a set of functions from $a$ to $b$, or the type a -> b. So does the notation $[a, b]$ (the internal hom is the same as the external hom in $\mathbf{Set}$). The product $a \times b$ is the type of pairs (a, b).

The only tricky part is the sum over $n$. A sum corresponds exactly to an existential type. Our SomeVect, for instance, can be considered a sum over n of all vector types Vect a n.

Here’s the intuition: Consider that to construct a sum type like Either a b it’s enough to provide a value of either type a or type b. Once the Either is constructed, the information about which one was used is lost. If you want to use an Either, you have to provide two functions, one for each of the two branches of the case statement. Similarly, to construct SomeVect its enough to provide a vector of some particular lenght n. Instead of having two possibilities of Either, we have infinitely many possibilities corresponding to different n‘s. The information about what n was used is then promptly lost.

The sum in the definition of the polynomial lens:

$\sum_{n} a_n \times [b_n, t_k]$

can be encoded in this existential type:

data SomePair : Nt -> Nt -> Nt -> Type where
HidePair : {n : Nat} ->
(k : Nat) -> a n -> (b n -> t k) -> SomePair a b t

Notice that we are hiding n, but not k.

Taking it all together, we end up with the following type definition:

PolyLens : Nt -> Nt -> Nt -> Nt -> Type
PolyLens s t a b = {k : Nat} -> s k -> SomePair a b t

The way we read this definition is that PolyLens is a function polymorphic in k. Given a value of the type s k it produces and existential pair SomePair a b t. This pair contains a value of the type a n and a function b n -> t k. The important part is that the value of n is hidden from the caller inside the existential type.

Using the Lens

Because of the existential type, it’s not immediately obvious how one can use the polynomial lens. For instance, we would like to be able to extract the foci a n, but we don’t know what the value of n is. The trick is to hide n inside an existential Some. Here is the “getter” for this lens:

getLens :  PolyLens sn tn an bn -> sn n -> Some an
getLens lens t =
let  HidePair k v _ = lens t
in Hide v

We call lens with the argument t, pattern match on the constructor HidePair and immediately hide the contents back using the constructor Hide. The compiler is smart enough to know that the existential value of n hasn’t been leaked.

The second component of SomePair, the “setter”, is trickier to use because, without knowing the value of n, we don’t know what argument to pass to it. The trick is to take advantage of the match between the producer and the consumer that are the two components of the existential pair. Without disclosing the value of n we can take the a‘s and use a polymorphic function to transform them into b‘s.

transLens : PolyLens sn tn an bn -> ({n : Nat} -> an n -> bn n)
-> sn n -> Some tn
transLens lens f t =
let  HidePair k v vt = lens t
in  Hide (vt (f v))

The polymorphic function here is encoded as ({n : Nat} -> an n -> bn n). (An example of such a function is sortV.) Again, the value of n that’s hidden inside SomePair is never leaked.

Example

Let’s get back to our example: a polynomial lens that focuses on the leaves of a tree. The type signature of such a lens is:

treeLens : PolyLens (Tree a) (Tree b) (Vect a) (Vect b)

Using this lens we should be able to retrieve a vector of leaves Vect a n from a node-counted tree Tree a k and replace it with a new vector Vect b n to get a tree Tree b k. We should be able to do it without ever disclosing the number of leaves n.

To implement this lens, we have to write a function that takes a tree of a and produces a pair consisting of a vector of a‘s and a function that takes a vector of b‘s and produces a tree of b‘s. The type b is fixed in the signature of the lens. In fact we can pass this type to the function we are implementing. This is how it’s done:

treeLens : PolyLens (Tree a) (Tree b) (Vect a) (Vect b)
treeLens {b} t = replace b t

First, we bring b into the scope of the implementation as an implicit parameter {b}. Then we pass it as a regular type argument to replace. This is the signature of replace:

replace : (b : Type) -> Tree a n -> SomePair (Vect a) (Vect b) (Tree b)

We’ll implement it by pattern-matching on the tree.

The first case is easy:

replace b Empty = HidePair 0 VNil (\v => Empty)

For an empty tree, we return an empty vector and a function that takes and empty vector and recreates and empty tree.

The leaf case is also pretty straightforward, because we know that a leaf contains just one value:

replace b (Leaf x) = HidePair 1 (VCons x VNil)
(\(VCons y VNil) => Leaf y)

The node case is more tricky, because we have to recurse into the subtrees and then combine the results.

replace b (Node t1 t2) =
let (HidePair k1 v1 f1) = replace b t1
(HidePair k2 v2 f2) = replace b t2
v3 = concatV v1 v2
f3 = compose f1 f2
in HidePair (S (plus k2 k1)) v3 f3

Combining the two vectors is easy: we just concatenate them. Combining the two functions requires some thinking. First, let’s write the type signature of compose:

compose : (Vect b n -> Tree b k) -> (Vect b m -> Tree b j) ->
(Vect b (plus n m)) -> Tree b (S (plus j k))

The input is a pair of functions that turn vectors into trees. The result is a function that takes a larger vector whose size is the sume of the two sizes, and produces a tree that combines the two subtrees. Since it adds a new node, its node count is the sum of the node counts plus one.

Once we know the signature, the implementation is straightforward: we have to split the larger vector and pass the two subvectors to the two functions:

compose {n} f1 f2 v =
let (v1, v2) = splitV n v
in Node (f1 v1) (f2 v2)

The split is done by looking at the type of the first argument (Vect b n -> Tree b k). We know that we have to split at n, so we bring {n} into the scope of the implementation as an implicit parameter.

Besides the type-changing lens (that changes a to b), we can also implement a simple lens:

treeSimpleLens : PolyLens (Tree a) (Tree a) (Vect a) (Vect a)
treeSimpleLens {a} t = replace a t

We’ll need it later for testing.

Testing

To give it a try, let’s create a small tree with five nodes and three leaves:

t3 : Tree Char 5
t3 = (Node (Leaf 'z') (Node (Leaf 'a') (Leaf 'b')))

We can extract the leaves using our lens:

getLeaves : Tree a n -> SomeVect a
getLeaves t = getLens treeSimpleLens t

As expected, we get a vector containing 'z', 'a', and 'b'.

We can also transform the leaves using our lens and the polymorphic sort function:

trLeaves : ({n : Nat} -> Vect a n -> Vect b n) -> Tree a n -> SomeTree b
trLeaves f t = transLens treeLens f t
trLeaves sortV

The result is a new tree: ('a',('b','z'))

Complete code is available on github.

A PDF of this post is available on github

Motivation

Lenses seem to pop up in most unexpected places. Recently a new type of lens showed up as a set of morphisms between polynomial functors. This lens seemed to not fit the usual classification of optics, so it was not immediately clear that it had an existential representation using coends and, consequently a profunctor representation using ends. A profunctor representation of optics is of special interest since it lets us compose optics using standard function composition. In this post I will show how the polynomial lens fits into the framework of general optics.

Polynomial Functors

A polynomial functor in $\mathbf{Set}$ can be written as a sum (coproduct) of representables:

$P(y) = \sum_{n \in N} s_n \times \mathbf{Set}(t_n, y)$

The two families of sets, $s_n$ and $t_n$ are indexed by elements of the set $N$ (in particular, you may think of it as a set of natural numbers, but any set will do). In other words, they are fibrations of some sets $S$ and $T$ over $N$. In programming we call such families dependent types. We can also think of these fibrations as functors from a discrete category $\mathcal{N}$ to $\mathbf{Set}$.

Since, in $\mathbf{Set}$, the internal hom is isomorphic to the external hom, a polynomial functor is sometimes written in the exponential form, which makes it look more like an actual polynomial or a power series:

$P(y) = \sum_{n \in N} s_n \times y^{t_n}$

or, by representing all sets $s_n$ as sums of singlentons:

$P(y) = \sum_{n \in N} y^{t_n}$

I will also use the notation $[t_n, y]$ for the internal hom:

$P(y) = \sum_{n \in N} s_n \times [t_n, y]$

Polynomial functors form a category $\mathbf{Poly}$ in which morphisms are natural transformations.

Consider two polynomial functors $P$ and $Q$. A natural transformation between them can be written as an end. Let’s first expand the source functor:

$\mathbf{Poly}\left( \sum_k s_k \times [t_k, -], Q\right) = \int_{y\colon \mathbf{Set}} \mathbf{Set} \left(\sum_k s_k \times [t_k, y], Q(y)\right)$

The mapping out of a sum is isomorphic to a product of mappings:

$\cong \prod_k \int_y \mathbf{Set} \left(s_k \times [t_k, y], Q(y)\right)$

We can see that a natural transformation between polynomials can be reduced to a product of natural transformations out of monomials. So let’s consider a mapping out of a monomial:

$\int_y \mathbf{Set} \left( s \times [t, y], \sum_n a_n \times [b_n, y]\right)$

We can use the currying adjunction:

$\int_y \mathbf{Set} \left( [t, y], \left[s, \sum_n a_n \times [b_n, y]\right] \right)$

or, in $\mathbf{Set}$:

$\int_y \mathbf{Set} \left( \mathbf{Set}(t, y), \mathbf{Set} \left(s, \sum_n a_n \times [b_n, y]\right) \right)$

We can now use the Yoneda lemma to eliminate the end. This will simply replace $y$ with $t$ in the target of the natural transformation:

$\mathbf{Set}\left(s, \sum_n a_n \times [b_n, t] \right)$

The set of natural transformation between two arbitrary polynomials $\sum_k s_k \times [t_k, y]$ and $\sum_n a_n \times [b_n, y]$ is called a polynomial lens. Combining the previous results, we see that it can be written as:

$\mathbf{PolyLens}\langle s, t\rangle \langle a, b\rangle = \prod_{k \in K} \mathbf{Set}\left(s_k, \sum_{n \in N} a_n \times [b_n, t_k] \right)$

Notice that, in general, the sets $K$ and $N$ are different.

Using dependent-type language, we can describe the polynomial lens as acting on a whole family of types at once. For a given value of type $s_k$ it determines the index $n$. The interesting part is that this index and, consequently, the type of the focus $a_n$ and the type on the new focus $b_n$ depends not only on the type but also on the value of the argument $s_k$.

Here’s a simple example: consider a family of node-counted trees. In this case $s_k$ is a type of a tree with $k$ nodes. For a given node count we can still have trees with a different number of leaves. We can define a poly-lens for such trees that focuses on the leaves. For a given tree it produces a counted vector $a_n$ of leaves and a function that takes a counted vector $b_n$ (same size, but different type of leaf) and returns a new tree $t_k$.

Lenses and Kan Extensions

After publishing an Idris implementation of the polynomial lens, Baldur Blöndal (Iceland Jack) made an interesting observation on Twitter: The sum type in the definition of the lens looks like a left Kan extension. Indeed, if we treat $a$ and $b$ as co-presheaves, the left Kan extension of $a$ along $b$ is given by the coend:

$Lan_b a \cong \int^{n \colon \mathcal{N}} a \times [b, -]$

A coend over a discrete category is a sum (coproduct), since the co-wedge condition is trivially satisfied.

Similarly, an end over a discrete category $\mathcal{K}$ becomes a product. An end of hom-sets becomes a natural transformation. A polynomial lens can therefore be rewritten as:

$\prod_{k \in K} \mathbf{Set}\left(s_k, \sum_{n \in N} a_n \times [b_n, t_k] \right) \cong [\mathcal{K}, \mathbf{Set}](s, (Lan_b a) \circ t)$

Finally, since the left Kan extension is the left adjoint of functor pre-composition, we get this very compact formula:

$\mathbf{PolyLens}\langle s, t\rangle \langle a, b\rangle \cong [\mathbf{Set}, \mathbf{Set}](Lan_t s, Lan_b a)$

which works for arbitrary categories $\mathcal{N}$ and $\mathcal{K}$ for which the relevant Kan extensions exist.

Existential Representation

A lens is just a special case of optics. Optics have a very general representation as existential types or, categorically speaking, as coends.

The general idea is that optics describe various modes of decomposing a type into the focus (or multiple foci) and the residue. This residue is an existential type. Its only property is that it can be combined with a new focus (or foci) to produce a new composite.

The question is, what’s the residue in the case of a polynomial lens? The intuition from the counted-tree example tells us that such residue should be parameterized by both, the number of nodes, and the number of leaves. It should encode the shape of the tree, with placeholders replacing the leaves.

In general, the residue will be a doubly-indexed family $c_{m n}$ and the existential form of poly-lens will be implemented as a coend over all possible residues:

$\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle \cong$

$\int^{c_{k i}} \prod_{k \in K} \mathbf{Set} \left(s_k, \sum_{n \in N} a_n \times c_{n k} \right) \times \prod_{i \in K} \mathbf{Set} \left(\sum_{m \in N} b_m \times c_{m i}, t_i \right)$

To see that this representation is equivalent to the previous one let’s first rewrite a mapping out of a sum as a product of mappings:

$\prod_{i \in K} \mathbf{Set} \left(\sum_{m \in N} b_m \times c_{m i}, t_i \right) \cong \prod_{i \in K} \prod_{m \in N} \mathbf{Set}\left(b_m \times c_{m i}, t_i \right)$

and use the currying adjunction to get:

$\prod_{i \in K} \prod_{m \in N} \mathbf{Set}\left(c_{m i}, [b_m, t_i ]\right)$

The main observation is that, if we treat the sets $N$ and $K$ as a discrete categories $\mathcal{N}$ and $\mathcal{K}$, a product of mappings can be considered a natural transformation between functors. Functors from a discrete category are just mappings of objects, and naturality conditions are trivial.

A double product can be considered a natural transformation from a product category. And since a discrete category is its own opposite, we can (anticipating the general profunctor case) rewrite our mappings as natural transformations:

$\prod_{i \in K} \prod_{m \in N} \mathbf{Set} \left(c_{m i}, [b_m, t_i] \right) \cong [\mathcal{N}^{op} \times \mathcal{K}, \mathbf{Set}]\left(c_{= -}, [b_=, t_- ]\right)$

The indexes were replaced by placeholders. This notation underscores the interpretation of $b$ as a functor (co-presheaf) from $\mathcal{N}$ to $\mathbf{Set}$, $t$ as a functor from $\mathcal{K}$ to $\mathbf{Set}$, and $c$ as a profunctor on $\mathcal{N}^{op} \times \mathcal{K}$.

We can therefore use the co-Yoneda lemma to eliminate the coend over $c_{ki}$. The result is that $\mathbf{Pl}\langle s, t\rangle \langle a, b\rangle$ can be wrtitten as:

$\int^{c_{k i}} \prod_{k \in K} \mathbf{Set} \left(s_k, \sum_{n \in N} a_n \times c_{n k} \right) \times [\mathcal{N}^{op} \times \mathcal{K}, \mathbf{Set}]\left(c_{= -}, [b_=, t_- ]\right)$

$\cong \prod_{k \in K} \mathbf{Set}\left(s_k, \sum_{n \in N} a_n \times [b_n, t_k] \right)$

which is exactly the original polynomial-to-polynomial transformation.

Acknowledgments

I’m grateful to David Spivak, Jules Hedges and his collaborators for sharing their insights and unpublished notes with me, especially for convincing me that, in general, the two sets $N$ and $K$ should be different.

A PDF of this post is available on github.

Motivation

A lens is a reification of the concept of object composition. In a nutshell, it describes the process of decomposing the source object $s$ into a focus $a$ and a residue $c$ and recomposing a new object $t$ from a new focus $b$ and the same residue $c$.

The key observation is that we don’t care what the residue is as long as it exists. This is why a lens can be implemented, in Haskell, as an existential type:

data Lens s t a b where
Lens :: (s -> (c, a)) -> ((c, b) -> t) -> Lens s t a b


In category theory, the existential type is represented as a coend—essentially a gigantic sum over all objects $c$ in the category $\mathcal{C}$:

$\mathcal{L}\langle s, t\rangle \langle a, b \rangle = \int^{c \colon \mathcal{C}} \mathcal{C}(s, c \times a) \times \mathcal{C}(c \times b, t)$

There is a simple recipe to turn this representation into the more familiar one. The first step is to use the currying adjunction on the second hom-functor:

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

Here, $[b, t]$ is the internal hom, or the function object (b->t).
Once the object $c$ appears as the source in the hom-set, we can use the co-Yoneda lemma to eliminate the coend. This is the formula we use:

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

It works for any functor $F$ from the category $\mathcal{C}$ to $\mathbf{Set}$ so, in particular we have:

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

The result is a pair of arrows:

$\mathcal{C}(s, [b, t]) \times \mathcal{C}(s, a)$

the first corrsponding to:

set :: s -> b -> t


and the second to:

get :: s -> a


It turns out that this trick works for more general optics. It all depends on being able to isolate the object $c$ as the source of the second hom-set.

We’ve been able to do it case-by-case for lenses, prisms, traversals, and the whole zoo of optics. It turns out that the same problem was studied in all its generality by Australian category theorists Janelidze and Kelly in a context that had nothing to do with optics.

Monoidal Actions

Here’s the most general definition of an optic:

$\mathcal{O}\langle s, t\rangle \langle a, b \rangle = \int^{c \colon \mathcal{M}} \mathcal{D}(s, K_c a) \times \mathcal{C}(L_c b, t)$

This definition involves three separate categories. The category $\mathcal{M}$ is monoidal, and it has an action defiend on both $\mathcal{C}$ and $\mathcal{D}$. A category with a monoidal action is called an actegory.

We get the definition of a lens by having all three categories be the same—a cartesian closed category $\mathcal{C}$. We define the action of this category on itself:

$L_c a = c \times a$

(and the same for $K_c$).

There are two equivalent ways of defining the action of a monoidal category. One is as the mapping

$\bullet \colon \mathcal{M} \times \mathcal{C} \to \mathcal{C}$

written in infix notation as $c \bullet a$. It has to be equipped with two additional structure maps—isomorphisms that relate the tensor product $\otimes$ in $\mathcal{M}$ and its unit $I$ to the action in $\mathcal{C}$:

$\alpha_{d c a} \colon (d \otimes c) \bullet a \to d \bullet (c \bullet a)$

$\lambda_a \colon I \bullet a \to a$

plus some coherence conditions corresponding to associativity and unit laws.

Using this notation, we can rewrite the definition of an optic as:

$\mathcal{O}\langle s, t\rangle \langle a, b \rangle = \int^{c \colon \mathcal{M}} \mathcal{D}(s, c \bullet a) \times \mathcal{C}(c \bullet b, t)$

with the understanding that we use the same symbol for two different actions in $\mathcal{C}$ and $\mathcal{D}$.

Alternatively, we can curry the action $\bullet$, and use the mapping:

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

The target category here is the category of endofunctors $[\mathcal{C}, \mathcal{C}]$, which is naturally equipped with a monoidal structure given by functor composition (and, as we well know, a monad is just a monoid in that category).

The two structure maps from the definition of $\bullet$ translate to the requirement that $L$ be a strict monoidal functor, mapping tensor product to functor composition and unit object to identity functor.

The Adjunction

When we were eliminating the coend in the definition of a lens, we used the currying adjunction. This particular adjunction works inside a single category but, in general, an adjunction relates two functors between a pair of categories. Therefore, to eliminate the end from the optic, we need an adjunction that looks like this:

$\mathcal{C}(c \bullet a, t) \cong \mathcal{M}(c, R_a t)$

The category on the right is the monoidal category $\mathcal{M}$, because $c$ is the object from this category.

Using the adjunction and the co-Yoneda lemma we get:

$\mathcal{O}\langle s, t\rangle \langle a, b \rangle = \int^{c \colon \mathcal{M}} \mathcal{D}(s, c \bullet a) \times \mathcal{M}(c, R_b t) \cong \mathcal{D}(s, R_b t \bullet a)$

There is a whole slew of monoidal actions that have the right adjoint of this type. Let’s look at an example.

The category of sets is a monoidal category, and we can define its action on another category using the formula:

$\mathcal{C}(c \cdot a, t) \cong \mathbf{Set}(c, \mathcal{C}(a, t))$

This is the definition of a copower. A category in which this adjunction holds for all objects is called copowered or tensored over $\mathbf{Set}$.

The intuition is that a copower is like an iterated sum (hence the multiplication sign). Indeed, a mapping out of a coproduct of $c$ copies of $a$, where $c$ is a set, is equivalent to $c$ mappings out of $a$.

This formula generalizes to the case in which $\mathcal{C}$ is a category enriched over a monoidal category $\mathcal{V}$. In that case we have:

$\mathcal{C}(c \cdot a, t) \cong \mathcal{V}(c, \mathcal{C}(a, t))$

Here, the hom $\mathcal{C}(a, t)$ is an object in $\mathcal{V}$.

Relation to Enrichment

We are interested in the adjunction:

$\mathcal{C}( c \bullet a, t) \cong \mathcal{M}(c, R_a t)$

The functor $R$ is covariant in $t$ and contravariant in $a$:

$R \colon \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{M}$

In other words, it’s a profunctor. In fact, it has the right signature to be a hom-functor. And this is what Janelidze and Kelly show: the functor $R$ can serve as the hom-functor that generates the enrichment for $\mathcal{C}$. If we call the enriched category $\mathbf{C}$, we can define the hom-object as:

$\mathbf{C}(a, t) = R_a t$

Our adjunction can be rewritten as:

$\mathcal{C}( c \bullet a, t) \cong \mathcal{M}(c, \mathbf{C}(a, t))$

The counit of this adjunction is a mapping:

$\epsilon_{a t} \colon \mathbf{C}(a, t) \bullet a \to t$

which is analogous to function application.

The hom-object $\mathbf{C}(a, t)$ in an enriched category must satisfy the composition and identity laws. In an enriched category, composition is a mapping:

$\circ \colon \mathbf{C}(b, t) \otimes \mathbf{C}(a, b) \to \mathbf{C}(a, t)$

Let’s see if we can get this mapping from our adjunction by replacing $c$ with $\mathbf{C}(b, t) \otimes \mathbf{C}(a, b)$. We get:

$\mathcal{C}( (\mathbf{C}(b, t) \otimes \mathbf{C}(a, b)) \bullet a, t) \cong \mathcal{M}(\mathbf{C}(b, t) \otimes \mathbf{C}(a, b), \mathbf{C}(a, t))$

The right-hand side should contain the mapping we’re looking for. All we need is to point at a morphism on the left. Indeed, we can define it as the following composite:

$\big( \mathbf{C}(b, t) \otimes \mathbf{C}(a, b)\big) \bullet a \xrightarrow{\alpha}$

$\mathbf{C}(b, c) \bullet \big(\mathbf{C}(a, b)) \bullet a\big) \xrightarrow{id \; \bullet \; \epsilon}$

$\mathbf{C}(b, t) \bullet b \xrightarrow{\epsilon} t$

We used the structure map $\alpha$ and (twice) the counit of the adjunction $\epsilon$.

Similarly, the identity of composition, which is the mapping:

$id_a \colon I \to \mathbf{C}(a, a)$

is adjoint to the other structure map $\lambda_a$.

Janelidze and Kelly go on to prove that the action of a monoidal right-closed category having the right adjoint is equivalent to the existence of the tensored enrichment of the category on which this action is defined.

The two examples of monoidal actions we’ve seen so far are indeed equivalent to enrichments. A cartesian closed category in which we defined the action $L_c a = c \times a$ is automatically self-enriched. The copower action $L_c a = c \cdot a$ is equivalent to enrichment over $\mathbf{Set}$ (which doesn’t mean much, since regular categories are naturally $\mathbf{Set}$-enriched; but not all of them are tensored).

Acknowledgments

I’m grateful to Jules Hedges and his collaborators for sharing their insights and unpublished notes with me.

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) $F \colon \mathcal{C} \to \mathbf{Set}$:

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

A PDF version of this post is available on GitHub.

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:

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

which, in components, reads:

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

Previously: Profunctors.

Traversals

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

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


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

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

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

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

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

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


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

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

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

$a \to a \times c$

For prisms, it’s the “translation”

$a \to a + c$

For grates it’s the exponentiation

$a \to a^c$

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

$a \to c_0 + a \times c_1$

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

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

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

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

The existential version of a traversal is thus given by:

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

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

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

and use the currying adjunction

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

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

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

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

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

which is exactly the formula for traversals.

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

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

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

Generalizations

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

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

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

These actions define the most general existential optic:

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

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

We have also generalized Tambara modules:

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

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

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

The Physicist’s Explanation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$f \colon a' \to a$

and another one in $\mathcal{D}$

$g \colon b \to b'$

the profunctor uses them to extend the path:

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

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

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

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

The profunctor representation is like a path integral:

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

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

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

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

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

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

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

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

Conclusion

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

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