### Category Theory

I wanted to do category theory, not geometry, so the idea of studying simplexes didn’t seem very attractive at first. But as I was getting deeper into it, a very different picture emerged. Granted, the study of simplexes originated in geometry, but then category theorists took interest in it and turned it into something completely different. The idea is that simplexes define a very peculiar scheme for composing things. The way you compose lower dimensional simplexes in order to build higher dimensional simplexes forms a pattern that shows up in totally unrelated areas of mathematics… and programming. Recently I had a discussion with Edward Kmett in which he hinted at the simplicial structure of cumulative edits in a source file.

## Geometric picture

Let’s start with a simple idea, and see what we can do with it. The idea is that of triangulation, and it almost goes back to the beginning of the Agricultural Era. Somebody smart noticed long time ago that we can measure plots of land by subdividing them into triangles.

Why triangles and not, say, rectangles or quadrilaterals? Well, to begin with, a quadrilateral can be always divided into triangles, so triangles are more fundamental as units of composition in 2-d. But, more importantly, triangles also work when you embed them in higher dimensions, and quadrilaterals don’t. You can take any three points and there is a unique flat triangle that they span (it may be degenerate, if the points are collinear). But four points will, in general, span a warped quadrilateral. Mind you, rectangles work great on flat screens, and we use them all the time for selecting things with the mouse. But on a curved or bumpy surface, triangles are the only option.

Surveyors have covered the whole Earth, mountains and all, with triangles. In computer games, we build complex models, including human faces or dolphins, using wireframes. Wireframes are just systems of triangles that share some of the vertices and edges. So triangles can be used to approximate complex 2-d surfaces in 3-d.

## More dimensions

How can we generalize this process? First of all, we could use triangles in spaces that have more than 3 dimensions. This way we could, for instance, build a Klein bottle in 4-d without it intersecting itself.

We can also consider replacing triangles with higher-dimensional objects. For instance, we could approximate 3-d volumes by filling them with cubes. This technique is used in computer graphics, where we often organize lots of cubes in data structures called octrees. But just like squares or quadrilaterals don’t work very well on non-flat surfaces, cubes cannot be used in curved spaces. The natural generalization of a triangle to something that can fill a volume without any warping is a tetrahedron. Any four points in space span a tetrahedron.

We can go on generalizing this construction to higher and higher dimensions. To form an n-dimensional simplex we can pick $n+1$ points. We can draw a segment between any two points, a triangle between any three points, a tetrahedron between any four points, and so on. It’s thus natural to define a 1-dimensional simplex to be a segment, and a 0-dimensional simplex to be a point.

Simplexes (or simplices, as they are sometimes called) have very regular recursive structure. An n-dimensional simplex has $n+1$ faces, which are all $n-1$ dimensional simplexes. A tetrahedron has four triangular faces, a triangle has three sides (one-dimensional simplexes), and a segment has two endpoints. (A point should have one face–and it does, in the “augmented” theory). Every higher-dimensional simplex can be decomposed into lower-dimensional simplexes, and the process can be repeated until we get down to individual vertexes. This constitutes a very interesting composition scheme that will come up over and over again in unexpected places.

Notice that you can always construct a face of a simplex by deleting one point. It’s the point opposite to the face in question. This is why there are as many faces as there are points in a simplex.

## Look Ma! No coordinates!

So far we’ve been secretly thinking of points as elements of some n-dimensional linear space, presumably $\mathbb{R}^n$. Time to make another leap of abstraction. Let’s abandon coordinate systems. Can we still define simplexes and, if so, how would we use them?

Consider a wireframe built from triangles. It defines a particular shape. We can deform this shape any way we want but, as long as we don’t break connections or fuse points, we cannot change its topology. A wireframe corresponding to a torus can never be deformed into a wireframe corresponding to a sphere.

The information about topology is encoded in connections. The connections don’t depend on coordinates. Two points are either connected or not. Two triangles either share a side or they don’t. Two tetrahedrons either share a triangle or they don’t. So if we can define simplexes without resorting to coordinates, we’ll have a new language to talk about topology.

But what becomes of a point if we discard its coordinates? It becomes an element of a set. An arrangement of simplexes can be built from a set of points or 0-simplexes, together with a set of 1-simplexes, a set of 2-simplexes, and so on. Imagine that you bought a piece of furniture from Ikea. There is a bag of screws (0-simplexes), a box of sticks (1-simplexes), a crate of triangular planks (2-simplexes), and so on. All parts are freely stretchable (we don’t care about sizes).

You have no idea what the piece of furniture will look like unless you have an instruction booklet. The booklet tells you how to arrange things: which sticks form the edges of which triangles, etc. In general, you want to know which lower-order simplexes are the “faces” of higher-order simplexes. This can be determined by defining functions between the corresponding sets, which we’ll call face maps.

For instance, there should be two function from the set of segments to the set of points; one assigning the beginning, and the other the end, to each segment. There should be three functions from the set of triangles to the set of segments, and so on. If the same point is the end of one segment and the beginning of another, the two segments are connected. A segment may be shared between multiple triangles, a triangle may be shared between tetrahedrons, and so on.

You can compose these functions–for instance, to select a vertex of a triangle, or a side of a tetrahedron. Composable functions suggest a category, in this case a subcategory of Set. Selecting a subcategory suggests a functor from some other, simpler, category. What would that category be?

## The Simplicial category

The objects of this simpler category, let’s call it the simplicial category $\Delta$, would be mapped by our functor to corresponding sets of simplexes in Set. So, in $\Delta$, we need one object corresponding to the set of points, let’s call it $[0]$; another for segments, $[1]$; another for triangles, $[2]$; and so on. In other words, we need one object called $[n]$ per one set of n-dimensional simplexes.

What really determines the structure of this category is its morphisms. In particular, we need morphisms that would be mapped, under our functor, to the functions that define faces of our simplexes–the face maps. This means, in particular, that for every $n$ we need $n+1$ distinct functions from the image of $[n]$ to the image of $[n-1]$. These functions are themselves images of morphisms that go between $[n]$ and $[n-1]$ in $\Delta$; we do, however, have a choice of the direction of these morphisms. If we choose our functor to be contravariant, the face maps from the image of $[n]$ to the image of $[n-1]$ will be images of morphisms going from $[n-1]$ to $[n]$ (the opposite direction). This contravariant functor from $\Delta$ to Set (such functors are called pre-sheafe) is called the simplicial set.

What’s attractive about this idea is that there is a category that has exactly the right types of morphisms. It’s a category whose objects are ordinals, or ordered sets of numbers, and morphisms are order-preserving functions. Object $[0]$ is the one-element set $\{0\}$, $[1]$ is the set $\{0, 1\}$, $[2]$ is $\{0, 1, 2\}$, and so on. Morphisms are functions that preserve order, that is, if $n < m$ then $f(n) \leq f(m)$. Notice that the inequality is non-strict. This will become important in the definition of degeneracy maps.

The description of simplicial sets using a functor follows a very common pattern in category theory. The simpler category defines the primitives and the grammar for combining them. The target category (often the category of sets) provides models for the theory in question. The same trick is used, for instance, in defining abstract algebras in Lawvere theories. There, too, the syntactic category consists of a tower of objects with a very regular set of morphisms, and the models are contravariant Set-valued functors.

Because simplicial sets are functors, they form a functor category, with natural transformations as morphisms. A natural transformation between two simplicial sets is a family of functions that map vertices to vertices, edges to edges, triangles to triangles, and so on. In other words, it embeds one simplicial set in another.

## Face maps

We will obtain face maps as images of injective morphisms between objects of $\Delta$. Consider, for instance, an injection from $[1]$ to $[2]$. Such a morphism takes the set $\{0, 1\}$ and maps it to $\{0, 1, 2\}$. In doing so, it must skip one of the numbers in the second set, preserving the order of the other two. There are exactly three such morphisms, skipping either $0$, $1$, or $2$.

And, indeed, they correspond to three face maps. If you think of the three numbers as numbering the vertices of a triangle, the three face maps remove the skipped vertex from the triangle leaving the opposing side free. The functor is contravariant, so it reverses the direction of morphisms.

The same procedure works for higher order simplexes. An injection from $[n-1]$ to $[n]$ maps $\{0, 1,...,n-1\}$ to $\{0, 1,...,n\}$ by skipping some $k$ between $0$ and $n$.

The corresponding face map is called $d_{n, k}$, or simply $d_k$, if $n$ is obvious from the context.

Such face maps automatically satisfy the obvious identities for any $i < j$:

$d_i d_j = d_{j-1} d_i$

The change from $j$ to $j-1$ on the right compensates for the fact that, after removing the $i$th number, the remaining indexes are shifted down.

These injections generate, through composition, all the morphisms that strictly preserve the ordering (we also need identity maps to form a category). But, as I mentioned before, we are also interested in those maps that are non-strict in the preservation of ordering (that is, they can map two consecutive numbers into one). These generate the so called degeneracy maps. Before we get to definitions, let me provide some motivation.

## Homotopy

One of the important application of simplexes is in homotopy. You don’t need to study algebraic topology to get a feel of what homotopy is. Simply said, homotopy deals with shrinking and holes. For instance, you can always shrink a segment to a point. The intuition is pretty obvious. You have a segment at time zero, and a point at time one, and you can create a continuous “movie” in between. Notice that a segment is a 1-simplex, whereas a point is a 0-simplex. Shrinking therefore provides a bridge between different-dimensional simplexes.

Similarly, you can shrink a triangle to a segment–in particular the segment that is one of its sides.

You can also shrink a triangle to a point by pasting together two shrinking movies–first shrinking the triangle to a segment, and then the segment to a point. So shrinking is composable.

But not all higher-dimensional shapes can be shrunk to all lower-dimensional shapes. For instance, an annulus (a.k.a., a ring) cannot be shrunk to a segment–this would require tearing it. It can, however, be shrunk to a circular loop (or two segments connected end to end to form a loop). That’s because both, the annulus and the circle, have a hole. So continuous shrinking can be used to classify shapes according to how many holes they have.

We have a problem, though: You can’t describe continuous transformations without using coordinates. But we can do the next best thing: We can define degenerate simplexes to bridge the gap between dimensions. For instance, we can build a segment, which uses the same vertex twice. Or a collapsed triangle, which uses the same side twice (its third side is a degenerate segment).

## Degeneracy maps

We model operations on simplexes, such as face maps, through morphisms from the category opposite to $\Delta$. The creation of degenerate simplexes will therefore corresponds to mappings from $[n+1]$ to $[n]$. They obviously cannot be injective, but we may chose them to be surjective. For instance, the creation of a degenerate segment from a point corresponds to the (opposite) mapping of $\{0, 1\}$ to $\{0\}$, which collapses the two numbers to one.

We can construct a degenerate triangle from a segment in two ways. These correspond to the two surjections from $\{0, 1, 2\}$ to $\{0, 1\}$.

The first one called $\sigma_{1, 0}$ maps both $0$ and $1$ to $0$ and $2$ to $1$. Notice that, as required, it preserves the order, albeit weakly. The second, $\sigma_{1, 1}$ maps $0$ to $0$ but collapses $1$ and $2$ to $1$.

In general, $\sigma_{n, k}$ maps $\{0, 1, ... k, k+1 ... n+1\}$ to $\{0, 1, ... k ... n\}$ by collapsing $k$ and $k+1$ to $k$.

Our contravariant functor maps these order-preserving surjections to functions on sets. The resulting functions are called degeneracy maps: each $\sigma_{n, k}$ mapped to the corresponding $s_{n, k}$. As with face maps, we usually omit the first index, as it’s either arbitrary or easily deducible from the context.

Two degeneracy maps. In the triangles, two of the sides are actually the same segment. The third side is a degenerate segment whose ends are the same point.

There is an obvious identity for the composition of degeneracy maps:

$s_i s_j = s_{j+1} s_i$

for $i \leq j$.

The interesting identities relate degeneracy maps to face maps. For instance, when $i = j$ or $i = j + 1$, we have:

$d_i s_j = id$

(that’s the identity morphism). Geometrically speaking, imagine creating a degenerate triangle from a segment, for instance by using $s_0$. The first side of this triangle, which is obtained by applying $d_0$, is the original segment. The second side, obtained by $d_1$, is the same segment again.

The third side is degenerate: it can be obtained by applying $s_0$ to the vertex obtained by $d_1$.

In general, for $i > j + 1$:

$d_i s_j = s_j d_{i-1}$

Similarly:

$d_i s_j = s_{j-1} d_i$

for $i < j$.

All the face- and degeneracy-map identities are relevant because, given a family of sets and functions that satisfy them, we can reproduce the simplicial set (contravariant functor from $\Delta$ to Set) that generates them. This shows the equivalence of the geometric picture that deals with triangles, segments, faces, etc., with the combinatorial picture that deals with rearrangements of ordered sequences of numbers.

## Monoidal structure

A triangle can be constructed by adjoining a point to a segment. Add one more point and you get a tetrahedron. This process of adding points can be extended to adding together arbitrary simplexes. Indeed, there is a binary operator in $\Delta$ that combines two ordered sequences by stacking one after another.

This operation can be lifted to morphisms, making it a bifunctor. It is associative, so one might ask the question whether it can be used as a tensor product to make $\Delta$ a monoidal category. The only thing missing is the unit object.

The lowest dimensional simplex in $\Delta$ is $[0]$, which represents a point, so it cannot be a unit with respect to our tensor product. Instead we are obliged to add a new object, which is called $[-1]$, and is represented by an empty set. (Incidentally, this is the object that may serve as “the face” of a point.)

With the new object $[-1]$, we get the category $\Delta_a$, which is called the augmented simplicial category. Since the unit and associativity laws are satisfied “on the nose” (as opposed to “up to isomorphism”), $\Delta_a$ is a strict monoidal category.

Note: Some authors prefer to name the objects of $\Delta_a$ starting from zero, rather than minus one. They rename $[-1]$ to $\bold{0}$, $[0]$ to $\bold{1}$, etc. This convention makes even more sense if you consider that $\bold{0}$ is the initial object and $\bold{1}$ the terminal object in $\Delta_a$.

Monoidal categories are a fertile breeding ground for monoids. Indeed, the object $[0]$ in $\Delta_a$ is a monoid. It is equipped with two morphisms that act like unit and multiplication. It has an incoming morphism from the monoidal unit $[-1]$–the morphism that’s the precursor of the face map that assigns the empty set to every point. This morphism can be used as the unit $\eta$ of our monoid. It also has an incoming morphism from $[1]$ (which happens to be the tensorial square of $[0]$). It’s the precursor of the degeneracy map that creates a segment from a single point. This morphism is the multiplication $\mu$ of our monoid. Unit and associativity laws follow from the standard identities between morphisms in $\Delta_a$.

It turns out that this monoid $([0], \eta, \mu)$ in $\Delta_a$ is the mother of all monoids in strict monoidal categories. It can be shown that, for any monoid $m$ in any strict monoidal category $C$, there is a unique strict monoidal functor $F$ from $\Delta_a$ to $C$ that maps the monoid $[0]$ to the monoid $m$. The category $\Delta_a$ has exactly the right structure, and nothing more, to serve as the pattern for any monoid we can come up within a (strict) monoidal category. In particular, since a monad is just a monoid in the (strictly monoidal) category of endofunctors, the augmented simplicial category is behind every monad as well.

## One more thing

Incidentally, since $\Delta_a$ is a monoidal category, (contravariant) functors from it to Set are automatically equipped with monoidal structure via Day convolution. The result of Day convolution is a join of simplicial sets. It’s a generalized cone: two simplicial sets together with all possible connections between them. In particular, if one of the sets is just a single point, the result of the join is an actual cone (or a pyramid).

## Different shapes

If we are willing to let go of geometric interpretations, we can replace the target category of sets with an arbitrary category. Instead of having a set of simplexes, we’ll end up with an object of simplexes. Simplicial sets become simplicial objects.

Alternatively, we can generalize the source category. As I mentioned before, simplexes are a good choice of primitives because of their geometrical properties–they don’t warp. But if we don’t care about embedding these simplexes in $\mathbb{R}^n$, we can replace them with cubes of varying dimensions (a one dimensional cube is a segment, a two dimensional cube is a square, and so on). Functors from the category of n-cubes to Set are called cubical sets. An even further generalization replaces simplexes with shapeless globes producing globular sets.

All these generalizations become important tools in studying higher category theory. In an n-category, we naturally encounter various shapes, as reflected in the naming convention: objects are called 0-cells; morphisms, 1-cells; morphisms between morphisms, 2-cells, and so on. These “cells” are often visualized as n-dimensional shapes. If a 1-cell is an arrow, a 2-cell is a (directed) surface spanning two arrows; a 3-cell, a volume between two surfaces; e.t.c. In this way, the shapeless hom-set that connects two objects in a regular category turns into a topologically rich blob in an n-category.

This is even more pronounced in infinity groupoids, which became popularized by homotopy type theory, where we have an infinite tower of bidirectional n-morphisms. The presence or the absence of higher order morphisms between any two morphisms can be visualized as the existence of holes that prevent the morphing of one cell into another. This kind of morphing can be described by homotopies which, in turn, can be described using simplicial, cubical, globular, or even more exotic sets.

## Conclusion

I realize that this post might seem a little rambling. I have two excuses: One is that, when I started looking at simplexes, I had no idea where I would end up. One thing led to another and I was totally fascinated by the journey. The other is the realization how everything is related to everything else in mathematics. You start with simple triangles, you compose and decompose them, you see some structure emerging. Suddenly, the same compositional structure pops up in totally unrelated areas. You see it in algebraic topology, in a monoid in a monoidal category, or in a generalization of a hom-set in an n-category. Why is it so? It seems like there aren’t that many ways of composing things together, and we are forced to keep reusing them over and over again. We can glue them, nail them, or solder them. The way simplicial category is put together provides a template for one of the universal patterns of composition.

## Bibliography

1. John Baez, A Quick Tour of Basic Concepts in Simplicial Homotopy Theory
2. Greg Friedman, An elementary illustrated introduction to simplicial sets.
3. N J Wildberger, Algebraic Topology. An excellent series of videos.

## Acknowledgments

I’m grateful to Edward Kmett and Derek Elkins for reviewing the draft and for providing helpful suggestions.

There is a lot of folklore about various data types that pop up in discussions about lenses. For instance, it’s known that FunList and Bazaar are equivalent, although I haven’t seen a proof of that. Since both data structures appear in the context of Traversable, which is of great interest to me, I decided to do some research. In particular, I was interested in translating these data structures into constructs in category theory. This is a continuation of my previous blog posts on free monoids and free applicatives. Here’s what I have found out:

• FunList is a free applicative generated by the Store functor. This can be shown by expressing the free applicative construction using Day convolution.
• Using Yoneda lemma in the category of applicative functors I can show that Bazaar is equivalent to FunList

Let’s start with some definitions. FunList was first introduced by Twan van Laarhoven in his blog. Here’s a (slightly generalized) Haskell definition:

data FunList a b t = Done t
| More a (FunList a b (b -> t))

It’s a non-regular inductive data structure, in the sense that its data constructor is recursively called with a different type, here the function type b->t. FunList is a functor in t, which can be written categorically as:

$L_{a b} t = t + a \times L_{a b} (b \to t)$

where $b \to t$ is a shorthand for the hom-set $Set(b, t)$.

Strictly speaking, a recursive data structure is defined as an initial algebra for a higher-order functor. I will show that the higher order functor in question can be written as:

$A_{a b} g = I + \sigma_{a b} \star g$

where $\sigma_{a b}$ is the (indexed) store comonad, which can be written as:

$\sigma_{a b} s = \Delta_a s \times C(b, s)$

Here, $\Delta_a$ is the constant functor, and $C(b, -)$ is the hom-functor. In Haskell, this is equivalent to:

newtype Store a b s = Store (a, b -> s)

The standard (non-indexed) Store comonad is obtained by identifying a with b and it describes the objects of the slice category $C/s$ (morphisms are functions $f : a \to a'$ that make the obvious triangles commute).

If you’ve read my previous blog posts, you may recognize in $A_{a b}$ the functor that generates a free applicative functor (or, equivalently, a free monoidal functor). Its fixed point can be written as:

$L_{a b} = I + \sigma_{a b} \star L_{a b}$

The star stands for Day convolution–in Haskell expressed as an existential data type:

data Day f g s where
Day :: f a -> g b -> ((a, b) -> s) -> Day f g s

Intuitively, $L_{a b}$ is a “list of” Store functors concatenated using Day convolution. An empty list is the identity functor, a one-element list is the Store functor, a two-element list is the Day convolution of two Store functors, and so on…

In Haskell, we would express it as:

data FunList a b t = Done t
| More ((Day (Store a b) (FunList a b)) t)

To show the equivalence of the two definitions of FunList, let’s expand the definition of Day convolution inside $A_{a b}$:

$(A_{a b} g) t = t + \int^{c d} (\Delta_b c \times C(a, c)) \times g d \times C(c \times d, t)$

The coend $\int^{c d}$ corresponds, in Haskell, to the existential data type we used in the definition of Day.

Since we have the hom-functor $C(a, c)$ under the coend, the first step is to use the co-Yoneda lemma to “perform the integration” over $c$, which replaces $c$ with $a$ everywhere. We get:

$t + \int^d \Delta_b a \times g d \times C(a \times d, t)$

We can then evaluate the constant functor and use the currying adjunction:

$C(a \times d, t) \cong C(d, a \to t)$

to get:

$t + \int^d b \times g d \times C(d, a \to t)$

Applying the co-Yoneda lemma again, we replace $d$ with $a \to t$:

$t + b \times g (a \to t)$

This is exactly the functor that generates FunList. So FunList is indeed the free applicative generated by Store.

All transformations in this derivation were natural isomorphisms.

Now let’s switch our attention to Bazaar, which can be defined as:

type Bazaar a b t = forall f. Applicative f => (a -> f b) -> f t

(The actual definition of Bazaar in the lens library is even more general–it’s parameterized by a profunctor in place of the arrow in a -> f b.)

The universal quantification in the definition of Bazaar immediately suggests the application of my favorite double Yoneda trick in the functor category: The set of natural transformations (morphisms in the functor category) between two functors (objects in the functor category) is isomorphic, through Yoneda embedding, to the following end in the functor category:

$Nat(h, g) \cong \int_{f \colon [C, Set]} Set(Nat(g, f), Nat(h, f))$

The end is equivalent (modulo parametricity) to Haskell forall. Here, the sets of natural transformations between pairs of functors are just hom-functors in the functor category and the end over $f$ is a set of higher-order natural transformations between them.

In the double Yoneda trick we carefully select the two functors $g$ and $h$ to be either representable, or somehow related to representables.

The universal quantification in Bazaar is limited to applicative functors, so we’ll pick our two functors to be free applicatives. We’ve seen previously that the higher-order functor that generates free applicatives has the form:

$F g = Id + g \star F g$

Here’s the version of the Yoneda embedding in which $f$ varies over all applicative functors in the category $App$, and $g$ and $h$ are arbitrary functors in $[C, Set]$:

$App(F h, F g) \cong \int_{f \colon App} Set(App(F g, f), App(F h, f))$

The free functor $F$ is the left adjoint to the forgetful functor $U$:

$App(F g, f) \cong [C, Set](g, U f)$

Using this adjunction, we arrive at:

$[C, Set](h, U (F g)) \cong \int_{f \colon App} Set([C, Set](g, U f), [C, Set](h, U f))$

We’re almost there–we just need to carefuly pick the functors $g$ and $h$. In order to arrive at the definition of Bazaar we want:

$g = \sigma_{a b} = \Delta_a \times C(b, -)$

$h = C(t, -)$

The right hand side becomes:

$\int_{f \colon App} Set\big(\int_c Set (\Delta_a c \times C(b, c), (U f) c)), \int_c Set (C(t, c), (U f) c)\big)$

where I represented natural transformations as ends. The first term can be curried:

$Set \big(\Delta_a c \times C(b, c), (U f) c)\big) \cong Set\big(C(b, c), \Delta_a c \to (U f) c \big)$

and the end over $c$ can be evaluated using the Yoneda lemma. So can the second term. Altogether, the right hand side becomes:

$\int_{f \colon App} Set\big(a \to (U f) b)), (U f) t)\big)$

In Haskell notation, this is just the definition of Bazaar:

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

The left hand side can be written as:

$\int_c Set(h c, (U (F g)) c)$

Since we have chosen $h$ to be the hom-functor $C(t, -)$, we can use the Yoneda lemma to “perform the integration” and arrive at:

$(U (F g)) t$

With our choice of $g = \sigma_{a b}$, this is exactly the free applicative generated by Store–in other words, FunList.

This proves the equivalence of Bazaar and FunList. Notice that this proof is only valid for $Set$-valued functors, although a generalization to the enriched setting is relatively straightforward.

There is another family of functors, Traversable, that uses universal quantification over applicatives:

class (Functor t, Foldable t) => Traversable t where
traverse :: forall f. Applicative f => (a -> f b) -> t a -> f (t b)

The same double Yoneda trick can be applied to it to show that it’s related to Bazaar. There is, however, a much simpler derivation, suggested to me by Derek Elkins, by changing the order of arguments:

traverse :: t a -> (forall f. Applicative f => (a -> f b) -> f (t b))

which is equivalent to:

traverse :: t a -> Bazaar a b (t b)

In view of the equivalence between Bazaar and FunList, we can also write it as:

traverse :: t a -> FunList a b (t b)

Note that this is somewhat similar to the definition of toList:

toList :: Foldable t => t a -> [a]

In a sense, FunList is able to freely accumulate the effects from traversable, so that they can be interpreted later.

## Acknowledgments

I’m grateful to Edward Kmett and Derek Elkins for many discussions and valuable insights.

# Abstract

The use of free monads, free applicatives, and cofree comonads lets us separate the construction of (often effectful or context-dependent) computations from their interpretation. In this paper I show how the ad hoc process of writing interpreters for these free constructions can be systematized using the language of higher order algebras (coalgebras) and catamorphisms (anamorphisms).

# Introduction

Recursive schemes [meijer] are an example of successful application of concepts from category theory to programming. The idea is that recursive data structures can be defined as initial algebras of functors. This allows a separation of concerns: the functor describes the local shape of the data structure, and the fixed point combinator builds the recursion. Operations over data structures can be likewise separated into shallow, non-recursive computations described by algebras, and generic recursive procedures described by catamorphisms. In this way, data structures often replace control structures in driving computations.

Since functors also form a category, it’s possible to define functors acting on functors. Such higher order functors show up in a number of free constructions, notably free monads, free applicatives, and cofree comonads. These free constructions have good composability properties and they provide means of separating the creation of effectful computations from their interpretation.

This paper’s contribution is to systematize the construction of such interpreters. The idea is that free constructions arise as fixed points of higher order functors, and therefore can be approached with the same algebraic machinery as recursive data structures, only at a higher level. In particular, interpreters can be constructed as catamorphisms or anamorphisms of higher order algebras/coalgebras.

# Initial Algebras and Catamorphisms

The canonical example of a data structure that can be described as an initial algebra of a functor is a list. In Haskell, a list can be defined recursively:

data List a = Nil | Cons a (List a)


There is an underlying non-recursive functor:

data ListF a x = NilF | ConsF a x
instance Functor (ListF a) where
fmap f NilF = NilF
fmap f (ConsF a x) = ConsF a (f x)


Once we have a functor, we can define its algebras. An algebra consist of a carrier c and a structure map (evaluator). An algebra can be defined for an arbitrary functor f:

type Algebra f c = f c -> c


Here’s an example of a simple list algebra, with Int as its carrier:

sum :: Algebra (ListF Int) Int
sum NilF = 0
sum (ConsF a c) = a + c


Algebras for a given functor form a category. The initial object in this category (if it exists) is called the initial algebra. In Haskell, we call the carrier of the initial algebra Fix f. Its structure map is a function:

f (Fix f) -> Fix f


By Lambek’s lemma, the structure map of the initial algebra is an isomorphism. In Haskell, this isomorphism is given by a pair of functions: the constructor In and the destructor out of the fixed point combinator:

newtype Fix f = In { out :: f (Fix f) }


When applied to the list functor, the fixed point gives rise to an alternative definition of a list:

type List a = Fix (ListF a)


The initiality of the algebra means that there is a unique algebra morphism from it to any other algebra. This morphism is called a catamorphism and, in Haskell, can be expressed as:

cata :: Functor f => Algebra f a -> Fix f -> a
cata alg = alg . fmap (cata alg) . out


A list catamorphism is known as a fold. Since the list functor is a sum type, its algebra consists of a value—the result of applying the algebra to NilF—and a function of two variables that corresponds to the ConsF constructor. You may recognize those two as the arguments to foldr:

foldr :: (a -> c -> c) -> c -> [a] -> c


The list functor is interesting because its fixed point is a free monoid. In category theory, monoids are special objects in monoidal categories—that is categories that define a product of two objects. In Haskell, a pair type plays the role of such a product, with the unit type as its unit (up to isomorphism).

As you can see, the list functor is the sum of a unit and a product. This formula can be generalized to an arbitrary monoidal category with a tensor product $\otimes$ and a unit $1$:

$L\, a\, x = 1 + a \otimes x$

Its initial algebra is a free monoid .

# Higher Algebras

In category theory, once you performed a construction in one category, it’s easy to perform it in another category that shares similar properties. In Haskell, this might require reimplementing the construction.

We are interested in the category of endofunctors, where objects are endofunctors and morphisms are natural transformations. Natural transformations are represented in Haskell as polymorphic functions:

type f :~> g = forall a. f a -> g a
infixr 0 :~>


In the category of endofunctors we can define (higher order) functors, which map functors to functors and natural transformations to natural transformations:

class HFunctor hf where
hfmap :: (g :~> h) -> (hf g :~> hf h)
ffmap :: Functor g => (a -> b) -> hf g a -> hf g b


The first function lifts a natural transformation; and the second function, ffmap, witnesses the fact that the result of a higher order functor is again a functor.

An algebra for a higher order functor hf consists of a functor f (the carrier object in the functor category) and a natural transformation (the structure map):

type HAlgebra hf f = hf f :~> f


As with regular functors, we can define an initial algebra using the fixed point combinator for higher order functors:

newtype FixH hf a = InH { outH :: hf (FixH hf) a }


Similarly, we can define a higher order catamorphism:

hcata :: HFunctor h => HAlgebra h f -> FixH h :~> f
hcata halg = halg . hfmap (hcata halg) . outH


The question is, are there any interesting examples of higher order functors and algebras that could be used to solve real-life programming problems?

We’ve seen the usefulness of lists, or free monoids, for structuring computations. Let’s see if we can generalize this concept to higher order functors.

The definition of a list relies on the cartesian structure of the underlying category. It turns out that there are multiple cartesian structures of interest that can be defined in the category of functors. The simplest one defines a product of two endofunctors as their composition. Any two endofunctors can be composed. The unit of functor composition is the identity functor.

If you picture endofunctors as containers, you can easily imagine a tree of lists, or a list of Maybes.

A monoid based on this particular monoidal structure in the endofunctor category is a monad. It’s an endofunctor m equipped with two natural transformations representing unit and multiplication:

class Monad m where
eta :: Identity    :~> m
mu  :: Compose m m :~> m


In Haskell, the components of these natural transformations are known as return and join.

A straightforward generalization of the list functor to the functor category can be written as:

$L\, f\, g = 1 + f \circ g$

type FunctorList f g = Identity :+: Compose f g


where we used the operator :+: to define the coproduct of two functors:

data (f :+: g) e = Inl (f e) | Inr (g e)
infixr 7 :+:


Using more conventional notation, FunctorList can be written as:

data MonadF f g a =
DoneM a
| MoreM (f (g a))


We’ll use it to generate a free monoid in the category of endofunctors. First of all, let’s show that it’s indeed a higher order functor in the second argument g:

instance Functor f => HFunctor (MonadF f) where
hfmap _   (DoneM a)  = DoneM a
hfmap nat (MoreM fg) = MoreM $fmap nat fg ffmap h (DoneM a) = DoneM (h a) ffmap h (MoreM fg) = MoreM$ fmap (fmap h) fg


In category theory, because of size issues, this functor doesn’t always have a fixed point. For most common choices of f (e.g., for algebraic data types), the initial higher order algebra for this functor exists, and it generates a free monad. In Haskell, this free monad can be defined as:

type FreeMonad f = FixH (MonadF f)


We can show that FreeMonad is indeed a monad by implementing return and bind:

instance Functor f => Monad (FreeMonad f) where
return = InH . DoneM
(InH (DoneM a))    >>= k = k a
(InH (MoreM ffra)) >>= k =
InH (MoreM (fmap (>>= k) ffra))


Free monads have many applications in programming. They can be used to write generic monadic code, which can then be interpreted in different monads. A very useful property of free monads is that they can be composed using coproducts. This follows from the theorem in category theory, which states that left adjoints preserve coproducts (or, more generally, colimits). Free constructions are, by definition, left adjoints to forgetful functors. This property of free monads was explored by Swierstra [swierstra] in his solution to the expression problem. I will use an example based on his paper to show how to construct monadic interpreters using higher order catamorphisms.

A stack-based calculator can be implemented directly using the state monad. Since this is a very simple example, it will be instructive to re-implement it using the free monad approach.

We start by defining a functor, in which the free parameter k represents the continuation:

data StackF k  = Push Int k
| Top (Int -> k)
| Pop k
deriving Functor


We use this functor to build a free monad:

type FreeStack = FreeMonad StackF


You may think of the free monad as a tree with nodes that are defined by the functor StackF. The unary constructors, like Add or Pop, create linear list-like branches; but the Top constructor branches out with one child per integer.

The level of indirection we get by separating recursion from the functor makes constructing free monad trees syntactically challenging, so it makes sense to define a helper function:

liftF :: (Functor f) => f r -> FreeMonad f r
liftF fr = InH $MoreM$ fmap (InH . DoneM) fr


With this function, we can define smart constructors that build leaves of the free monad tree:

push :: Int -> FreeStack ()
push n = liftF (Push n ())

pop :: FreeStack ()
pop = liftF (Pop ())

top :: FreeStack Int
top = liftF (Top id)



All these preparations finally pay off when we are able to create small programs using do notation:

calc :: FreeStack Int
calc = do
push 3
push 4
x <- top
pop
return x


Of course, this program does nothing but build a tree. We need a separate interpreter to do the calculation. We’ll interpret our program in the state monad, with state implemented as a stack (list) of integers:

type MemState = State [Int]


The trick is to define a higher order algebra for the functor that generates the free monad and then use a catamorphism to apply it to the program. Notice that implementing the algebra is a relatively simple procedure because we don’t have to deal with recursion. All we need is to case-analyze the shallow constructors for the free monad functor MonadF, and then case-analyze the shallow constructors for the functor StackF.

runAlg :: HAlgebra (MonadF StackF) MemState
runAlg (DoneM a)  = return a
runAlg (MoreM ex) =
case ex of
Top  ik  -> get >>= ik  . head
Pop  k   -> get >>= put . tail   >> k
Push n k -> get >>= put . (n : ) >> k
Add  k   -> do (a: b: s) <- get
put (a + b : s)
k


The catamorphism converts the program calc into a state monad action, which can be run over an empty initial stack:

runState (hcata runAlg calc) []


The real bonus is the freedom to define other interpreters by simply switching the algebras. Here’s an algebra whose carrier is the Const functor:

showAlg :: HAlgebra (MonadF StackF) (Const String)

showAlg (DoneM a) = Const "Done!"
showAlg (MoreM ex) = Const $case ex of Push n k -> "Push " ++ show n ++ ", " ++ getConst k Top ik -> "Top, " ++ getConst (ik 42) Pop k -> "Pop, " ++ getConst k Add k -> "Add, " ++ getConst k  Runing the catamorphism over this algebra will produce a listing of our program: getConst$ hcata showAlg calc

> "Push 3, Push 4, Add, Top, Pop, Done!"

# Free Applicative

There is another monoidal structure that exists in the category of functors. In general, this structure will work for functors from an arbitrary monoidal category $C$ to $Set$. Here, we’ll restrict ourselves to endofunctors on $Set$. The product of two functors is given by Day convolution, which can be implemented in Haskell using an existential type:

data Day f g c where
Day :: f a -> g b -> ((a, b) -> c) -> Day f g c


The intuition is that a Day convolution contains a container of some as, and another container of some bs, together with a function that can convert any pair (a, b) to c.

Day convolution is a higher order functor:

instance HFunctor (Day f) where
hfmap nat (Day fx gy xyt) = Day fx (nat gy) xyt
ffmap h   (Day fx gy xyt) = Day fx gy (h . xyt)


In fact, because Day convolution is symmetric up to isomorphism, it is automatically functorial in both arguments.

To complete the monoidal structure, we also need a functor that could serve as a unit with respect to Day convolution. In general, this would be the the hom-functor from the monoidal unit:

$C(1, -)$

In our case, since $1$ is the singleton set, this functor reduces to the identity functor.

We can now define monoids in the category of functors with the monoidal structure given by Day convolution. These monoids are equivalent to lax monoidal functors which, in Haskell, form the class:

class Functor f => Monoidal f where
unit  :: f ()
(>*<) :: f x -> f y -> f (x, y)


Lax monoidal functors are equivalent to applicative functors [mcbride], as seen in this implementation of pure and <*>:

  pure  :: a -> f a
pure a = fmap (const a) unit
(<*>) :: f (a -> b) -> f a -> f b
fs <*> as = fmap (uncurry ($)) (fs >*< as)  We can now use the same general formula, but with Day convolution as the product: $L\, f\, g = 1 + f \star g$ to generate a free monoidal (applicative) functor: data FreeF f g t = DoneF t | MoreF (Day f g t)  This is indeed a higher order functor: instance HFunctor (FreeF f) where hfmap _ (DoneF x) = DoneF x hfmap nat (MoreF day) = MoreF (hfmap nat day) ffmap f (DoneF x) = DoneF (f x) ffmap f (MoreF day) = MoreF (ffmap f day)  and it generates a free applicative functor as its initial algebra: type FreeA f = FixH (FreeF f)  ## Free Applicative Example The following example is taken from the paper by Capriotti and Kaposi [capriotti]. It’s an option parser for a command line tool, whose result is a user record of the following form: data User = User { username :: String , fullname :: String , uid :: Int } deriving Show  A parser for an individual option is described by a functor that contains the name of the option, an optional default value for it, and a reader from string: data Option a = Option { optName :: String , optDefault :: Maybe a , optReader :: String -> Maybe a } deriving Functor  Since we don’t want to commit to a particular parser, we’ll create a parsing action using a free applicative functor: userP :: FreeA Option User userP = pure User <*> one (Option "username" (Just "John") Just) <*> one (Option "fullname" (Just "Doe") Just) <*> one (Option "uid" (Just 0) readInt)  where readInt is a reader of integers: readInt :: String -> Maybe Int readInt s = readMaybe s  and we used the following smart constructors: one :: f a -> FreeA f a one fa = InH$ MoreF $Day fa (done ()) fst done :: a -> FreeA f a done a = InH$ DoneF a


We are now free to define different algebras to evaluate the free applicative expressions. Here’s one that collects all the defaults:

alg :: HAlgebra (FreeF Option) Maybe
alg (DoneF a) = Just a
alg (MoreF (Day oa mb f)) =
fmap f (optDefault oa >*< mb)


I used the monoidal instance for Maybe:

instance Monoidal Maybe where
unit = Just ()
Just x >*< Just y = Just (x, y)
_ >*< _ = Nothing


This algebra can be run over our little program using a catamorphism:

parserDef :: FreeA Option a -> Maybe a
parserDef = hcata alg


And here’s an algebra that collects the names of all the options:

alg2 :: HAlgebra (FreeF Option) (Const String)
alg2 (DoneF a) = Const "."
alg2 (MoreF (Day oa bs f)) =
fmap f (Const (optName oa) >*< bs)


Again, this uses a monoidal instance for Const:

instance Monoid m => Monoidal (Const m) where
unit = Const mempty
Const a >*< Const b = Const (a  b)


We can also define the Monoidal instance for IO:

instance Monoidal IO where
unit = return ()
ax >*< ay = do a <- ax
b <- ay
return (a, b)


This allows us to interpret the parser in the IO monad:

alg3 :: HAlgebra (FreeF Option) IO
alg3 (DoneF a) = return a
alg3 (MoreF (Day oa bs f)) = do
putStrLn $optName oa s <- getLine let ma = optReader oa s a = fromMaybe (fromJust (optDefault oa)) ma fmap f$ return a >*< bs


Every construction in category theory has its dual—the result of reversing all the arrows. The dual of a product is a coproduct, the dual of an algebra is a coalgebra, and the dual of a monad is a comonad.

Let’s start by defining a higher order coalgebra consisting of a carrier f, which is a functor, and a natural transformation:

type HCoalgebra hf f = f :~> hf f


An initial algebra is dualized to a terminal coalgebra. In Haskell, both are the results of applying the same fixed point combinator, reflecting the fact that the Lambek’s lemma is self-dual. The dual to a catamorphism is an anamorphism. Here is its higher order version:

hana :: HFunctor hf
=> HCoalgebra hf f -> (f :~> FixH hf)
hana hcoa = InH . hfmap (hana hcoa) . hcoa


The formula we used to generate free monoids:

$1 + a \otimes x$

dualizes to:

$1 \times a \otimes x$

and can be used to generate cofree comonoids .

A cofree functor is the right adjoint to the forgetful functor. Just like the left adjoint preserved coproducts, the right adjoint preserves products. One can therefore easily combine comonads using products (if the need arises to solve the coexpression problem).

Just like the monad is a monoid in the category of endofunctors, a comonad is a comonoid in the same category. The functor that generates a cofree comonad has the form:

type ComonadF f g = Identity :*: Compose f g


where the product of functors is defined as:

data (f :*: g) e = Both (f e) (g e)
infixr 6 :*:


Here’s the more familiar form of this functor:

data ComonadF f g e = e :< f (g e)


It is indeed a higher order functor, as witnessed by this instance:

instance Functor f => HFunctor (ComonadF f) where
hfmap nat (e :< fge) = e :< fmap nat fge
ffmap h (e :< fge) = h e :< fmap (fmap h) fge


A cofree comonad is the terminal coalgebra for this functor and can be written as a fixed point:

type Cofree f = FixH (ComonadF f)


Indeed, for any functor f, Cofree f is a comonad:

instance Functor f => Comonad (Cofree f) where
extract (InH (e :< fge)) = e
duplicate fr@(InH (e :< fge)) =
InH (fr :< fmap duplicate fge)


The canonical example of a cofree comonad is an infinite stream:

type Stream = Cofree Identity


We can use this stream to sample a function. We’ll encapsulate this function inside the following functor (in fact, itself a comonad):

data Store a x = Store a (a -> x)
deriving Functor


We can use a higher order coalgebra to unpack the Store into a stream:

streamCoa :: HCoalgebra (ComonadF Identity)(Store Int)
streamCoa (Store n f) =
f n :< (Identity \$ Store (n + 1) f)


The actual unpacking is a higher order anamorphism:

stream :: Store Int a -> Stream a
stream = hana streamCoa


We can use it, for instance, to generate a list of squares of natural numbers:

stream (Store 0 (^2))


Since, in Haskell, the same fixed point defines a terminal coalgebra as well as an initial algebra, we are free to construct algebras and catamorphisms for streams. Here’s an algebra that converts a stream to an infinite list:

listAlg :: HAlgebra (ComonadF Identity) []
listAlg(a :< Identity as) = a : as

toList :: Stream a -> [a]
toList = hcata listAlg


# Future Directions

In this paper I concentrated on one type of higher order functor:

$1 + a \otimes x$

and its dual. This would be equivalent to studying folds for lists and unfolds for streams. But the structure of the functor category is richer than that. Just like basic data types can be combined into algebraic data types, so can functors. Moreover, besides the usual sums and products, the functor category admits at least two additional monoidal structures generated by functor composition and Day convolution.

Another potentially fruitful area of exploration is the profunctor category, which is also equipped with two monoidal structures, one defined by profunctor composition, and another by Day convolution. A free monoid with respect to profunctor composition is the basis of Haskell Arrow library [jaskelioff]. Profunctors also play an important role in the Haskell lens library [kmett].

## Bibliography

1. Erik Meijer, Maarten Fokkinga, and Ross Paterson, Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire
2. Conor McBride, Ross Paterson, Idioms: applicative programming with effects
3. Paolo Capriotti, Ambrus Kaposi, Free Applicative Functors
4. Wouter Swierstra, Data types a la carte
5. Exequiel Rivas and Mauro Jaskelioff, Notions of Computation as Monoids
6. Edward Kmett, Lenses, Folds and Traversals
7. Richard Bird and Lambert Meertens, Nested Datatypes
8. Patricia Johann and Neil Ghani, Initial Algebra Semantics is Enough!

Functors from a monoidal category C to Set form a monoidal category with Day convolution as product. A monoid in this category is a lax monoidal functor. We define an initial algebra using a higher order functor and show that it corresponds to a free lax monoidal functor.

Recently I’ve been obsessing over monoidal functors. I have already written two blog posts, one about free monoidal functors and one about free monoidal profunctors. I followed some ideas from category theory but, being a programmer, I leaned more towards writing code than being preoccupied with mathematical rigor. That left me longing for more elegant proofs of the kind I’ve seen in mathematical literature.

I believe that there isn’t that much difference between programming and math. There is a whole spectrum of abstractions ranging from assembly language, weakly typed languages, strongly typed languages, functional programming, set theory, type theory, category theory, and homotopy type theory. Each language comes with its own bag of tricks. Even within one language one starts with some relatively low level encodings and, with experience, progresses towards higher abstractions. I’ve seen it in Haskell, where I started by hand coding recursive functions, only to realize that I can be more productive using bulk operations on types, then building recursive data structures and applying recursive schemes, eventually diving into categories of functors and profunctors.

I’ve been collecting my own bag of mathematical tricks, mostly by reading papers and, more recently, talking to mathematicians. I’ve found that mathematicians are happy to share their knowledge even with outsiders like me. So when I got stuck trying to clean up my monoidal functor code, I reached out to Emily Riehl, who forwarded my query to Alexander Campbell from the Centre for Australian Category Theory. Alex’s answer was a very elegant proof of what I was clumsily trying to show in my previous posts. In this blog post I will explain his approach. I should also mention that most of the results presented in this post have already been covered in a comprehensive paper by Rivas and Jaskelioff, Notions of Computation as Monoids.

## Lax Monoidal Functors

To properly state the problem, I’ll have to start with a lot of preliminaries. This will require some prior knowledge of category theory, all within the scope of my blog/book.

We start with a monoidal category $C$, that is a category in which you can “multiply” objects using some kind of a tensor product $\otimes$. For any pair of objects $a$ and $b$ there is an object $a \otimes b$; and this mapping is functorial in both arguments (that is, you can also “multiply” morphisms). A monoidal category will also have a special object $I$ that is the unit of multiplication. In general, the unit and associativity laws are satisfied up to isomorphism:

$\lambda : I \otimes a \cong a$

$\rho : a \otimes I \cong a$

$\alpha : (a \otimes b) \otimes c \cong a \otimes (b \otimes c)$

These isomorphisms are called, respectively, the left and right unitors, and the associator.

The most familiar example of a monoidal category is the category of types and functions, in which the tensor product is the cartesian product (pair type) and the unit is the unit type ().

Let’s now consider functors from $C$ to the category of sets, $Set$. These functors also form a category called $[C, Set]$, in which morphisms between any two functors are natural transformations.

In Haskell, a natural transformation is approximated by a polymorphic function:

type f ~> g = forall x. f x -> g x

The category $Set$ is monoidal, with cartesian product $\times$ serving as a tensor product, and the singleton set $1$ as the unit.

We are interested in functors in $[C, Set]$ that preserve the monoidal structure. Such a functor should map the tensor product in $C$ to the cartesian product in $Set$ and the unit $I$ to the singleton set $1$. Accordingly, a strong monoidal functor $F$ comes with two isomorphisms:

$F a \times F b \cong F (a \otimes b)$

$1 \cong F I$

We are interested in a weaker version of a monoidal functor called lax monoidal functor, which is equipped with a one-way natural transformation:

$\mu : F a \times F b \to F (a \otimes b)$

and a one-way morphism:

$\eta : 1 \to F I$

A lax monoidal functor must also preserve unit and associativity laws.

Associativity law: $\alpha$ is the associator in the appropriate category (top arrow, in Set; bottom arrow, in C).

In Haskell, a lax monoidal functor can be defined as:

class Monoidal f where
eta :: () -> f ()
mu  :: (f a, f b) -> f (a, b)

It’s also known as the applicative functor.

## Day Convolution and Monoidal Functors

It turns out that our category of functors $[C, Set]$ is also equipped with monoidal structure. Two functors $F$ and $G$ can be “multiplied” using Day convolution:

$(F \star G) c = \int^{a b} C(a \otimes b, c) \times F a \times G b$

Here, $C(a \otimes b, c)$ is the hom-set, or the set of morphisms from $a \otimes b$ to $c$. The integral sign stands for a coend, which can be interpreted as a generalization of an (infinite) coproduct (modulo some identifications). An element of this coend can be constructed by injecting a triple consisting of a morphism from $C(a \otimes b, c)$, an element of the set $F a$, and an element of the set $G b$, for some $a$ and $b$.

In Haskell, a coend corresponds to an existential type, so the Day convolution can be defined as:

data Day f g c where
Day :: ((a, b) -> c, f a, g b) -> Day f g c

(The actual definition uses currying.)

The unit with respect to Day convolution is the hom-functor:

$C(I, -)$

which assigns to every object $c$ the set of morphisms $C(I, c)$ and acts on morphisms by post-composition.

The proof that this is the unit is instructive, as it uses the standard trick: the co-Yoneda lemma. In the coend form, the co-Yoneda lemma reads, for a covariant functor $F$:

$\int^x C(x, a) \times F x \cong F a$

and for a contravariant functor $H$:

$\int^x C(a, x) \times H x \cong H a$

(The mnemonics is that the integration variable must appear twice, once in the negative, and once in the positive position. An argument to a contravariant functor is in a negative position.)

Indeed, substituting $C(I, -)$ for the first functor in Day convolution produces:

$(C(I, -) \star G) c = \int^{a b} C(a \otimes b, c) \times C(I, a) \times G b$

which can be “integrated” over $a$ using the Yoneda lemma to yield:

$\int^{b} C(I \otimes b, c) \times G b$

and, since $I$ is the unit of the tensor product, this can be further “integrated” over $b$ to give $G c$. The right unit law is analogous.

To summarize, we are dealing with three monoidal categories: $C$ with the tensor product $\otimes$ and unit $I$, $Set$ with the cartesian product and singleton $1$, and a functor category $[C, Set]$ with Day convolution and unit $C(I, -)$.

## A Monoid in [C, Set]

A monoidal category can be used to define monoids. A monoid is an object $m$ equipped with two morphisms — unit and multiplication:

$\eta : I \to m$

$\mu : m \otimes m \to m$

These morphisms must satisfy unit and associativity conditions, which are best illustrated using commuting diagrams.

Unit laws. λ and ρ are the unitors.

Associativity law: α is the associator.

This definition of a monoid can be translated directly to Haskell:

class Monoid m where
eta :: () -> m
mu  :: (m, m) -> m

It so happens that a lax monoidal functor is exactly a monoid in our functor category $[C, Set]$. Since objects in this category are functors, a monoid is a functor $F$ equipped with two natural transformations:

$\eta : C(I, -) \to F$

$\mu : F \star F \to F$

At first sight, these don’t look like the morphisms in the definition of a lax monoidal functor. We need some new tricks to show the equivalence.

Let’s start with the unit. The first trick is to consider not one natural transformation but the whole hom-set:

$[C, Set](C(I, -), F)$

The set of natural transformations can be represented as an end (which, incidentally, corresponds to the forall quantifier in the Haskell definition of natural transformations):

$\int_c Set(C(I, c), F c)$

The next trick is to use the Yoneda lemma which, in the end form reads:

$\int_c Set(C(a, c), F c) \cong F a$

In more familiar terms, this formula asserts that the set of natural transformations from the hom-functor $C(a, -)$ to $F$ is isomorphic to $F a$.

There is also a version of the Yoneda lemma for contravariant functors:

$\int_c Set(C(c, a), H c) \cong H a$

The application of Yoneda to our formula produces $F I$, which is in one-to-one correspondence with morphisms $1 \to F I$.

We can use the same trick of bundling up natural transformations that define multiplication $\mu$.

$[C, Set](F \star F, F)$

and representing this set as an end over the hom-functor:

$\int_c Set((F \star F) c, F c)$

Expanding the definition of Day convolution, we get:

$\int_c Set(\int^{a b} C(a \otimes b, c) \times F a \times F b, F c)$

The next trick is to pull the coend out of the hom-set. This trick relies on the co-continuity of the hom-functor in the first argument: a hom-functor from a colimit is isomorphic to a limit of hom-functors. In programmer-speak: a function from a sum type is equivalent to a product of functions (we call it case analysis). A coend is a generalized colimit, so when we pull it out of a hom-functor, it turns into a limit, or an end. Here’s the general formula, in which $p x y$ is an arbitrary profunctor:

$Set(\int^x p x x, y) \cong \int_x Set(p x x, y)$

Let’s apply it to our formula:

$\int_c \int_{a b} Set(C(a \otimes b, c) \times F a \times F b, F c)$

We can combine the ends under one integral sign (it’s allowed by the Fubini theorem) and move to the next trick: hom-set adjunction:

$Set(a \times b, c) \cong Set(a, b \to c)$

In programming this is known as currying. This adjunction exists because $Set$ is a cartesian closed category. We’ll use this adjunction to move $F a \times F b$ to the right:

$\int_{a b c} Set(C(a \otimes b, c), (F a \times F b) \to F c)$

Using the Yoneda lemma we can “perform the integration” over $c$  to get:

$\int_{a b} (F a \times F b) \to F (a \otimes b))$

This is exactly the set of natural transformations used in the definition of a lax monoidal functor. We have established one-to-one correspondence between monoidal multiplication and lax monoidal mapping.

Of course, a complete proof would require translating monoid laws to their lax monoidal counterparts. You can find more details in Rivas and Jaskelioff, Notions of Computation as Monoids.

We’ll use the fact that a monoid in the category $[C, Set]$ is a lax monoidal functor later.

### Alternative Derivation

Incidentally, there are shorter derivations of these formulas that use the trick borrowed from the proof of the Yoneda lemma, namely, evaluating things at the identity morphism. (Whenever mathematicians speak of Yoneda-like arguments, this is what they mean.)

Starting from $F \star F \to F$ and plugging in the Day convolution formula, we get:

$\int^{a' b'} C(a' \otimes b', c) \times F a' \times F b' \to F c$

There is a component of this natural transformation at $(a \otimes b)$ that is the morphism:

$\int^{a' b'} C(a' \otimes b', a \otimes b) \times F a' \times F b' \to F (a \otimes b)$

This morphism must be defined for all possible values of the coend. In particular, it must be defined for the triple $(id_{a \otimes b}, F a, F b)$, giving us the $\mu$ we seek.

There is also an alternative derivation for the unit: Take the component of the natural transformation $\eta$ at $I$:

$\eta_I : C(I, I) \to L I$

$C(I, I)$ is guaranteed to contain at least one element, the identity morphism $id_I$. We can use $\eta_I \, id_I$ as the (only) value of the lax monoidal constraint at the singleton $1$.

## Free Monoid

Given a monoidal category $C$, we might be able to define a whole lot of monoids in it. These monoids form a category $Mon(C)$. Morphisms in this category correspond to those morphisms in $C$ that preserve monoidal structure.

Consider, for instance, two monoids $m$ and $m'$. A monoid morphism is a morphism $f : m \to m'$ in $C$ such that the unit of $m'$ is related to the unit of $m$:

$\eta' = f \circ \eta$

and similarly for multiplication:

$\mu' \circ (f \otimes f) = f \circ \mu$

Remember, we assumed that the tensor product is functorial in both arguments, so it can be used to lift a pair of morphisms.

There is an obvious forgetful functor $U$ from $Mon(C)$ to $C$ which, for every monoid, picks its underlying object in $C$ and maps every monoid morphism to its underlying morphism in $C$.

The left adjoint to this functor, if it exists, will map an object $a$ in $C$ to a free monoid $L a$.

The intuition is that a free monoid $L a$ is a list of $a$.

In Haskell, a list is defined recursively:

data List a = Nil | Cons a (List a)

Such a recursive definition can be formalized as a fixed point of a functor. For a list of a, this functor is:

data ListF a x = NilF | ConsF a x

Notice the peculiar structure of this functor. It’s a sum type: The first part is a singleton, which is isomorphic to the unit type (). The second part is a product of a and x. Since the unit type is the unit of the product in our monoidal category of types, we can rewrite this functor symbolically as:

$\Phi a x = I + a \otimes x$

It turns out that this formula works in any monoidal category that has finite coproducts (sums) that are preserved by the tensor product. The fixed point of this functor is the free functor that generates free monoids.

I’ll define what is meant by the fixed point and prove that it defines a monoid. The proof that it’s the result of a free/forgetful adjunction is a bit involved, so I’ll leave it for a future blog post.

### Algebras

Let’s consider algebras for the functor $F$. Such an algebra is defined as an object $x$ called the carrier, and a morphism:

$f : F x \to x$

called the structure map or the evaluator.

In Haskell, an algebra is defined as:

type Algebra f x = f x -> x

There may be a lot of algebras for a given functor. In fact there is a whole category of them. We define an algebra morphism between two algebras $(x, f : F x \to x)$ and $(x', f' : F x' \to x')$ as a morphism $\nu : x \to x'$ which commutes with the two structure maps:

$\nu \circ f = f' \circ F \nu$

The initial object in the category of algebras is called the initial algebra, or the fixed point of the functor that generates these algebras. As the initial object, it has a unique algebra morphism to any other algebra. This unique morphism is called a catamorphism.

In Haskell, the fixed point of a functor f is defined recursively:

newtype Fix f = In { out :: f (Fix f) }

with, for instance:

type List a = Fix (ListF a)

A catamorphism is defined as:

cata :: Functor f => Algebra f a -> Fix f -> a
cata alg = alg . fmap (cata alg) . out

A list catamorphism is called foldr.

We want to show that the initial algebra $L a$ of the functor:

$\Phi a x = I + a \otimes x$

is a free monoid. Let’s see under what conditions it is a monoid.

### Initial Algebra is a Monoid

In this section I will show you how to concatenate lists the hard way.

We know that function type $b \to c$ (a.k.a., the exponential $c^b$) is the right adjoint to the product:

$Set(a \times b, c) \cong Set(a, b \to c)$

The function type is also called the internal hom.

In a monoidal category it’s sometimes possible to define an internal hom-object, denoted $[b, c]$, as the right adjoint to the tensor product:

$curry : C(a \otimes b, c) \cong C(a, [b, c])$

If this adjoint exists, the category is called closed monoidal.

In a closed monoidal category, the initial algebra $L a$ of the functor $\Phi a x = I + a \otimes x$ is a monoid. (In particular, a Haskell list of a, which is a fixed point of ListF a, is a monoid.)

To show that, we have to construct two morphisms corresponding to unit and multiplication (in Haskell, empty list and concatenation):

$\eta : I \to L a$

$\mu : L a \otimes L a \to L a$

What we know is that $L a$ is a carrier of the initial algebra for $\Phi a$, so it is equipped with the structure map:

$I + a \otimes L a \to L a$

which is equivalent to a pair of morphisms:

$\alpha : I \to L a$

$\beta : a \otimes L a \to L a$

Notice that, in Haskell, these correspond the two list constructors: Nil and Cons or, in terms of the fixed point:

nil :: () -> List a
nil () = In NilF

cons :: a -> List a -> List a
cons a as = In (ConsF a as)

We can immediately use $\alpha$ to implement $\eta$.

The second one, $\beta$, one can be rewritten using the hom adjuncion as:

$\bar{\beta} = curry \, \beta$

$\bar{\beta} : a \to [L a, L a]$

Notice that, if we could prove that $[L a, L a]$ is a carrier for the same algebra generated by $\Phi a$, we would know that there is a unique catamorphism from the initial algebra $L a$:

$\kappa_{[L a, L a]} : L a \to [L a, L a]$

which, by the hom adjunction, would give us the desired multiplication:

$\mu : L a \otimes L a \to L a$

Let’s establish some useful lemmas first.

Lemma 1: For any object $x$ in a closed monoidal category, $[x, x]$ is a monoid.

This is a generalization of the idea that endomorphisms form a monoid, in which identity morphism is the unit and composition is multiplication. Here, the internal hom-object $[x, x]$ generalizes the set of endomorphisms.

Proof: The unit:

$\eta : I \to [x, x]$

follows, through adjunction, from the unit law in the monoidal category:

$\lambda : I \otimes x \to x$

(In Haskell, this is a fancy way of writing mempty = id.)

Multiplication takes the form:

$\mu : [x, x] \otimes [x, x] \to [x, x]$

which is reminiscent of composition of edomorphisms. In Haskell we would say:

mappend = (.)

$curry^{-1} \, \mu : [x, x] \otimes [x, x] \otimes x \to x$

We have at our disposal the counit $eval$ of the adjunction:

$eval : [x, x] \otimes x \cong x$

We can apply it twice to get:

$\mu = curry (eval \circ (id \otimes eval))$

In Haskell, we could express this as:

mu :: ((x -> x), (x -> x)) -> (x -> x)
mu (f, g) = \x -> f (g x)

Here, the counit of the adjunction turns into simple function application.

$\square$

Lemma 2: For every morphism $f : a \to m$, where $m$ is a monoid, we can construct an algebra of the functor $\Phi a$ with $m$ as its carrier.

Proof: Since $m$ is a monoid, we have two morphisms:

$\eta : I \to m$

$\mu : m \otimes m \to m$

To show that $m$ is a carrier of our algebra, we need two morphisms:

$\alpha : I \to m$

$\beta : a \otimes m \to m$

The first one is the same as $\eta$, the second can be implemented as:

$\beta = \mu \circ (f \otimes id)$

In Haskell, we would do case analysis:

mapAlg :: Monoid m => ListF a m -> m
mapAlg NilF = mempty
mapAlg (ConsF a m) = f a mappend m

$\square$

We can now build a larger proof. By lemma 1, $[L a, L a]$ is a monoid with:

$\mu = curry (eval \circ (id \otimes eval))$

We also have a morphism $\bar{\beta} : a \to [L a, L a]$ so, by lemma 2, $[L a, L a]$ is also a carrier for the algebra:

$\alpha = \eta$

$\beta = \mu \circ (\bar{\beta} \otimes id)$

It follows that there is a unique catamorphism $\kappa_{[L a, L a]}$ from the initial algebra $L a$ to it, and we know how to use it to implement monoidal multiplication for $L a$. Therefore, $L a$ is a monoid.

Translating this to Haskell, $\bar{\beta}$ is the curried form of Cons and what we have shown is that concatenation (multiplication of lists) can be implemented as a catamorphism:

concat :: List a -> List a -> List a
conc x y = cata alg x y
where alg NilF        = id
alg (ConsF a t) = (cons a) . t


The type:

List a -> (List a -> List a)

(parentheses added for emphasis) corresponds to $L a \to [L a, L a]$.

It’s interesting that concatenation can be described in terms of the monoid of list endomorphisms. Think of turning an element a of the list into a transformation, which prepends this element to its argument (that’s what $\bar{\beta}$ does). These transformations form a monoid. We have an algebra that turns the unit $I$ into an identity transformation on lists, and a pair $a \otimes t$ (where $t$ is a list transformation) into the composite $\bar{\beta} a \circ t$. The catamorphism for this algebra takes a list $L a$ and turns it into one composite list transformation. We then apply this transformation to another list and get the final result: the concatenation of two lists. $\square$

Incidentally, lemma 2 also works in reverse: If a monoid $m$ is a carrier of the algebra of $\Phi a$, then there is a morphism $f : a \to m$. This morphism can be thought of as inserting generators represented by $a$ into the monoid $m$.

Proof: if $m$ is both a monoid and a carrier for the algebra $\Phi a$, we can construct the morphism $a \to m$ by first applying the identity law to go from $a$ to $a \otimes I$, then apply $id_a \otimes \eta$ to get $a \otimes m$. This can be right-injected into the coproduct $I + a \otimes m$ and then evaluated down to $m$ using the structure map for the algebra on $m$.

$a \to a \otimes I \to a \otimes m \to I + a \otimes m \to m$

In Haskell, this corresponds to a construction and evaluation of:

ConsF a mempty

$\square$

## Free Monoidal Functor

Let’s go back to our functor category. We started with a monoidal category $C$ and considered a functor category $[C, Set]$. We have shown that $[C, Set]$ is itself a monoidal category with Day convolution as tensor product and the hom functor $C(I, -)$ as unit. A monoid is this category is a lax monoidal functor.

The next step is to build a free monoid in $[C, Set]$, which would give us a free lax monoidal functor. We have just seen such a construction in an arbitrary closed monoidal category. We just have to translate it to $[C, Set]$. We do this by replacing objects with functors and morphisms with natural transformations.

Our construction relied on defining an initial algebra for the functor:

$I + a \otimes b$

Straightforward translation of this formula to the functor category $[C, Set]$ produces a higher order endofunctor:

$A_F G = C(I, -) + F \star G$

It defines, for any functor $F$, a mapping from a functor $G$ to a functor $A_F G$. (It also maps natural transformations.)

We can now use $A_F$ to define (higher-order) algebras. An algebra consists of a carrier — here, a functor $T$ — and a structure map — here, a natural transformation:

$A_F T \to T$

The initial algebra for this higher-order endofunctor defines a monoid, and therefore a lax monoidal functor. We have shown it for an arbitrary closed monoidal category. So the only question is whether our functor category with Day convolution is closed.

We want to define the internal hom-object in $[C, Set]$ that satisfies the adjunction:

$[C, Set](F \star G, H) \cong [C, Set](F, [G, H])$

We start with the set of natural transformations — the hom-set in $[C, Set]$:

$[C, Set](F \star G, H)$

We rewrite it as an end over $c$, and use the formula for Day convolution:

$\int_c Set(\int^{a b} C(a \otimes b, c) \times F a \times G b, H c)$

We use the co-continuity trick to pull the coend out of the hom-set and turn it into an end:

$\int_{c a b} Set(C(a \otimes b, c) \times F a \times G b, H c)$

Keeping in mind that our goal is to end up with $F a$ on the left, we use the regular hom-set adjunction to shuffle the other two terms to the right:

$\int_{c a b} Set(F a, C(a \otimes b, c) \times G b \to H c)$

The hom-functor is continuous in the second argument, so we can sneak the end over $b c$ under it:

$\int_{a} Set(F a, \int_{b c} C(a \otimes b, c) \times G b \to H c)$

We end up with a set of natural transformations from the functor $F$ to the functor we will call:

$[G, H] = \int_{b c} (C(- \otimes b, c) \times G b \to H c)$

We therefore identify this functor as the right adjoint (internal hom-object) for Day convolution. We can further simplify it by using the hom-set adjunction:

$\int_{b c} (C(- \otimes b, c) \to (G b \to H c))$

and applying the Yoneda lemma to get:

$[G, H] = \int_{b} (G b \to H (- \otimes b))$

In Haskell, we would write it as:

newtype DayHom f g a = DayHom (forall b . f b -> g (a, b))

Since Day convolution has a right adjoint, we conclude that the fixed point of our higher order functor defines a free lax monoidal functor. We can write it in a recursive form as:

$Free_F = C(I, -) + F \star Free_F$

data FreeMonR f t =
Done t
| More (Day f (FreeMonR f) t)

This blog post wouldn’t be complete without mentioning that the same construction works for monads. Famously, a monad is a monoid in the category of endofunctors. Endofunctors form a monoidal category with functor composition as tensor product and the identity functor as unit. The fact that we can construct a free monad using the formula:

$FreeM_F = Id + F \circ FreeM_F$

is due to the observation that functor composition has a right adjoint, which is the right Kan extension. Unfortunately, due to size issues, this Kan extension doesn’t always exist. I’ll quote Alex Campbell here: “By making suitable size restrictions, we can give conditions for free monads to exist: for example, free monads exist for accessible endofunctors on locally presentable categories; a special case is that free monads exist for finitary endofunctors on $Set$, where finitary means the endofunctor preserves filtered colimits (more generally, an endofunctor is accessible if it preserves $\kappa$-filtered colimits for some regular cardinal number $\kappa$).”

## Conclusion

As we acquire experience in programming, we learn more tricks of trade. A seasoned programmer knows how to read a file, parse its contents, or sort an array. In category theory we use a different bag of tricks. We bunch morphisms into hom-sets, move ends and coends, use Yoneda to “integrate,” use adjunctions to shuffle things around, and use initial algebras to define recursive types.

Results derived in category theory can be translated to definitions of functions or data structures in programming languages. A lax monoidal functor becomes an Applicative. Free monoidal functor becomes:

data FreeMonR f t =
Done t
| More (Day f (FreeMonR f) t)

What’s more, since the derivation made very few assumptions about the category $C$ (other than that it’s monoidal), this result can be immediately applied to profunctors (replacing $C$ with $C^{op}\times C$) to produce:

data FreeMon p s t where
DoneFM :: t -> FreeMon p s t
MoreFM :: p a b -> FreeMon p c d ->
(b -> d -> t) ->
(s -> (a, c)) -> FreeMon p s t

Replacing Day convolution with endofunctor composition gives us a free monad:

data FreeMonadF f g a =
DoneFM a
| MoreFM (Compose f g a)

Category theory is also the source of laws (commuting diagrams) that can be used in equational reasoning to verify the correctness of programming constructs.

Writing this post has been a great learning experience. Every time I got stuck, I would ask Alex for help, and he would immediately come up with yet another algebra and yet another catamorphism. This was so different from the approach I would normally take, which would be to get bogged down in inductive proofs over recursive data structures.

Abstract: I derive free monoidal profunctors as fixed points of a higher order functor acting on profunctors. Monoidal profunctors play an important role in defining traversals.

The beauty of category theory is that it lets us reuse concepts at all levels. In my previous post I have derived a free monoidal functor that goes from a monoidal category $C$ to $Set$. The current post may then be shortened to: Since profunctors are just functors from $C^{op} \times C$ to $Set$, with the obvious monoidal structure induced by the tensor product in $C$, we automatically get free monoidal profunctors.

Let me fill in the details.

Here’s the definition of a profunctor from Data.Profunctor:

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

The idea is that, just like a functor acts on objects, a profunctor p acts on pairs of objects $\langle a, b \rangle$. In other words, it’s a type constructor that takes two types as arguments. And just like a functor acts on morphisms, a profunctor acts on pairs of morphisms. The only tricky part is that the first morphism of the pair is reversed: instead of going from $a$ to $s$, as one would expect, it goes from $s$ to $a$. This is why we say that the first argument comes from the opposite category $C^{op}$, where all morphisms are reversed with respect to $C$. Thus a morphism from $\langle a, b \rangle$ to $\langle s, t \rangle$ in $C^{op} \times C$ is a pair of morphisms $\langle s \to a, b \to t \rangle$.

Just like functors form a category, profunctors form a category too. In this category profunctors are objects, and natural transformations are morphisms. A natural transformation between two profunctors $p$ and $q$ is a family of functions which, in Haskell, can be approximated by a polymorphic function:

type p ::~> q = forall a b. p a b -> q a b

If the category $C$ is monoidal (has a tensor product $\otimes$ and a unit object $1$), then the category $C^{op} \times C$ has a trivially induced tensor product:

$\langle a, b \rangle \otimes \langle c, d \rangle = \langle a \otimes c, b \otimes d \rangle$

and unit $\langle 1, 1 \rangle$

In Haskell, we’ll use cartesian product (pair type) as the underlying tensor product, and () type as the unit.

Notice that the induced product does not have the usual exponential as the right adjoint. Indeed, the hom-set:

$(C^{op} \times C) \, ( \langle a, b \rangle \otimes \langle c, d \rangle, \langle s, t \rangle )$

is a set of pairs of morphisms:

$\langle s \to a \otimes c, b \otimes d \to t \rangle$

If the right adjoint existed, it would be a pair of objects $\langle X, Y \rangle$, such that the following hom-set would be isomorphic to the previous one:

$\langle X \to a, b \to Y \rangle$

While $Y$ could be the internal hom, there is no candidate for $X$ that would produce the isomorphism:

$s \to a \otimes c \cong X \to a$

(Consider, for instance, unit $()$ for $a$.) This lack of the right adjoint is the reason why we can’t define an analog of Applicative for profunctors. We can, however, define a monoidal profunctor:

class Monoidal p where
punit :: p () ()
(>**<) :: p a b -> p c d -> p (a, c) (b, d)

This profunctor is a map between two monoidal structures. For instance, punit can be seen as mapping the unit in $Set$ to the unit in $C^{op} \times C$:

punit :: () -> p <1, 1>

Operator >**< maps the product in $Set$ to the induced product in $C^{op} \times C$:

(>**<) :: (p <a, b>, p <c, d>) -> p (<a, b> × <c, d>)

Day convolution, which works with monoidal structures, generalizes naturally to the profunctor category:

data PDay p q s t = forall a b c d.
PDay (p a b) (q c d) ((b, d) -> t) (s -> (a, c))

## Higher Order Functors

Since profunctors form a category, we can define endofunctors in that category. This is a no-brainer in category theory, but it requires some new definitions in Haskell. Here’s a higher-order functor that maps a profunctor to another profunctor:

class HPFunctor pp where
hpmap :: (p ::~> q) -> (pp p ::~> pp q)
ddimap :: (s -> a) -> (b -> t) -> pp p a b -> pp p s t

The function hpmap lifts a natural transformation, and ddimap shows that the result of the mapping is also a profunctor.

An endofunctor in the profunctor category may have a fixed point:

newtype FixH pp a b = InH { outH :: pp (FixH pp) a b }

which is also a profunctor:

instance HPFunctor pp => Profunctor (FixH pp) where
dimap f g (InH pp) = InH (ddimap f g pp)

Finally, our Day convolution is a higher-order endofunctor in the category of profunctors:

instance HPFunctor (PDay p) where
hpmap nat (PDay p q from to) = PDay p (nat q) from to
ddimap f g (PDay p q from to) = PDay p q (g . from) (to . f)

We’ll use this fact to construct a free monoidal profunctor next.

## Free Monoidal Profunctor

In the previous post, I defined the free monoidal functor as a fixed point of the following endofunctor:

data FreeF f g t =
DoneF t
| MoreF (Day f g t)

Replacing the functors f and g with profunctors is straightforward:

data FreeP p q s t =
DoneP (s -> ()) (() -> t)
| MoreP (PDay p q s t)

The only tricky part is realizing that the first term in the sum comes from the unit of Day convolution, which is the type () -> t, and it generalizes to an appropriate pair of functions (we’ll simplify this definition later).

FreeP is a higher order endofunctor acting on profunctors:

instance HPFunctor (FreeP p) where
hpmap _ (DoneP su ut) = DoneP su ut
hpmap nat (MoreP day) = MoreP (hpmap nat day)
ddimap f g (DoneP au ub) = DoneP (au . f) (g . ub)
ddimap f g (MoreP day) = MoreP (ddimap f g day)

We can, therefore, define its fixed point:

type FreeMon p = FixH (FreeP p)

and show that it is indeed a monoidal profunctor. As before, the trick is to fist show the following property of Day convolution:

cons :: Monoidal q => PDay p q a b -> q c d -> PDay p q (a, c) (b, d)
cons (PDay pxy quv yva bxu) qcd =
PDay pxy (quv >**< qcd) (bimap yva id . reassoc)
(assoc . bimap bxu id)

where

assoc ((a,b),c) = (a,(b,c))
reassoc (a, (b, c)) = ((a, b), c)

Using this function, we can show that FreeMon p is monoidal for any p:

instance Profunctor p => Monoidal (FreeMon p) where
punit = InH (DoneP id id)
(InH (DoneP au ub)) >**< frcd = dimap snd (\d -> (ub (), d)) frcd
(InH (MoreP dayab)) >**< frcd = InH (MoreP (cons dayab frcd))

FreeMon can also be rewritten as a recursive data type:

data FreeMon p s t where
DoneFM :: t -> FreeMon p s t
MoreFM :: p a b -> FreeMon p c d ->
(b -> d -> t) ->
(s -> (a, c)) -> FreeMon p s t

## Categorical Picture

As I mentioned before, from the categorical point of view there isn’t much to talk about. We define a functor in the category of profunctors:

$A_p q = (C^{op} \times C) (1, -) + \int^{ a b c d } p a b \times q c d \times (C^{op} \times C) (\langle a, b \rangle \otimes \langle c, d \rangle, -)$

As previously shown in the general case, its initial algebra defines a free monoidal profunctor.

## Acknowledgments

I’m grateful to Eugenia Cheng not only for talking to me about monoidal profunctors, but also for getting me interested in category theory in the first place through her Catsters video series. Thanks also go to Edward Kmett for numerous discussions on this topic.

In my category theory blog posts, I stated many theorems, but I didn’t provide many proofs. In most cases, it’s enough to know that the proof exists. We trust mathematicians to do their job. Granted, when you’re a professional mathematician, you have to study proofs, because one day you’ll have to prove something, and it helps to know the tricks that other people used before you. But programmers are engineers, and are therefore used to taking advantage of existing solutions, trusting that somebody else made sure they were correct. So it would seem that mathematical proofs are irrelevant to programming. Or so it may seem, until you learn about the Curry-Howard isomorphism–or propositions as types, as it is sometimes called–which says that there is a one to one correspondence between logic and programs, and that every function can be seen as a proof of a theorem. And indeed, I found that a lot of proofs in category theory turn out to be recipes for implementing functions. In most cases the problem can be reduced to this: Given some morphisms, implement another morphism, usually using simple composition. This is very much like using point-free notation to define functions in Haskell. The other ingredient in categorical proofs is diagram chasing, which is very much like equational resoning in Haskell. Of course, mathematicians use different notation, and they use lots of different categories, but the principles are the same.

I want to illustrate these points with the example from Emily Riehl’s excellent book Category Theory in Context. It’s a book for mathematicians, so it’s not an easy read. I’m going to concentrate on theorem 6.2.1, which derives a formula for left Kan extensions using colimits. I picked this theorem because it has calculational content: it tells you how to calculate a particular functor.

It’s not a short proof, and I have made it even longer by unpacking every single step. These steps are not too hard, it’s mostly a matter of understanding and using definitions of functoriality, naturality, and universality.

There is a bonus at the end of this post for Haskell programmers.

## Kan Extensions

I wrote about Kan extensions before, so here I’ll only recap the definition using the notation from Emily’s book. Here’s the setup: We want to extend a functor $F$, which goes from category $C$ to $E$, along another functor $K$, which goes from $C$ to $D$. This extension is a new functor from $D$ to $E$.

To give you some intuition, imagine that the functor $F$ is the Rosetta Stone. It’s a functor that maps the Ancient Egyptian text of a royal decree to the same text written in Ancient Greek. The functor $K$ embeds the Rosetta Stone hieroglyphics into the know corpus of Egyptian texts from various papyri and inscriptions on the walls of temples. We want to extend the functor $F$ to the whole corpus. In other words, we want to translate new texts from Egyptian to Greek (or whatever other language that’s isomorphic to it).

In the ideal case, we would just want $F$ to be isomorphic to the composition of the new functor after $K$. That’s usually not possible, so we’ll settle for less. A Kan extension is a functor which, when composed with $K$ produces something that is related to $F$ through a natural transformation. In particular, the left Kan extension, $Lan_K F$, is equipped with a natural transformation $\eta$ from $F$ to $Lan_K F \circ K$.

(The right Kan extension has this natural transformation reversed.)

There are usually many such functors, so there is the standard trick of universal construction to pick the best one.

In our analogy, we would ideally like the new functor, when applied to the hieroglyphs from the Rosetta stone, to exactly reproduce the original translation, but we’ll settle for something that has the same meaning. We’ll try to translate new hieroglyphs by looking at their relationship with known hieroglyphs. That means we should look closely at morphism in $D$.

## Comma Category

The key to understanding how Kan extensions work is to realize that, in general, the functor $K$ embeds $C$ in $D$ in a lossy way.

There may be objects (and morphisms) in $D$ that are not in the image of $K$. We have to somehow define the action of $Lan_K F$ on those objects. What do we know about such objects?

We know from the Yoneda lemma that all the information about an object is encoded in the totality of morphisms incoming to or outgoing from that object. We can summarize this information in the form of a functor, the hom-functor. Or we can define a category based on this information. This category is called the slice category. Its objects are morphisms from the original category. Notice that this is different from Yoneda, where we talked about sets of morphisms — the hom-sets. Here we treat individual morphisms as objects.

This is the definition: Given a category $C$ and a fixed object $c$ in it, the slice category $C/c$ has as objects pairs $(x, f)$, where $x$ is an object of $C$ and $f$ is a morphism from $x$ to $c$. In other words, all the arrows whose codomain is $c$ become objects in $C/c$.

A morphism in $C/c$ between two objects, $(x, f)$ and $(y, g)$ is a morphism $h : x \to y$ in $C$ that makes the following triangle commute:

In our case, we are interested in an object $d$ in $D$, and the slice category $D/d$ describes it in terms of morphisms. Think of this category as a holographic picture of $d$.

But what we are really interested in, is how to transfer this information about $d$ to $E$. We do have a functor $F$, which goes from $C$ to $E$. We need to somehow back-propagate the information about $d$ to $C$ along $K$, and then use $F$ to move it to $E$.

So let’s try again. Instead of using all morphisms impinging on $d$, let’s only pick the ones that originate in the image of $K$, because only those can be back-propagated to $C$.

This gives us limited information about $d$, but it’ll have to do. We’ll just use a partial hologram of $d$. Instead of the slice category, we’ll use the comma category $K \downarrow d$.

Here’s the definition: Given a functor $K : C \to D$ and an object $d$ of $D$, the comma category $K \downarrow d$ has as objects pairs $(c, f)$, where $c$ is an object of $C$ and $f$ is a morphism from $K c$ to $d$. So, indeed, this category describes the totality of morphisms coming to $d$ from the image of $K$.

A morphism in the comma category from $(c, f)$ to $(c', g)$ is a morphism $h : c \to c'$ such that the following triangle commutes:

So how does back-propagation work? There is a projection functor $\Pi_d : K \downarrow d \to C$ that maps an object $(c, f)$ to $c$ (with obvious action on morphisms). This functor loses a lot of information about objects (completely forgets the $f$ part), but it keeps a lot of the information in morphisms — it only picks those morphisms in $C$ that preserve the structure of the comma category. And it lets us “upload” the hologram of $d$ into $C$

Next, we transport all this information to $E$ using $F$. We get a mapping

$F \circ \Pi_d : K \downarrow d \to E$

Here’s the main observation: We can look at this functor $F \circ \Pi_d$ as a diagram in $E$ (remember, when constructing limits, diagrams are defined through functors). It’s just a bunch of objects and morphisms in $E$ that somehow approximate the image of $d$. This holographic information was extracted by looking at morphisms impinging on $d$. In our search for $(Lan_K F) d$ we should therefore look for an object in $E$ together with morphisms impinging on it from the diagram we’ve just constructed. In particular, we could look at cones under that diagram (or co-cones, as they are sometimes called). The best such cone defines a colimit. If that colimit exists, it is indeed the left Kan extension $(Lan_K F) d$. That’s the theorem we are going to prove.

To prove it, we’ll have to go through several steps:

1. Show that the mapping we have just defined on objects is indeed a functor, that is, we’ll have to define its action on morphisms.
2. Construct the transformation $\eta$ from $F$ to $Lan_K F \circ K$ and show the naturality condition.
3. Prove universality: For any other functor $G$ together with a natural transformation $\gamma$, show that $\gamma$ uniquely factorizes through $\eta$.

All of these things can be shown using clever manipulations of cones and the universality of the colimit. Let’s get to work.

## Functoriality

We have defined the action of $Lan_K F$ on objects of $D$. Let’s pick a morphism $g : d \to d'$. Just like $d$, the object $d'$ defines its own comma category $K \downarrow d'$, its own projection $\Pi_{d'}$, its own diagram $F \circ \Pi_{d'}$, and its own limiting cone. Parts of this new cone, however, can be reinterpreted as a cone for the old object $d$. That’s because, surprisingly, the diagram $F \circ \Pi_{d'}$ contains the diagram $F \circ \Pi_{d}$.

Take, for instance, an object $(c, f) : K \downarrow d$, with $f: K c \to d$. There is a corresponding object $(c, g \circ f)$ in $K \downarrow d'$. Both get projected down to the same $c$. That shows that every object in the diagram for the $d$ cone is automatically an object in the diagram for the $d'$ cone.

Now take a morphism $h$ from $(c, f)$ to $(c', f')$ in $K \downarrow d$. It is also a morphism in $K \downarrow d'$ between $(c, g \circ f)$ and $(c', g \circ f')$. The commuting triangle condition in $K \downarrow d$

$f = f' \circ K h$

ensures the commuting condition in $K \downarrow d'$

$g \circ f = g \circ f' \circ K h$

All this shows that the new cone that defines the colimit of $F \circ \Pi_{d'}$ contains a cone under $F \circ \Pi_{d}$.

But that diagram has its own colimit $(Lan_K F) d$. Because that colimit is universal, there must be a unique morphism from $(Lan_K F) d$ to $(Lan_K F) d'$, which makes all triangles between the two cones commute. We pick this morphism as the lifting of our $g$, which ensures the functoriality of $Lan_K F$.

The commuting condition will come in handy later, so let’s spell it out. For every object $(c, f:Kc \to d)$ we have a leg of the cone, a morphism $V_d^{(c, f)}$ from $F c$ to $(Lan_K F) d$; and another leg of the cone $V_{d'}^{(c, g \circ f)}$ from $F c$ to $(Lan_K F) d'$. If we denote the lifting of $g$ as $(Lan_K F) g$ then the commuting triangle is:

$(Lan_K F) g \circ V_{d}^{(c, f)} = V_{d'}^{(c, g \circ f)}$

In principle, we should also check that this newly defined functor preserves composition and identity, but this pretty much follows automatically whenever lifting is defined using composition of morphisms, which is indeed the case here.

## Natural Transformation

We want to show that there is a natural transformation $\eta$ from $F$ to $Lan_K F \circ K$. As usual, we’ll define the component of this natural transformation at some arbitrary object $c$. It’s a morphism between $F c$ and $(Lan_K F) (K c)$. We have lots of morphisms at our disposal, with all those cones lying around, so it shouldn’t be a problem.

First, observe that, because of the pre-composition with $K$, we are only looking at the action of $Lan_K F$ on objects which are inside the image of $K$.

The objects of the corresponding comma category $K \downarrow K c'$ have the form $(c, f)$, where $f : K c \to K c'$. In particular, we can pick $c' = c$, and $f = id_{K c}$, which will give us one particular vertex of the diagram $F \circ \Pi_{K c}$. The object at this vertex is $F c$ — exactly what we need as a starting point for our natural transformation. The leg of the colimiting cone we are interested in is:

$V_{K c}^{(c, id)} : F c \to (Lan_K F) (K c)$

We’ll pick this leg as the component of our natural transformation $\eta_c$.

What remains is to show the naturality condition. We pick a morphism $h : c \to c'$. We know how to lift this morphism using $F$ and using $Lan_K F \circ K$. The other two sides of the naturality square are $\eta_c$ and $\eta_{c'}$.

The bottom left composition is $(Lan_K F) (K h) \circ V_{K c}^{(c, id)}$. Let’s take the commuting triangle that we used to show the functoriality of $Lan_K F$:

$(Lan_K F) g \circ V_{d}^{(c, f)} = V_{d'}^{(c, g \circ f)}$

and replace $f$ by $id_{K c}$, $g$ by $K h$, $d$ by $K c$, and $d'$ by $K c'$, to get:

$(Lan_K F) (K h) \circ V_{K c}^{(c, id)} = V_{K c'}^{(c, K h)}$

Let’s draw this as the diagonal in the naturality square, and examine the upper right composition:

$V_{K c'}^{(c', id)} \circ F h$.

This is also equal to the diagonal $V_{K c'}^{(c, K h)}$. That’s because these are two legs of the same colimiting cone corresponding to $(c', id_{K c'})$ and $(c, K h)$. These vertices are connected by $h$ in $K \downarrow K c'$.

But how do we know that $h$ is a morphism in $K \downarrow K c'$? Not every morphism in $C$ is a morphism in the comma category. In this case, however, the triangle identity is automatic, because one of its sides is an identity $id_{K c'}$.

We have shown that our naturality square is composed of two commuting triangles, with $V_{K c'}^{(c, K h)}$ as its diagonal, and therefore commutes as a whole.

## Universality

Kan extension is defined using a universal construction: it’s the best of all functors that extend $F$ along $K$. It means that any other extension will factor through ours. More precisely, if there is another functor $G$ and another natural transformation:

$\gamma : F \to G \circ K$

then there is a unique natural transformation $\alpha$, such that

$(\alpha \cdot K) \circ \eta = \gamma$

(where we have a horizontal composition of a natural transformation $\alpha$ with the functor $K$)

As always, we start by clearly stating the goals. The proof proceeds in these steps:

1. Implement $\alpha$.
2. Prove that it’s natural.
3. Show that it factorizes $\gamma$ through $\eta$.
4. Show that it’s unique.

We are defining a natural transformation $\alpha$ between two functors: $Lan_K F$, which we have defined as a colimit, and $G$. Both are functors from $D$ to $E$. Let’s implement the component of $\alpha$ at some $d$. It must be a morphism:

$\alpha_d : (Lan_K F) d \to G d$

Notice that $d$ varies over all of $D$, not just the image of $K$. We are comparing the two extensions where it really matters.

We have at our disposal the natural transformation:

$\gamma : F \to G \circ K$

or, in components:

$\gamma_c : F c \to G (K c)$

More importantly, though, we have the universal property of the colimit. If we can construct a cone with the nadir at $G d$ then we can use its factorizing morphism to define $\alpha_d$.

This cone has to be built under the same diagram as $(Lan_K F) d$. So let’s start with some $(c, f : K c \to d)$. We want to construct a leg of our new cone going from $F c$ to $G d$. We can use $\gamma_c$ to get to $G (K c)$ and then hop to $G d$ using $G f$. Thus we can define the new cone’s leg as:

$W_c = G f \circ \gamma_c$

Let’s make sure that this is really a cone, meaning, its sides must be commuting triangles.

Let’s pick a morphism $h$ in $K \downarrow d$ from $(c, f)$ to $(c', f')$. A morphism in $K \downarrow d$ must satisfy the triangle condition, $f = f' \circ K h$:

We can lift this triangle using $G$:

$G f = G f' \circ G (K h)$

Naturality condition for $\gamma$ tells us that:

$\gamma_{c'} \circ F h = G (K h) \circ \gamma_c$

By combining the two, we get the pentagon:

whose outline forms a commuting triangle:

$G f \circ \gamma_c = G f' \circ \gamma_{c'} \circ F h$

Now that we have a cone with the nadir at $G d$, universality of the colimit tells us that there is a unique morphism from the colimiting cone to it that factorizes all triangles between the two cones. We make this morphism our $\alpha_d$. The commuting triangles are between the legs of the colimiting cone $V_d^{(c, f)}$ and the legs of our new cone $W_c$:

$\alpha_d \circ V_d^{(c, f)} = G f \circ \gamma_c$

Now we have to show that so defined $\alpha$ is a natural transformation. Let’s pick a morphism $g : d \to d'$. We can lift it using $Lan_K F$ or using $G$ in order to construct the naturality square:

$G g \circ \alpha_d = \alpha_{d'} \circ (Lan_K F) g$

Remember that the lifting of a morphism by $Lan_K F$ satisfies the following commuting condition:

$(Lan_K F) g \circ V_{d}^{(c, f)} = V_{d'}^{(c, g \circ f)}$

We can combine the two diagrams:

The two arms of the large triangle can be replaced using the diagram that defines $\alpha$, and we get:

$G (g \circ f) \circ \gamma_c = G g \circ G f \circ \gamma_c$

which commutes due to functoriality of $G$.

Now we have to show that $\alpha$ factorizes $\gamma$ through $\eta$. Both $\eta$ and $\gamma$ are natural transformations between functors going from $C$ to $E$, whereas $\alpha$ connects functors from $D$ to $E$. To extend $\alpha$, we horizontally compose it with the identity natural transformation from $K$ to $K$. This is called whiskering and is written as $\alpha \cdot K$. This becomes clear when expressed in components. Let’s pick an object $c$ in $C$. We want to show that:

$\gamma_c = \alpha_{K c} \circ \eta_c$

Let’s go back all the way to the definition of $\eta$:

$\eta_c = V_{K c}^{(c, id)}$

where $id$ is the identity morphism at $K c$. Compare this with the definition of $\alpha$:

$\alpha_d \circ V_d^{(c, f)} = G f \circ \gamma_c$

If we replace $f$ with $id$ and $d$ with $K c$, we get:

$\alpha_{K c} \circ V_{K c}^{(c, id)} = G id \circ \gamma_c$

which, due to functoriality of $G$ is exactly what we need:

$\alpha_{K c} \circ \eta_c = \gamma_c$

Finally, we have to prove the uniqueness of $\alpha$. Here’s the trick: assume that there is another natural transformation $\alpha'$ that factorizes $\gamma$. Let’s rewrite the naturality condition for $\alpha'$:

$G g \circ \alpha'_d = \alpha'_{d'} \circ (Lan_K F) g$

Replacing $g : d \to d'$ with $f : K c \to d$, we get:

$G f \circ \alpha'_{K c} = \alpha'_d \circ (Lan_K F) f$

The lifiting of $f$ by $Lan_K F$ satisfies the triangle identity:

$V_d^{(c, f)} = (Lan_K F) f \circ V_{K c}^{(c, id)}$

where we recognize $V_{K c}^{(c, id)}$ as $\eta_c$.

We said that $\alpha'$ factorizes $\gamma$ through $\eta$:

$\gamma_c = \alpha'_{K c} \circ \eta_c$

which let us straighten the left side of the pentagon.

This shows that $\alpha'$ is another factorization of the cone with the nadir at $G d$ through the colimit cone with the nadir $(Lan_K F) d$. But that would contradict the universality of the colimit, therefore $\alpha'$ must be the same as $\alpha$.

This completes the proof.

This post would not be complete if I didn’t mention a Haskell implementation of Kan extensions by Edward Kmett, which you can find in the library Data.Functor.Kan.Lan. At first look you might not recognize the definition given there:

data Lan g h a where
Lan :: (g b -> a) -> h b -> Lan g h a

To make it more in line with the previous discussion, let’s rename some variables:

data Lan k f d where
Lan :: (k c -> d) -> f c -> Lan k f d

This is an example of GADT syntax, which is a Haskell way of implementing existential types. It’s equivalent to the following pseudo-Haskell:

Lan k f d = exists c. (k c -> d, f c)

This is more like it: you may recognize (k c -> d) as an object of the comma category $K \downarrow d$, and f c as the mapping of $c$ (which is the projection of this object back to $C$) under the functor $F$. In fact, the Haskell representation is based on the encoding of the colimit using a coend:

$(Lan_K F) d = \int^{c \in C} C(K c, d) \times F c$

The Haskell library also contains the proof that Kan extension is a functor:

instance Functor (Lan k f) where
fmap g (Lan kcd fc) = Lan (g . kcd) fc

The natural transformation $\eta$ that is part of the definition of the Kan extension can be extracted from the Haskell definition:

eta :: f c -> Lan k f (k c)
eta = Lan id

In Haskell, we don’t have to prove naturality, as it is a consequence of parametricity.

The universality of the Kan extension is witnessed by the following function:

toLan :: Functor g => (forall c. f c -> g (k c)) -> Lan k f d -> g d
toLan gamma (Lan kcd fc) = fmap kcd (gamma fc)

It takes a natural transformation $\gamma$ from $F$ to $G \circ K$, and produces a natural transformation we called $\alpha$ from $Lan_K F$ to $G$.

This is $\gamma$ expressed as a composition of $\alpha$ and $\eta$:

fromLan :: (forall d. Lan k f d -> g d) -> f c -> g (k c)
fromLan alpha = alpha . eta

As an example of equational reasoning, let’s prove that $\alpha$ defined by toLan indeed factorizes $\gamma$. In other words, let’s prove that:

fromLan (toLan gamma) = gamma

Let’s plug the definition of toLan into the left hand side:

fromLan (\(Lan kcd fc) -> fmap kcd (gamma fc))

then use the definition of fromLan:

(\(Lan kcd fc) -> fmap kcd (gamma fc)) . eta

Let’s apply this to some arbitrary function g and expand eta:

(\(Lan kcd fc) -> fmap kcd (gamma fc)) (Lan id g)

Beta-reduction gives us:

fmap id (gamma g)

which is indeed equal to the right hand side acting on g:

gamma g

The proof of toLan . fromLan = id is left as an exercise to the reader (hint: you’ll have to use naturality).

## Acknowledgments

I’m grateful to Emily Riehl for reviewing the draft of this post and for writing the book from which I borrowed this proof.

La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.

― Galileo Galilei, Il Saggiatore (The Assayer)

Joan was quizzical; studied pataphysical science in the home. Late nights all alone with a test tube.

— The Beatles, Maxwell’s Silver Hammer

Unless you’re a member of the Flat Earth Society, I bet you’re pretty confident that the Earth is round. In fact, you’re so confident that you don’t even ask yourself the question why you are so confident. After all, there is overwhelming scientific evidence for the round-Earth hypothesis. There is the old “ships disappearing behind the horizon” proof, there are satellites circling the Earth, there are even photos of the Earth seen from the Moon, the list goes on and on. I picked this particular theory because it seems so obviously true. So if I try to convince you that the Earth is flat, I’ll have to dig very deep into the foundation of your belief systems. Here’s what I’ve found: We believe that the Earth is round not because it’s the truth, but because we are lazy and stingy (or, to give it a more positive spin, efficient and parsimonious). Let me explain…

## The New Flat Earth Theory

Let’s begin by stressing how useful the flat-Earth model is in everyday life. I use it all the time. When I want to find the nearest ATM or a gas station, I take out my cell phone and look it up on its flat screen. I’m not carrying a special spherical gadget in my pocket. The screen on my phone is not bulging in the slightest when it’s displaying a map of my surroundings. So, at least within the limits of my city, or even the state, flat-Earth theory works just fine, thank you!

I’d like to make parallels with another widely accepted theory, Einstein’s special relativity. We believe that it’s true, but we never use it in everyday life. The vast majority of objects around us move much slower than the speed of light, so traditional Newtonian mechanics works just fine for us. When was the last time you had to reset your watch after driving from one city to another to account for the effects of time dilation?

The point is that every physical theory is only valid within a certain range of parameters. Physicists have always been looking for the Holy Grail of theories — the theory of everything that would be valid for all values of parameters with no exceptions. They haven’t found one yet.

But, obviously, special relativity is better than Newtonian mechanics because it’s more general. You can derive Newtonian mechanics as a low velocity approximation to special relativity. And, sure enough, the flat-Earth theory is an approximation to the round-Earth theory for small distances. Or, equivalently, it’s the limit as the radius of the Earth goes to infinity.

But suppose that we were prohibited (for instance, by a religion or a government) from ever considering the curvature of the Earth. As explorers travel farther and farther, they discover that the “naive” flat-Earth theory gives incorrect answers. Unlike present-day flat-earthers, who are not scientifically sophisticated, they would actually put some effort to refine their calculations to account for the “anomalies.” For instance, they could postulate that, as you get away from the North Pole, which is the center of the flat Earth, something funny keeps happening to measuring rods. They get elongated when positioned along the parallels (the circles centered at the North Pole). The further away you get from the North Pole, the more they elongate, until at a certain distance they become infinite. Which means that the distances (measured using those measuring rods) along the big circles get smaller and smaller until they shrink to zero.

I know this theory sounds weird at first, but so does special and, even more so, general relativity. In special relativity, weird things happen when your speed is close to the speed of light. Time slows down, distances shrink in the direction of flight (but not perpendicular to it!), and masses increase. In general relativity, similar things happen when you get closer to a black hole’s event horizon. In both theories things diverge as you hit the limit — the speed of light, or the event horizon, respectively.

Back to flat Earth — our explorers conquer space. They have to extend their weird geometry to three dimensions. They find out that horizontally positioned measuring rods shrink as you go higher (they un-shrink when you point them vertically). The intrepid explorers also dig into the ground, and probe the depths with seismographs. They find another singularity at a particular depth, where the horizontal dilation of measuring rods reaches infinity (round-Earthers call this the center of the Earth).

This generalized flat-Earth theory actually works. I know that, because I have just described the spherical coordinate system. We use it when we talk about degrees of longitude and latitude. We just never think of measuring distances using spherical coordinates — it’s too much work, and we are lazy. But it’s possible to express the metric tensor in those coordinates. It’s not constant — it varies with position — and it’s not isotropic — distances vary with direction. In fact, because of that, flat Earthers would be better equipped to understand general relativity than we are.

So is the Earth flat or spherical? Actually it’s neither. Both theories are just approximations. In cartesian coordinates, the Earth is the shape of a flattened ellipsoid, but as you increase the resolution, you discover more and more anomalies (we call them mountains, canyons, etc.). In spherical coordinates, the Earth is flat, but again, only approximately. The biggest difference is that the math is harder in spherical coordinates.

Have I confused you enough? On one level, unless you’re an astronaut, your senses tell you that the Earth is flat. On the other level, unless you’re a conspiracy theorist who believes that NASA is involved in a scam of enormous proportions, you believe that the Earth is pretty much spherical. Now I’m telling you that there is a perfectly consistent mathematical model in which the Earth is flat. It’s not a cult, it’s science! So why do you feel that the round Earth theory is closer to the truth?

## The Occam’s Razor

The round Earth theory is just simpler. And for some reason we cling to the belief that nature abhors complexity (I know, isn’t it crazy?). We even express this belief as a principle called the Occam’s razor. In a nutshell, it says that:

Among competing hypotheses, the one with the fewest assumptions should be selected.

Notice that this is not a law of nature. It’s not even scientific: there is no way to falsify it. You can argue for the Occam’s razor on the grounds of theology (William of Ockham was a Franciscan friar) or esthetics (we like elegant theories), but ultimately it boils down to pragmatism: A simpler theory is easier to understand and use.

It’s a mistake to think that Occam’s razor tells us anything about the nature of things, whatever that means. It simply describes the limitations of our mind. It’s not nature that abhors complexity — it’s our brains that prefer simplicity.

Unless you believe that physical laws have an independent existence of their own.

## The Layered Cake Hypothesis

Scientists since Galileo have a picture of the Universe that consists of three layers. The top layer is nature that we observe and interact with. Below are laws of physics — the mechanisms that drive nature and make it predictable. Still below is mathematics — the language of physics (that’s what Galileo’s quote at the top of this post is about). According to this view, physics and mathematics are the hidden components of the Universe. They are the invisible cogwheels and pulleys whose existence we can only deduce indirectly. According to this view, we discover the laws of physics. We also discover mathematics.

Notice that this is very different from art. We don’t say that Beethoven discovered the Fifth Symphony (although Igor Stravinsky called it “inevitable”) or that Leonardo da Vinci discovered the Mona Lisa. The difference is that, had not Beethoven composed his symphony, nobody would; but if Cardano hadn’t discovered complex numbers, somebody else probably would. In fact there were many cases of the same mathematical idea being discovered independently by more than one person. Does this prove that mathematical ideas exist the same way as, say, the moons of Jupiter?

Physical discoveries have a very different character than mathematical discoveries. Laws of physics are testable against physical reality. We perform experiments in the real world and if the results contradict a theory, we discard the theory. A mathematical theory, on the other hand, can only be tested against itself. We discard a theory when it leads to internal contradictions.

The belief that mathematics is discovered rather than invented has its roots in Platonism. When we say that the Earth is spherical, we are talking about the idea of a sphere. According to Plato, these ideas do exist independently of the observer — in this case, a mathematician who studies them. Most mathematicians are Platonists, whether they admit it or not.

Being able to formulate laws of physics in terms of simple mathematical equations is a thing of beauty and elegance. But you have to realize that history of physics is littered with carcasses of elegant theories. There was a very elegant theory, which postulated that all matter was made of just four elements: fire, air, water, and earth. The firmament was a collection of celestial spheres (spheres are so Platonic). Then the orbits of planets were supposed to be perfect circles — they weren’t. They aren’t even elliptical, if you study them close enough.

Celestial spheres. An elegant theory, slightly complicated by the need to introduce epicycles to describe the movements of planets

## The Impass

But maybe at the level of elementary particles and quantum fields some of this presumed elegance of the Universe shines through? Well, not really. If the Universe obeyed the Occam’s razor, it would have stopped at two quarks, up and down. Nobody needs the strange and the charmed quarks, not to mention the bottom and the top quarks. The Standard Model of particle physics looks like a kitchen sink filled with dirty dishes. And then there is gravity that resists all attempts at grand unification. Strings were supposed to help but they turned out to be as messy as the rest of it.

Of course the current state of impasse in physics might be temporary. After all we’ve been making tremendous progress up until about the second half of the twentieth century (the most recent major theoretical breakthroughs were the discovery of the Higgs mechanism in 1964 and the proof or renormalizability of the Standard Model in 1971).

On the other hand, it’s possible that we might be reaching the limits of human capacity to understand the Universe. After all, there is no reason to believe that the structure of the Universe is simple enough for the human brain to analyze. There is no guarantee that it can be translated into the language of physics and mathematics.

## Is the Universe Knowable?

In fact, if you think about it, our expectation that the Universe is knowable is quite arbitrary. On the one hand you have the vast complex Universe, on the other hand you have slightly evolved monkey brains that have only recently figured out how to use tools and communicate using speech. The idea that these brains could produce and store a model of the Universe is preposterous. Granted, our monkey brains are a product of evolution, and our survival depends on those brains being able to come up with workable models of our environment. These models, however, do not include the microcosm or the macrocosm — just the narrow band of phenomena in between. Our senses can perceive space and time scales within about 8 orders of magnitude. For comparison, the Universe is about 40 orders of magnitude larger than the size of the atomic nucleus (not to mention another 20 orders of magnitude down to Planck length).

The evolution came up with an ingenious scheme to deal with the complexities of our environment. Since it is impossible to store all information about the Universe in the very limited amount of memory at our disposal, and it’s impossible to run the simulation in real time, we have settled for the next best thing: creating simplified partial models that are composable.

The idea is that, in order to predict the trajectory of a spear thrown at a mammoth, it’s enough to roughly estimate the influence of a constant downward pull of gravity and the atmospheric drag on the idealized projectile. It is perfectly safe to ignore a lot of subtle effects: the non-uniformity of the gravitational field, air-density fluctuations, imperfections of the spear, not to mention relativistic effects or quantum corrections.

And this is the key to understanding our strategy: we build a simple model and then calculate corrections to it. The idea is that corrections are small enough as not to destroy the premise of the model.

## Celestial Mechanics

A great example of this is celestial mechanics. To the lowest approximation, the planets revolve around the Sun along elliptical orbits. The ellipse is a solution of the one body problem in a central gravitational field of the Sun; or a two body problem, if you also take into account the tiny orbit of the Sun. But planets also interact with each other — in particular the heaviest one, Jupiter, influences the orbits of other planets. We can treat these interactions as corrections to the original solution. The more corrections we add, the better predictions we can make. Astronomers came up with some ingenious numerical methods to make such calculations possible. And yet it’s known that, in the long run, this procedure fails miserably. That’s because even the tiniest of corrections may lead to a complete change of behavior in the far future. This is the property of chaotic systems, our Solar System being just one example of such. You must have heard of the butterfly effect — the Universe is filled with this kind of butterflies.

Ephemerides: Tables showing positions of planets on the firmament.

## The Microcosm

Anyone who is not shocked by quantum
theory has not understood a single word.

— Niels Bohr

At the other end of the spectrum we have atoms and elementary particles. We call them particles because, to the lowest approximation, they behave like particles. You might have seen traces made by particles in a bubble chamber.

Elementary particles might, at first sight, exhibit some properties of macroscopic objects. They follow paths through the bubble chamber. A rock thrown in the air also follows a path — so elementary particles can’t be much different from little rocks. This kind of thinking led to the first model of the atom as a miniature planetary system. As it turned out, elementary particles are nothing like little rocks. So maybe they are like waves on a lake? But waves are continuous and particles can be counted by Geiger counters. We would like elementary particles to either behave like particles or like waves but, despite our best efforts, they refuse to nicely fall into one of the categories.

There is a good reason why we favor particle and wave explanations: they are composable. A two-particle system is a composition of two one-particle systems. A complex wave can be decomposed into a superposition of simpler waves. A quantum system is neither. We might try to separate a two-particle system into its individual constituents, but then we have to introduce spooky action at a distance to explain quantum entanglement. A quantum system is an alien entity that does not fit our preconceived notions, and the main characteristic that distinguishes it from classical phenomena is that it’s not composable. If quantum phenomena were composable in some other way, different from particles or waves, we could probably internalize it. But non-composable phenomena are totally alien to our way of thinking. You might think that physicists have some deeper insight into quantum mechanics, but they don’t. Richard Feynman, who was a no-nonsense physicist, famously said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” The problem with understanding quantum mechanics is not that it’s too complex. The problem is that our brains can only deal with concepts that are composable.

It’s interesting to notice that by accepting quantum mechanics we gave up on composability on one level in order to decompose something at another level. The periodic table of elements was the big challenge at the beginning of the 20th century. We already knew that earth, water, air, and fire were not enough. We understood that chemical compounds were combinations of atoms; but there were just too many kinds of atoms, and they could be grouped into families that shared similar properties. Atom was supposed to be indivisible (the Greek word ἄτομος [átomos] means indivisible), but we could not explain the periodic table without assuming that there was some underlying structure. And indeed, there is structure there, but the way the nucleus and the electrons compose in order to form an atom is far from trivial. Electrons are not like planets orbiting the nucleus. They form shells and orbitals. We had to wait for quantum mechanics and the Fermi exclusion principle to describe the structure of an atom.

Every time we explain one level of complexity by decomposing it in terms of simpler constituents we seem to trade off some of the simplicity of the composition itself. This happened again in the sixties, when physicists were faced with a confusing zoo of elementary particles. It seemed like there were hundreds of strongly interacting particles, hadrons, and every year was bringing new discoveries. This mess was finally cleaned up by the introduction of quarks. It was possible to categorize all hadrons as composed of just six types of quarks. This simplification didn’t come without a price, though. When we say an atom is composed of the nucleus and electrons, we can prove it by knocking off a few electrons and studying them as independent particles. We can even split the nucleus into protons and neutrons, although the neutrons outside of a nucleus are short lived. But no matter how hard we try, we cannot split a proton into its constituent quarks. In fact we know that quarks cannot exist outside of hadrons. This is called quark- or color-confinement. Quarks are supposed to come in three “colors,” but the only composites we can observe are colorless. We have stretched the idea of composition by accepting the fact that a composite structure can never be decomposed into its constituents.

## I’m Slightly Perturbed

How do physicists deal with quantum mechanics? They use mathematics. Richard Feynman came up with ingenious ways to perform calculations in quantum electrodynamics using perturbation theory. The idea of perturbation theory is that you start with the simple approximation and keep adding corrections to it, just like with celestial mechanics. The terms in the expansion can be visualized as Feynman diagrams. For instance, the lowest term in the interaction between two electrons corresponds to a diagram in which the electrons exchange a virtual photon.

This terms gives the classical repulsive force between two charged particles. The first quantum correction to it involves the exchange of two virtual photons. And here’s the kicker: this correction is not only larger than the original term — it’s infinite! So much for small corrections. Yes, there are tricks to shove this infinity under the carpet, but everybody who’s not fooling themselves understands that the so called renormalization is an ugly hack. We don’t understand what the world looks like at very small scales and we try to ignore it using tricks that make mathematicians faint.

Physicists are very pragmatic. As long as there is a recipe for obtaining results that can be compared with the experiment, they are happy with a theory. In this respect, the Standard Model is the most successful theory in the Universe. It’s a unified quantum field theory of electromagnetism, strong, and weak interactions that produces results that are in perfect agreement with all high-energy experiments we were able to perform to this day. Unfortunately, the Standard Model does not give us the understanding of what’s happening. It’s as if physicists were given an alien cell phone and figured out how to use various applications on it but have no idea about the internal workings of the gadget. And that’s even before we try to involve gravity in the model.

The “periodic table” of elementary particles.

The prevailing wisdom is that these are just little setbacks on the way toward the ultimate theory of everything. We just have to figure out the correct math. It may take us twenty years, or two hundred years, but we’ll get there. The hope that math is the answer led theoretical physicists to study more and more esoteric corners of mathematics and to contribute to its development. One of the most prominent theoretical physicists, Edward Witten, the father of M-theory that unified a number of string theories, was awarded the prestigious Fields Medal for his contribution to mathematics (Nobel prizes are only awarded when a theory is confirmed by experiment which, in the case of string theory, may be a be long way off, if ever).

If mathematics is discoverable, then we might indeed be able to find the right combination of math and physics to unlock the secrets of the Universe. That would be extremely lucky, though.

There is one property of all of mathematics that is really striking, and it’s most clearly visible in foundational theories, such as logic, category theory, and lambda calculus. All these theories are about composability. They all describe how to construct more complex things from simpler elements. Logic is about combining simple predicates using conjunctions, disjunctions, and implications. Category theory starts by defining a composition of arrows. It then introduces ways of combining objects using products, coproducts, and exponentials. Typed lambda calculus, the foundation of computer languages, shows us how to define new types using product types, sum types, and functions. In fact it can be shown that constructive logic, cartesian closed categories, and typed lambda calculus are three different formulations of the same theory. This is known as the Curry Howard Lambek isomorphism. We’ve been discovering the same thing over and over again.

It turns out that most mathematical theories have a skeleton that can be captured by category theory. This should not be a surprise considering how the biggest revolutions in mathematics were the result of realization that two or more disciplines were closely related to each other. The latest such breakthrough was the proof of the Fermat’s last theorem. This proof was based on the Taniyama-Shimura conjecture that related the study of elliptic curves to modular forms — two radically different branches of mathematics.

Earlier, geometry was turned upside down when it became obvious that one can define shapes using algebraic equations in cartesian coordinates. This retooling of geometry turned out to be very advantageous, because algebra has better compositional qualities than Euclidean-style geometry.

Finally, any mathematical theory starts with a set of axioms, which are combined using proof systems to produce theorems. Proof systems are compositional which, again, supports the view that mathematics is all about composition. But even there we hit a snag when we tried to decompose the space of all statements into true and false. Gödel has shown that, in any non-trivial theory, we can formulate a statement that can neither be proved to be right or wrong, and thus the Hilbert’s great project of defining one grand mathematical theory fell apart. It’s as if we have discovered that the Lego blocks we were playing with were not part of a giant Lego spaceship.

## Where Does Composability Come From?

It’s possible that composability is the fundamental property of the Universe, which would make it comprehensible to us humans, and it would validate our physics and mathematics. Personally, I’m very reluctant to accept this point of view, because it would give intelligent life a special place in the grand scheme of things. It’s as if the laws of the Universe were created in such a way as to be accessible to the brains of the evolved monkeys that we are.

It’s much more likely that mathematics describes the ways our brains are capable of composing simpler things into more complex systems. Anything that we can comprehend using our brains must, by necessity, be decomposable — and there are only so many ways of putting things together. Discovering mathematics means discovering the structure of our brains. Platonic ideals exist only as patterns of connections between neurons.

The amazing scientific progress that humanity has been able to make to this day was possible because there were so many decomposable phenomena available to us. Granted, as we progressed, we had to come up with more elaborate composition schemes. We have discovered differential equations, Hilbert spaces, path integrals, Lie groups, tensor calculus, fiber bundles, etc. With the combination of physics and mathematics we have tapped into a gold vein of composable phenomena. But research takes more and more resources as we progress, and it’s possible that we have reached the bedrock that may be resistant to our tools.

We have to seriously consider the possibility that there is a major incompatibility between the complexity of the Universe and the simplicity of our brains. We are not without recourse, though. We have at our disposal tools that multiply the power of our brains. The first such tool is language, which helps us combine brain powers of large groups of people. The invention of the printing press and then the internet helped us record and gain access to vast stores of information that’s been gathered by the combined forces of teams of researchers over long periods of time. But even though this is quantitative improvement, the processing of this information still relies on composition because it has to be presented to human brains. The fact that work can be divided among members of larger teams is proof of its decomposability. This is also why we sometimes need a genius to make a major breakthrough, when a task cannot be easily decomposed into smaller, easier, subtasks. But even genius has to start somewhere, and the ability to stand on the shoulders of giants is predicated on decomposability.

## Can Computers Help?

The role of computers in doing science is steadily increasing. To begin with, once we have a scientific theory, we can write computer programs to perform calculations. Nobody calculates the orbits of planets by hand any more — computers can do it much faster and error free. We are also beginning to use computers to prove mathematical theorems. The four-color problem is an example of a proof that would be impossible without the help of computers. It was decomposable, but the number of special cases was well over a thousand (it was later reduced to 633 — still too many, even for a dedicated team of graduate students).

Every planar map can be colored using only four colors.

Computer programs that are used in theorem proving provide a level of indirection between the mind of a scientist and formal manipulations necessary to prove a theorem. A programmer is still in control, and the problem is decomposable, but the number of components may be much larger, often too large for a human to go over one by one. The combined forces of humans and computers can stretch the limits of composability.

But how can we tackle problems that cannot be decomposed? First, let’s observe that in real life we rarely bother to go through the process of detailed analysis. In fact the survival of our ancestors depended on the ability to react quickly to changing circumstances, to make instantaneous decisions. When you see a tiger, you don’t decompose the image into individual parts, analyze them, and put together a model of a tiger. Image recognition is one of these areas where the analytic approach fails miserably. People tried to write programs that would recognize faces using separate subroutines to detect eyes, noses, lips, ears, etc., and composing them together, but they failed. And yet we instinctively recognize faces of familiar people at a glance.

## Neural Networks and the AI

We are now able to teach computers to classify images and recognize faces. We do it not by designing dedicated algorithms; we do it by training artificial neural networks. A neural network doesn’t start with a subsystem for recognizing eyes or noses. It’s possible that, in the process of training, it will develop the notions of lines, shadows, maybe even eyes and noses. But by no means is this necessary. Those abstractions, if they evolve, would be encoded in the connections between its neurons. We might even help the AI develop some general abstractions by tweaking its architecture. It’s common, for instance, to include convolutional layers to pre-process the input. Such a layer can be taught to recognize local features and compress the input to a more manageable size. This is very similar to how our own vision works: the retina in our eye does this kind of pre-processing before sending compressed signals through the optic nerve.

Compression is the key to matching the complexity of the task at hand to the simplicity of the system that is processing it. Just like our sensory organs and brains compress the inputs, so do neural networks. There are two kinds of compression: the kind that doesn’t lose any information, just removing the redundancy in the original signal; and the lossy kind that throws away irrelevant information. The task of deciding what information is irrelevant is in itself a process of discovery. The difference between the Earth and a sphere is the size of the Himalayas, but we ignore it when when we look at the globe. When calculating orbits around the Sun, we shrink all planets to points. That’s compression by elimination of details that we deem less important for the problem we are solving. In science, this kind of compression is called abstraction.

We are still way ahead of neural networks in our capacity to create abstractions. But it’s possible that, at some point, they’ll catch up with us. The problem is: Will we be able to understand machine-generated abstractions? We are already at the limits of understanding human-generated abstractions. You may count yourself a member of a very small club if you understand the statement “monad is a monoid in the category of endofunctors” that is chock full of mathematical abstractions. If neural networks come up with new abstractions/compression schemes, we might not be able to reverse engineer them. Unlike a human scientist, an AI is unlikely to be able to explain to us how it came up with a particular abstraction.

I’m not scared about a future AI trying to eliminate human kind (unless that’s what its design goals are). I’m afraid of the scenario in which we ask the AI a question like, “Can quantum mechanics be unified with gravity?” and it will answer, “Yes, but I can’t explain it to you, because you don’t have the brain capacity to understand the explanation.”

And this is the optimistic scenario. It assumes that such questions can be answered within the decomposition/re-composition framework. That the Universe can be decomposed into particles, waves, fields, strings, branes, and maybe some new abstractions that we haven’t even though about. We would at least get the satisfaction that we were on the right path but that the number of moving parts was simply too large for us to assimilate — just like with the proof of the four-color theorem.

But it’s possible that this reductionist scenario has its limits. That the complexity of the Universe is, at some level, irreducible and cannot be captured by human brains or even the most sophisticated AIs.

There are people who believe that we live in a computer simulation. But if the Universe is irreducible, it would mean that the smallest computer on which such a simulation could be run is the Universe itself, in which case it doesn’t make sense to call it a simulation.

## Conclusion

The scientific method has been tremendously successful in explaining the workings of our world. It led to exponential expansion of science and technology that started in the 19th century and continues to this day. We are so used to its successes that we are betting the future of humanity on it. Usually when somebody attacks the scientific method, they are coming from the background of obscurantism. Such attacks are easily rebuffed or dismissed. What I’m arguing is that science is not a property of the Universe, but rather a construct of our limited brains. We have developed some very sophisticated tools to create models of the Universe based on the principle of composition. Mathematics is the study of various ways of composing things and physics is applied composition. There is no guarantee, however, that the Universe is decomposable. Assuming that would be tantamount to postulating that its structure revolves around human brains, just like we used to believe that the Universe revolves around Earth.

You can also watch my talk on this subject.

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