## Flies

It was a hot evening and, as is usual at that time of the year, there were quite a few flies buzzing around us. Understandably, I was annoyed, as they were interfering with my meditation.

“Master,” I said, “You keep telling me that everything in this world has a purpose, but I can’t figure out the purpose of these flies. All they do is break my concentration. Can we move indoors already, behind the screens, so that we can continue the lessons in peace?”

The Master looked at me the way he usually does when I say something that shows my lack of understanding — which unfortunately happens a lot.

“The flies are here to teach us about meditation,” he said.

“How so?” I said. “Are you trying to tell me that I should be able to quiet my mind even when there’s constant interference?”

“That would be the ultimate goal,” said the Master, “but for now I’d like you to observe the way these flies move. Can you do that?”

“Of course, Master,” I said and started watching the flies criss-crossing the air in front of me.

“What do you see?” asked the Master after a while.

“I see them zig-zagging constantly. They never seem to fly in a straight line for longer than a fraction of a second.”

“You are very astute, my Disciple,” said the Master. “Now, why would you say they’re doing that?”

“I think they are doing that to avoid being caught” I said. “Those flies that were, long ago, flying in straight lines were eliminated by predators, and only those that employed more elaborate movement schemes survived long enough to produce offspring. Evolution in action!” I said not without some pride at my cleverness.

“Quite so, my Disciple,” said the Master, “quite so…”

“But suppose,” he continued, “that, in some other universe, there is a colony of flies that live confined within the boundaries a hostile environment. Their life is short and full of suffering. But there is a benevolent being that can set individual flies free, to live a happy and productive life. The trouble is that she has to catch them first. And, in the beginning, it was easy, since they were all flying in straight lines. Almost all. The benevolent being was able to remove the straight-flying flies and make them happy. But there remained a few flies that, for one reason or another, kept zig-zagging. They survived long enough to produce offspring, some of which also kept zig-zagging. Soon enough, all flies in that hostile and unhappy environment developed this new survival strategy that prevented them from escaping their horrible fate. That’s evolution in action, too.”

“That’s a pretty sad story,” I said. “It shows that evolution is a cruel mistress. It doesn’t care if we are happy or not, as long as we produce offspring.”

“But what does it have to do with meditation?” I said, a little confused.

“The flies are our thoughts,” said the Master.

## The Pipe

It was a pleasant evening and I was enjoying the warm breeze coming from the mountains bringing with it the smell of pine and something else.

“Why are you smoking a pipe, Master? It’s bad for your health.”

“This is not a pipe,” said the Master.

“This is most definitely a pipe. You must have bought it in a pipe shop,” I said.

“This is not a pipe,” said the Master.

I thought for a moment.

“Oh, I see. You are making reference to the famous painting by Rene Magritte, right? Ceci n’est pas une pipe! But what Magritte meant was that it wasn’t a pipe–it was a picture of a pipe. He played on our confusion between the object itself and its representation. But here you are holding an actual pipe.”

“What makes you think it’s a pipe?” asked the Master.

“Well, I can look up the definition of a pipe and you’ll see that it describes the object you are holding. Do you want me to do that?” I asked.

I pulled out my tablet and started tapping.

“That which cannot be named,” said the Master.

“Oh, I know that one,” I said and continued tapping. I stopped after a few tries and looked up.

“I tried Tao and Dao, upper- and lowercase, but it didn’t work. Is it ‘the Tao,’ with ‘the’?”

“You are much too clever, my Disciple,” said the Master.

“Oh, you mean it’s literally ‘that which cannot be named’?” I started tapping again.

“Okay, here it is. According to the OED, a pipe is…” I hesitated for a moment.

“Wait, you don’t really want me to read the definition,” I said.

“No,” said the Master.

After a moment of silence, I said:
“You have corrected me, Master, because by naming the pipe I focused on just one small aspect of it. Its relationship to other pipes. By doing that I ignored its relationship to you, to me, to our conversation, to this lovely sunset, to Magritte and–now I get it–to the Tao.”

“But, Master, I’m confused,” I said after a while. “When you say that ‘The Tao that can be named is not the eternal Tao,’ you are giving it a name, aren’t you? You are calling it the Tao.”

“This is not a name,” said the Master.

“I have the feeling that if I say that this is indeed a name, I would be hitting a dead end,” I said.

We sat there for a moment while I was organizing my thoughts. Then it occurred to me.

“By calling it the Tao, you are not separating it from everything else, because the Tao is in everything. Neither are you ignoring its relationship to yourself or to me, because you are the Tao and I am the Tao. And the lovely sunset, and the pipe, and Magritte, it’s all the eternal Tao.

“This is not the Tao,” said the Master.

## Fallout

“I watched a movie last night,” I said. “And it made me think.”

“Movies often make us think,” said the Master. “Good movies, like life itself, ask a lot of questions, but rarely provide answers.”

“Well, that’s the thing, Master” I said. “Maybe you know the answer to this question, or maybe you can steer me towards the answer. I’m sure this problem has been analyzed before by many people much wiser than yours truly.

“It’s a problem of moral nature. In the movie, agent Hunt faces a dilemma. His friend is in immediate mortal danger. Hunt can save him, but at the risk of endangering the lives of thousands of innocent people. He makes a choice, saves the friend but, in the process, the terrorists get hold of plutonium, which they use to make nuclear bombs. Of course, in the movie, he’s ultimately able to avert the disaster, disabling the bombs literally one second before they’re about to go off.

“Sorry if I spoiled the movie for you, Master.”

“Don’t worry, I’ve seen the movie,” said the Master.

“So what do you think, Master? Was agent Hunt acting recklessly, risking uncountable lives to save one?”

“I think the answer is clear. It’s just simple math: one life against thousands. I would probably feel guilty for the rest of my life for sacrificing a friend, but what right do I have to risk thousands of innocent lives?”

“You say it’s simple math,” said the Master. “I presume there is an equation that calculates the moral value of an act, based on the number of lives saved or lost.”

“It’s not an exact science, but I guess one could make some rough estimates,” I said. “I’ve read some articles that mostly deal with pulling levers to divert trolleys. So this seems like one of these problems, where your friend is tied up on one track, and thousands of people on another. A runaway trolley is going to kill your friend, and you pull the switch to divert it to the other track, possibly killing thousands of people.”

“If this is simple math, then why do you say you’d feel guilty? Shouldn’t you feel satisfied, like when you solve a difficult equation?”

“I don’t know. I think I would always speculate: What if? What if I saved my friend and, just like in the movie, were able to avert the disaster? I’d never know.”

“And what if you saved your friend’s life and the bomb exploded?” asked the Master.

“I guess I’d feel terrible for the rest of my life. And I would probably be the most despised person on Earth.”

“And what if that explosion prevented an even bigger disaster in the future?” asked the Master.

“And what if that bigger disaster prevented an even bigger disaster?” I asked. “Where does this end? Are you saying that, since we cannot predict the results of our actions on a global scale, then there is no moral imperative?”

“Would that satisfy you?” asked the Master.

“No, it wouldn’t!”

“Would you like to have a small set of simple rules to guide all moral decisions in your life?” asked the Master.

“When you put it this way, I’m not sure. I think there’s been many attempts at rule-based ethics, and they all have exhibited some pretty disastrous failure modes. It vaguely reminds me of the Goedel’s incompleteness theorem. No matter what moral axioms you choose, there will be a situation in which they fail.

“On the other hand, rejecting the axioms may lead to an even bigger tragedy, like in the case of Raskolnikov.”

“Do you see similarities between Raskolnikov and agent Hunt?” asked the Master.

“They both reject the ‘Thou shalt not kill’ commandment. They both feel intense loyalty to their friends and family. But Raskolnikov had a lot of time to think about his choices, he even published an article about it; whereas Hunt acted impulsively, following his gut feelings. One was rational, the other irrational.”

“But you said that Raskolnikov had no axioms,” said the Master. “So how could he rationally justify his actions?”

“I see your point,” I said. “He was trying to do the math. Solve the ethical equation. His hubris was not in rejecting the accepted axioms, but in believing that he can come up with a better set. So, in a way, agent Hunt had the advantage of being a moral simpleton.”

“He was the uncarved wood,” said the Master.

Yes, it’s this time of the year again! I started a little tradition a year ago with Stalking a Hylomorphism in the Wild. This year I was reminded of the Advent of Code by a tweet with this succint C++ program:

This piece of code is probably unreadable to a regular C++ programmer, but makes perfect sense to a Haskell programmer.

Here’s the description of the problem: You are given a list of equal-length strings. Every string is different, but two of these strings differ only by one character. Find these two strings and return their matching part. For instance, if the two strings were “abcd” and “abxd”, you would return “abd”.

What makes this problem particularly interesting is its potential application to a much more practical task of matching strands of DNA while looking for mutations. I decided to explore the problem a little beyond the brute force approach. And, of course, I had a hunch that I might encounter my favorite wild beast–the hylomorphism.

## Brute force approach

First things first. Let’s do the boring stuff: read the file and split it into lines, which are the strings we are supposed to process. So here it is:

main = do
let cs = lines txt
print $findMatch cs The real work is done by the function findMatch, which takes a list of strings and produces the answer, which is a single string. findMatch :: [String] -> String First, let’s define a function that calculates the distance between any two strings. distance :: (String, String) -> Int We’ll define the distance as the count of mismatched characters. Here’s the idea: We have to compare strings (which, let me remind you, are of equal length) character by character. Strings are lists of characters. The first step is to take two strings and zip them together, producing a list of pairs of characters. In fact we can combine the zipping with the next operation–in this case, comparison for inequality, (/=)–using the library function zipWith. However, zipWith is defined to act on two lists, and we will want it to act on a pair of lists–a subtle distinction, which can be easily overcome by applying uncurry: uncurry :: (a -> b -> c) -> ((a, b) -> c) which turns a function of two arguments into a function that takes a pair. Here’s how we use it: uncurry (zipWith (/=)) The comparison operator (/=) produces a Boolean result, True or False. We want to count the number of differences, so we’ll covert True to one, and False to zero: fromBool :: Num a => Bool -> a fromBool False = 0 fromBool True = 1 (Notice that such subtleties as the difference between Bool and Int are blisfully ignored in C++.) Finally, we’ll sum all the ones using sum. Altogether we have: distance = sum . fmap fromBool . uncurry (zipWith (/=))  Now that we know how to find the distance between any two strings, we’ll just apply it to all possible pairs of strings. To generate all pairs, we’ll use list comprehension: let ps = [(s1, s2) | s1 <- ss, s2 <- ss] (In C++ code, this was done by cartesian_product.) Our goal is to find the pair whose distance is exactly one. To this end, we’ll apply the appropriate filter: filter ((== 1) . distance) ps For our purposes, we’ll assume that there is exactly one such pair (if there isn’t one, we are willing to let the program fail with a fatal exception). (s, s') = head$ filter ((== 1) . distance) ps

The final step is to remove the mismatched character:

filter (uncurry (==)) $zip s s' We use our friend uncurry again, because the equality operator (==) expects two arguments, and we are calling it with a pair of arguments. The result of filtering is a list of identical pairs. We’ll fmap fst to pick the first components. findMatch :: [String] -> String findMatch ss = let ps = [(s1, s2) | s1 <- ss, s2 <- ss] (s, s') = head$ filter ((== 1) . distance) ps
in fmap fst $filter (uncurry (==))$ zip s s'

This program produces the correct result and we could stop right here. But that wouldn’t be much fun, would it? Besides, it’s possible that other algorithms could perform better, or be more flexible when applied to a more general problem.

## Data-driven approach

The main problem with our brute-force approach is that we are comparing everything with everything. As we increase the number of input strings, the number of comparisons grows like a factorial. There is a standard way of cutting down on the number of comparison: organizing the input into a neat data structure.

We are comparing strings, which are lists of characters, and list comparison is done recursively. Assume that you know that two strings share a prefix. Compare the next character. If it’s equal in both strings, recurse. If it’s not, we have a single character fault. The rest of the two strings must now match perfectly to be considered a solution. So the best data structure for this kind of algorithm should batch together strings with equal prefixes. Such a data structure is called a prefix tree, or a trie (pronounced try).

At every level of our prefix tree we’ll branch based on the current character (so the maximum branching factor is, in our case, 26). We’ll record the character, the count of strings that share the prefix that led us there, and the child trie storing all the suffixes.

data Trie = Trie [(Char, Int, Trie)]
deriving (Show, Eq)

Here’s an example of a trie that stores just two strings, “abcd” and “abxd”. It branches after b.

   a 2
b 2
c 1    x 1
d 1    d 1

When inserting a string into a trie, we recurse both on the characters of the string and the list of branches. When we find a branch with the matching character, we increment its count and insert the rest of the string into its child trie. If we run out of branches, we create a new one based on the current character, give it the count one, and the child trie with the rest of the string:

insertS :: Trie -> String -> Trie
insertS t "" = t
insertS (Trie bs) s = Trie (inS bs s)
where
inS ((x, n, t) : bs) (c : cs) =
if c == x
then (c, n + 1, insertS t cs) : bs
else (x, n, t) : inS bs (c : cs)
inS [] (c : cs) = [(c, 1, insertS (Trie []) cs)]

We convert our input to a trie by inserting all the strings into an (initially empty) trie:

mkTrie :: [String] -> Trie
mkTrie = foldl insertS (Trie [])

Of course, there are many optimizations we could use, if we were to run this algorithm on big data. For instance, we could compress the branches as is done in radix trees, or we could sort the branches alphabetically. I won’t do it here.

I won’t pretend that this implementation is simple and elegant. And it will get even worse before it gets better. The problem is that we are dealing explicitly with recursion in multiple dimensions. We recurse over the input string, the list of branches at each node, as well as the child trie. That’s a lot of recursion to keep track of–all at once.

Now brace yourself: We have to traverse the trie starting from the root. At every branch we check the prefix count: if it’s greater than one, we have more than one string going down, so we recurse into the child trie. But there is also another possibility: we can allow to have a mismatch at the current level. The current characters may be different but, since we allow only one mismatch, the rest of the strings have to match exactly. That’s what the function exact does. Notice that exact t is used inside foldMap, which is a version of fold that works on monoids–here, on strings.

match1 :: Trie -> [String]
match1 (Trie bs) = go bs
where
go :: [(Char, Int, Trie)] -> [String]
go ((x, n, t) : bs) =
let a1s = if n > 1
then fmap (x:) $match1 t else [] a2s = foldMap (exact t) bs a3s = go bs -- recurse over list in a1s ++ a2s ++ a3s go [] = [] exact t (_, _, t') = matchAll t t' Here’s the function that finds all exact matches between two tries. It does it by generating all pairs of branches in which top characters match, and then recursing down. matchAll :: Trie -> Trie -> [String] matchAll (Trie bs) (Trie bs') = mAll bs bs' where mAll :: [(Char, Int, Trie)] -> [(Char, Int, Trie)] -> [String] mAll [] [] = [""] mAll bs bs' = let ps = [ (c, t, t') | (c, _, t) <- bs , (c', _', t') <- bs' , c == c'] in foldMap go ps go (c, t, t') = fmap (c:) (matchAll t t') When mAll reaches the leaves of the trie, it returns a singleton list containing an empty string. Subsequent actions of fmap (c:) will prepend characters to this string. Since we are expecting exactly one solution to the problem, we’ll extract it using head: findMatch1 :: [String] -> String findMatch1 cs = head$ match1 (mkTrie cs)

## Recursion schemes

As you hone your functional programming skills, you realize that explicit recursion is to be avoided at all cost. There is a small number of recursive patterns that have been codified, and they can be used to solve the majority of recursion problems (for some categorical background, see F-Algebras). Recursion itself can be expressed in Haskell as a data structure: a fixed point of a functor:

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

In particular, our trie can be generated from the following functor:

data TrieF a = TrieF [(Char, a)]
deriving (Show, Functor)

Notice how I have replaced the recursive call to the Trie type constructor with the free type variable a. The functor in question defines the structure of a single node, leaving holes marked by the occurrences of a for the recursion. When these holes are filled with full blown tries, as in the definition of the fixed point, we recover the complete trie.

I have also made one more simplification by getting rid of the Int in every node. This is because, in the recursion scheme I’m going to use, the folding of the trie proceeds bottom-up, rather than top-down, so the multiplicity information can be passed upwards.

The main advantage of recursion schemes is that they let us use simpler, non-recursive building blocks such as algebras and coalgebras. Let’s start with a simple coalgebra that lets us build a trie from a list of strings. A coalgebra is a fancy name for a particular type of function:

type Coalgebra f x = x -> f x

Think of x as a type for a seed from which one can grow a tree. A colagebra tells us how to use this seed to create a single node described by the functor f and populate it with (presumably smaller) seeds. We can then pass this coalgebra to a simple algorithm, which will recursively expand the seeds. This algorithm is called the anamorphism:

ana :: Functor f => Coalgebra f a -> a -> Fix f
ana coa = In . fmap (ana coa) . coa

Let’s see how we can apply it to the task of building a trie. The seed in our case is a list of strings (as per the definition of our problem, we’ll assume they are all equal length). We start by grouping these strings into bunches of strings that start with the same character. There is a library function called groupWith that does exactly that. We have to import the right library:

import GHC.Exts (groupWith)

This is the signature of the function:

groupWith :: Ord b => (a -> b) -> [a] -> [[a]]

It takes a function a -> b that converts each list element to a type that supports comparison (as per the typeclass Ord), and partitions the input into lists that compare equal under this particular ordering. In our case, we are going to extract the first character from a string using head and bunch together all strings that share that first character.

let sss = groupWith head ss

The tails of those strings will serve as seeds for the next tier of the trie. Eventually the strings will be shortened to nothing, triggering the end of recursion.

fromList :: Coalgebra TrieF [String]
fromList ss =
-- are strings empty? (checking one is enough)
then TrieF [] -- leaf
else
let sss = groupWith head ss
in TrieF $fmap mkBranch sss The function mkBranch takes a bunch of strings sharing the same first character and creates a branch seeded with the suffixes of those strings. mkBranch :: [String] -> (Char, [String]) mkBranch sss = let c = head (head sss) -- they're all the same in (c, fmap tail sss) Notice that we have completely avoided explicit recursion. The next step is a little harder. We have to fold the trie. Again, all we have to define is a step that folds a single node whose children have already been folded. This step is defined by an algebra: type Algebra f x = f x -> x Just as the type x described the seed in a coalgebra, here it describes the accumulator–the result of the folding of a recursive data structure. We pass this algebra to a special algorithm called a catamorphism that takes care of the recursion: cata :: Functor f => Algebra f a -> Fix f -> a cata alg = alg . fmap (cata alg) . out Notice that the folding proceeds from the bottom up: the algebra assumes that all the children have already been folded. The hardest part of designing an algebra is figuring out what information needs to be passed up in the accumulator. We obviously need to return the final result which, in our case, is the list of strings with one mismatched character. But when we are in the middle of a trie, we have to keep in mind that the mismatch may still happen above us. So we also need a list of strings that may serve as suffixes when the mismatch occurs. We have to keep them all, because they might be matched later with strings from other branches. In other words, we need to be accumulating two lists of strings. The first list accumulates all suffixes for future matching, the second accumulates the results: strings with one mismatch (after the mismatch has been removed). We therefore should implement the following algebra: Algebra TrieF ([String], [String]) To understand the implementation of this algebra, consider a single node in a trie. It’s a list of branches, or pairs, whose first component is the current character, and the second a pair of lists of strings–the result of folding a child trie. The first list contains all the suffixes gathered from lower levels of the trie. The second list contains partial results: strings that were matched modulo single-character defect. As an example, suppose that you have a node with two branches: [ ('a', (["bcd", "efg"], ["pq"])) , ('x', (["bcd"], []))] First we prepend the current character to strings in both lists using the function prep with the following signature: prep :: (Char, ([String], [String])) -> ([String], [String]) This way we convert each branch to a pair of lists. [ (["abcd", "aefg"], ["apq"]) , (["xbcd"], [])] We then merge all the lists of suffixes and, separately, all the lists of partial results, across all branches. In the example above, we concatenate the lists in the two columns. (["abcd", "aefg", "xbcd"], ["apq"])  Now we have to construct new partial results. To do this, we create another list of accumulated strings from all branches (this time without prefixing them): ss = concat$ fmap (fst . snd) bs

In our case, this would be the list:

["bcd", "efg", "bcd"]

To detect duplicate strings, we’ll insert them into a multiset, which we’ll implement as a map. We need to import the appropriate library:

import qualified Data.Map as M

and define a multiset Counts as:

type Counts a = M.Map a Int

Every time we add a new item, we increment the count:

add :: Ord a => Counts a -> a -> Counts a
add cs c = M.insertWith (+) c 1 cs

To insert all strings from a list, we use a fold:

mset = foldl add M.empty ss

We are only interested in items that have multiplicity greater than one. We can filter them and extract their keys:

dups = M.keys $M.filter (> 1) mset Here’s the complete algebra: accum :: Algebra TrieF ([String], [String]) accum (TrieF []) = ([""], []) accum (TrieF bs) = -- b :: (Char, ([String], [String])) let -- prepend chars to string in both lists pss = unzip$ fmap prep bs
(ss1, ss2) = both concat pss
-- find duplicates
ss = concat $fmap (fst . snd) bs mset = foldl add M.empty ss dups = M.keys$ M.filter (> 1) mset
in (ss1, dups ++ ss2)
where
prep :: (Char, ([String], [String])) -> ([String], [String])
prep (c, pss) = both (fmap (c:)) pss

I used a handy helper function that applies a function to both components of a pair:

both :: (a -> b) -> (a, a) -> (b, b)
both f (x, y) = (f x, f y)

And now for the grand finale: Since we create the trie using an anamorphism only to immediately fold it using a catamorphism, why don’t we cut the middle person? Indeed, there is an algorithm called the hylomorphism that does just that. It takes the algebra, the coalgebra, and the seed, and returns the fully charged accumulator.

hylo :: Functor f => Algebra f a -> Coalgebra f b -> b -> a
hylo alg coa = alg . fmap (hylo alg coa) . coa

And this is how we extract and print the final result:

print $head$ snd $hylo accum fromList cs ## Conclusion The advantage of using the hylomorphism is that, because of Haskell’s laziness, the trie is never wholly constructed, and therefore doesn’t require large amounts of memory. At every step enough of the data structure is created as is needed for immediate computation; then it is promptly released. In fact, the definition of the data structure is only there to guide the steps of the algorithm. We use a data structure as a control structure. Since data structures are much easier to visualize and debug than control structures, it’s almost always advantageous to use them to drive computation. In fact, you may notice that, in the very last step of the computation, our accumulator recreates the original list of strings (actually, because of laziness, they are never fully reconstructed, but that’s not the point). In reality, the characters in the strings are never copied–the whole algorithm is just a choreographed dance of internal pointers, or iterators. But that’s exactly what happens in the original C++ algorithm. We just use a higher level of abstraction to describe this dance. I haven’t looked at the performance of various implementations. Feel free to test it and report the results. The code is available on github. ## Acknowledgments I’m grateful to the participants of the Seattle Haskell Users’ Group for many helpful comments during my presentation. 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? # Free Monad 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: $L\, f\, g = 1 + f \circ g$ or, in Haskell, 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. ## Free Monad Example 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 | Add 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 $fmap (InH . DoneM) fr  With this function, we can define smart constructors that build leaves of the free monad tree: push :: Int -> FreeStack () push n = liftF (Push n ()) pop :: FreeStack () pop = liftF (Pop ()) top :: FreeStack Int top = liftF (Top id) add :: FreeStack () add = liftF (Add ())  All these preparations finally pay off when we are able to create small programs using do notation: calc :: FreeStack Int calc = do push 3 push 4 add x <- top pop return x  Of course, this program does nothing but build a tree. We need a separate interpreter to do the calculation. We’ll interpret our program in the state monad, with state implemented as a stack (list) of integers: type MemState = State [Int]  The trick is to define a higher order algebra for the functor that generates the free monad and then use a catamorphism to apply it to the program. Notice that implementing the algebra is a relatively simple procedure because we don’t have to deal with recursion. All we need is to case-analyze the shallow constructors for the free monad functor MonadF, and then case-analyze the shallow constructors for the functor StackF. runAlg :: HAlgebra (MonadF StackF) MemState runAlg (DoneM a) = return a runAlg (MoreM ex) = case ex of Top ik -> get >>= ik . head Pop k -> get >>= put . tail >> k Push n k -> get >>= put . (n : ) >> k Add k -> do (a: b: s) <- get put (a + b : s) k  The catamorphism converts the program calc into a state monad action, which can be run over an empty initial stack: runState (hcata runAlg calc) []  The real bonus is the freedom to define other interpreters by simply switching the algebras. Here’s an algebra whose carrier is the Const functor: showAlg :: HAlgebra (MonadF StackF) (Const String) showAlg (DoneM a) = Const "Done!" showAlg (MoreM ex) = Const$
case ex of
Push n k ->
"Push " ++ show n ++ ", " ++ getConst k
Top ik ->
"Top, " ++ getConst (ik 42)
Pop k ->
"Pop, " ++ getConst k


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
, 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


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  # Cofree Comonad 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)  ## Cofree Comonad Example 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!

# Part II: Free Monoids

## String Diagrams

The utility of diagrams in formulating and proving theorems in category theory cannot be overemphasized. While working my way through the construction of free monoids, I noticed that there was a particular set of manipulations that had to be done algebraically, with little help from diagrams. These were operations involving a mix of tensor products and composition of morphisms. Tensor product is a bifunctor, so it preserves composition; which means you can often slide products through junctions of arrows—but the rules are not immediately obvious. Diagrams in which objects are nodes and morphisms are arrows have no immediate graphical representation for tensor products. A thought occurred to me that maybe a dual representation, where morphisms are nodes and objects are edges would be more accommodating. And indeed, a quick search for “string diagrams in monoidal categories” produced a paper by Joyal and Street, “The geometry of tensor calculus.”

The idea is very simple: if you represent morphisms as points on a plane, you have two additional dimensions to play with composition and tensoring. Two morphism—represented as points—can be composed if they share an object, which can be represented as a line connecting them. By convention, we read composition from the bottom of the diagram up. We follow lines as they go through points—that’s composition. Two lines ascending in parallel represent a tensor product. The geometry of the diagram just works!

Let me explain it on a simple example—the left unit law for a monoid $(m, \pi, \mu)$:

$\mu \circ (\pi \otimes m) = id$

The left hand side is a composition of two morphisms. The first morphism $\pi \otimes m$ starts from the object $I \otimes m$ (see Fig. 12).

Fig. 12. Left unit law

In principle, there should be two parallel lines at the bottom, one for $I$ and one for $m$; but $I \otimes m$ is isomorphic to $m$, so the $I$ line is redundant and can be omitted. Scanning the diagram from the bottom up, we encounter the morphism $\pi$ in parallel with the $m$ line. That’s exactly the graphical representation of $\pi \otimes m$. The output of $\pi$ is also $m$, so we now have two upward moving $m$ lines, corresponding to $m \otimes m$. That’s the input of the next morphism $\mu$. Its output is the single upward moving $m$. The unit law may be visualized as pulling the two $m$ strings in opposite direction until the whole diagram is straightened to one vertical $m$ string corresponding to $id_m$.

Here’s the right unit law:

$\mu \circ (m \otimes \pi) = id$

It works like a mirror image of the left unit law:

Fig. 13. Right unit law

The associativity law can be illustrated by the following diagram:

Fig. 14. Associativity law

The important property of a string diagram is that, because of functoriality, its value—the compound morphism it represents—doesn’t change under continuous transformations.

## Monoid

First we’d like to show that the carrier of the free $h$-algebra $(m, \sigma)$ which, as we’ve seen before, is also the initial algebra for the list functor $I + h \otimes -$, is automatically a monoid. To show that, we need to define its unit and multiplication—two morphisms that satisfy monoid laws. The obvious candidate for unit is the universal mapping $\pi \colon I \to m$. It’s the morphism in the definition of the free algebra from the previous post (see Fig 15).

Fig. 15. Free $h$-algebra $(m, \sigma)$ generated by $I$

Multiplication is a morphism:

$\mu \colon m \otimes m \to m$

which, if you think of a free monoid as a list, is the generalization of list concatenation.

The trick is to show that $m \otimes m$ is also a free $h$-algebra whose generator is $m$ itself. We could then use the universality of $m \otimes m$ to generate the unique algebra morphism from it to $m$ (which is also an $h$-algebra). That will be our $\mu$.

Proposition. {Monoid.}

The free $h$-algebra $(m, \sigma)$ generated by the unit object $I$ is a monoid whose unit is:

$\pi \colon I \to m$

and whose multiplication:

$\mu \colon m \otimes m \to m$

is the unique $h$-algebra morphism

$(m \otimes m, \sigma \otimes m) \to (m, \sigma)$

induced by the identity morphism $id_m$.

Proof.
In the previous post we’ve shown that, if $(m, \sigma)$ is a free algebra generated by the unit object $I$ with the universal map $\pi$, then $(m \otimes k, \sigma \otimes k)$ is a free algebra generated by $k$ with the universal map $\pi \otimes k$ (see Fig. 16).

Fig. 16. Free $h$-algebra generated by $k \cong I \otimes k$

We get $\mu$ by redrawing this diagram: using $m$ as both the generator and the target algebra, and replacing $f$ with $id_m$ (see Fig. 17):

Fig. 17. Monoid multiplication as an $h$-algebra morphism

Since so defined $\mu$ is an $h$-algebra morphism, it makes the following diagram, Fig. 18, commute.

Fig. 18. $\mu$ is an $h$-algebra morphism

This commuting condition can be redrawn as the identity of two string diagrams (Fig. 19) corresponding to the two paths through the original diagram.

Fig. 19. String diagram showing that $\mu$ is an algebra morphism

The universal condition in Fig. 17:

$\mu \circ (\pi \otimes m) = id_m$

gives us immediately the left unit law for the monoid.

The right unit law:

$\mu \circ (m \otimes \pi) = id_m$

requires a little more work.

There is a standard trick that we can use to show that two morphisms, whose source (in this case $m$) is a free algebra, are equal. It’s enough to prove that they are algebra morphisms, and that they are both induced by the same morphism (in this case $\pi$). Their equality then follows from the uniqueness of the universal construction.

We know that $\mu$ is an algebra morphism so, if we can show that $m \otimes \pi$ is also an algebra morphism, their composition will be an algebra morphism too. Trivially, $id_m$ is an algebra morphism so, if we can show that the two are induced by the same regular morphism $\pi$, then they must be equal.

To show that $m \otimes \pi$ is an $h$-algebra morphism, we have to show that the diagram in Fig. 20 commutes.

Fig. 20. $m \otimes \pi$ as an $h$-algebra morphism

We can redraw the two paths through Fig. 20 as two string diagrams in Fig. 21. They are equal because they can be deformed into each other.

Fig. 21. String diagram showing that $m \otimes \pi$ is an algebra morphism

Therefore the composition $\mu \circ (m \otimes \pi)$ is also an $h$-algebra morphism. The string diagram that illustrates this fact is shown in Fig. 22.

Fig. 22. String diagram showing that $\mu \circ (m \otimes \pi)$ is an algebra morphism

Since the identity $h$-algebra morphism is induced by $\pi$, we’d like to show that $\mu \circ (m \otimes \pi)$ is also induced by $\pi$ (Fig. 23).

Fig. 23. Universal property of the free $h$-algebra generated by $I$, with the algebra morphism induced by $\pi$

To do that, we have to prove the universal condition in Fig. 23:

$\mu \circ (m \otimes \pi) \circ \pi = \pi$

This is represented as a string diagram identity in Fig. 24. We can deform this diagram by sliding the left $\pi$ node up, past the right $\pi$ node, and then using the left identity.

Fig. 24. Universal condition in Fig. 23.

This concludes the proof of the right identity.

The proof of associativity is very similar, so I’ll just sketch it. We have to show that the two diagrams in Fig. 14 are equal. We’ll use the same trick as before. We’ll show that they are both algebra morphisms. Their source is a free algebra generated by $m \otimes m$ (see Fig. 25—the other diagram has $\mu \circ (\mu \otimes m)$ replaced by $\mu \circ (m \otimes \mu)$). The universal condition follows from the unit law for $m$. Associativity condition:

$\mu \circ (\mu \otimes m) = \mu \circ (m \otimes \mu)$

will then follow from the uniqueness of the universal construction.

Fig. 25. One part of associativity as an $h$-algebra morphism

You can easily convince yourself that showing that something is an $h$-algebra morphism can be done by first attaching the $h$ leg to the left of the string diagram and then sliding it to the top of the diagram, as illustrated in Fig. 26. This can be accomplished by repeatedly using the fact that $\mu$ is an $h$-algebra morphism.

Fig. 26. String diagram showing that one of the associativity diagrams is an $h$-algebra morphism

The same process can be applied to the second associativity diagram, thus completing the proof.

$\square$

For Haskell programmers, recall from the previous post our construction of the free $h$-algebra generated by $k$ and the derivation of the algebra morphism $g$ from it to the internal-hom algebra:

g :: Expr -> (k -> n)
g () = f
g a  = k -> nu (a, f k)
g (a, b) = k -> nu (a, nu (b, f k))
...


In the current proof we have replaced $k$ with $m$, which generalizes the list of $h$s, $f$ became $id$, and $\nu$ is a function that prepends an element to a list. In other words, $g$ concatenates its list-argument in front of the second list, and it does it in the correct order.

## Free Monoid

The monoid whe have just constructed from the free algebra is a free monoid. As we did with free algebras, instead of using the free/forgetful adjunction to prove it, we’ll use the free-object universal construction.

Theorem. {Free Monoid.}

The monoid $(m, \pi, \mu)$ is a free monoid generated by $h$, with a universal mapping given by $u = \sigma \circ (h \otimes \pi)$:

That is, for any monoid $(n, \eta, \nu)$ and any morphism $s \colon h \to n$, there is a unique monoid morphism $t$ from $(m, \pi, \mu)$ to $(n, \eta, \nu)$ such that the universal condition holds:

$t \circ u = s$

(see Fig. 27).

Fig. 27. Free monoid diagram

Proof. Recall that $(m, \sigma)$ is a free $h$-algebra generated by $I$. It turns out that any monoid $(n, \eta, \nu)$, for which there is a morphism $s \colon h \to n$, is automatically a carrier of an $h$-algebra. We construct its structure map $\lambda$ by combining $s$ with monoid multiplication $\nu$:

We can use $n$‘s monoidal unit $\eta$ to insert $I$ into $n$. Because $(m, \sigma)$ is a free $h$-algebra, there is a unique algebra morphism, let’s call it $t$, from it to $(n, \lambda)$, which is induced by $\eta$, such that $t \circ \pi = \eta$ (see Fig. 28). We want to show that this algebra morphism is also a monoid morphism. Furthermore, if we can show that this is the unique monoid morphism induced by $s$, we will have a proof that $m$ is a free monoid.

Fig. 28. Algebra morphism between monoids

Since $t$ is an algebra morphism, the rectangle in Fig. 29 commutes.

Fig. 29. $t$ is an $h$-algebra morphism

Let’s redraw it as an identity of string diagrams, Fig. 30. We’ll make use of it in a moment.

Fig. 30. $t$ is an $h$-algebra morphism

Going back to Fig. 27, we want to show that the universal condition holds, which means that we want the diagram in Fig. 31 to commute (I have expanded the definition of $u$).

Fig. 31. Free monoid universal condition

In other words we want show that the following two string diagrams are equal:

Fig. 32. Free monoid universal condition

Using the identity in Fig. 30, the left hand side can be rewritten as:

Fig. 33. Step 1 in transforming Fig 32

The right leg can be shrunk down to $\eta$ using the universal condition in Fig. 28:

$t \circ \pi = \eta$

which, incidentally, also expresses the fact that $t$ preserves the monoidal unit.

Finally, we can use the right unit law for the monoid $n$, Fig. 34,

Fig. 34. Right unit law for monoid $n$

to arrive at the right hand side of the identity in Fig. 32. This completes the proof of the universal condition in Fig. 27.

Now we have to show that $t$ is a full-blown monoid morphism, that is, it preserves multiplication (Fig. 35).

Fig. 35. Preservation of multiplication

The corresponding string diagrams are shown in Fig. 36.

Fig. 36. Preservation of multiplication

Let’s start with the fact that $m \otimes m$ is the free $h$-algebra generated by $m$. We will show that the two paths through the diagram in Fig. 35 are both $h$-algebra morphisms, and that they are induced by the same regular morphism $t \colon m \to n$. Therefore they must be equal.

The bottom path in Fig. 35, $t \circ \mu$, is an $h$-algebra morphism by virtue of being a composition of two $h$-algebra morphisms. This composite is induced by morphism $t$ in the diagram Fig. 37.

Fig. 37. $h$-algebra morphism $t \circ \mu$

The universal condition in Fig. 37 follows from the diagram in Fig. 38, which follows from the left unit law for the monoid $(m, \pi, \mu)$.

Fig. 38. Universal condition in Fig. 37

We want to show that the top path in Fig. 35 is also an $h$-algebra morphism, that is, the diagram in Fig. 39 commutes.

Fig. 39. $h$-algebra morphism diagram for $\nu \circ (t \otimes t)$

We can redraw this diagram as a string diagram identity in Fig. 40.

Fig. 40. $h$-algebra morphism diagram for $\nu \circ (t \otimes t)$

First, let’s use the associativity law for the monoid $n$ to transform the left hand side. We get the diagram in Fig. 41.

Fig. 41. After applying associativity, we can apply Fig. 30

We can now apply the identity in Fig. 30 to reproduce the right hand side of Fig. 40.

We have thus shown that both paths in Fig. 35 are algebra morphisms. We know that the bottom path is induced by morphism $t$. What remains is to show that the top path, which is given by $\nu \circ (t \otimes t)$ is induced by the same $t$. This will be true, if we can show the universal condition in Fig. 42.

Fig. 42. $h$-algebra morphism $\nu \circ (t \otimes t)$

This universal condition can be expanded to the diagram in Fig. 43.

Fig. 43. Universal condition in Fig. 42

Here’s the string diagram that traces the path around the square (Fig. 44).

Fig. 44. Path around Fig. 43 as a string diagram

First, let’s use the preservation of unit by $t$, Fig. 45, to shrink the left leg,

Fig. 45. Preservation of unit by $t$

and follow it with the left unit law for the monoid $(n, \eta, \nu)$ (Fig. 46). The result is that Fig. 44 shrinks to the single morphism $t$, thus making Fig. 43 commute.

Fig. 46. Left unit law for the monoid $n$

This completes the proof that $t$ is a monoid morphism.

The final step is to make sure that $t$ is the unique monoid morphism from $m$ to $n$. Suppose that there is another monoid morphism $t'$ (replacing $t$ in Fig. 28). If we can show that $t'$ is also an $h$-algebra morphism induced by the same $\eta$, it will have to, by universality, be equal to $t$. In other words, we have to show that the diagram in Fig. 28 also works for $t'$. Our assumptions are that both $t$ and $t'$ are monoid morphisms, that is, they preserve unit and multiplication; and they both satisfy the universal condition in Fig 27. In particular, $t'$ satisfies the condition in Fig. 47.

Fig. 47. Free monoid universal condition for $t'$ as a string diagram

Notice that, in the first part of the proof, we started with an $h$-algebra morphism $t$ and had to show that it’s a monoid morphism. Now we are going in the opposite direction: we know that $t'$ is a monoid morphism, and have to show that it’s an $h$-algebra morphism, and that the universal condition in Fig. 28

$t' \circ \pi = \eta$

holds. The latter simply restates our assumption that $t'$ preserves the unit.

To show that $t'$ is an algebra morphism, we have to show that the diagram in Fig 48 commutes.

Fig. 48. $t'$ as an $h$-algebra morphism

This diagram may be redrawn as a pair of string diagrams, Fig 49.

Fig. 49. $t'$ as an algebra morphism

The proof of this identity relies on redrawing string diagrams using the identities in Figs. 12, 19, 36, and 47. Before we continue, you might want to try it yourself. It’s an exercise well worth the effort.

We start by expanding the $s$ node using the diagram in Fig. 47 to get Fig. 50.

Fig. 50. After expanding the left leg of the diagram in Fig. 49, we can apply preservation of multiplication by $t'$.

We can now use the preservation of multiplication by $t'$ to obtain Fig 51.

Fig. 51. Applying the fact that $\mu$ is an $h$-algebra morphism

Next, we can use the fact that $\mu$ is an $h$-algebra morphism, Fig. 19, to slide the $\sigma$ node up, and obtain Fig. 52.

Fig. 52. Applying left unit law

We can now use the left unit law for the monoid $m$:

$\mu \circ (\pi \otimes m) = id$

as illustrated in Fig. 12, to arrive at the right hand side of Fig. 49.

This concludes the proof that $t'$ must be equal to $t$.

$\square$

## Conclusion

To summarize, we have shown that the free monoid can be constructed from a free algebra of the functor $h \otimes -$. This is a very general result that is valid in any monoidal closed category. Earlier we’ve seen that this free algebra is also the initial algebra of the list functor $I + h \otimes -$. The immediate consequence of this theorem is that it lets us construct free monoids in functor categories with interesting monoidal structures. In particular, we get a free monad as a free monoid in the category of endofunctors with functor composition as tensor product. We can also get a free applicative, or free lax monoidal functor, if we define the tensor product as Day convolution—the latter can be also constructed in the profunctor category.

## Preface

In my previous blog post I used, without proof, the fact that the initial algebra of the functor $I + h \otimes -$ is a free monoid. The proof of this statement is not at all trivial and, frankly, I would never have been able to work it out by myself. I was lucky enough to get help from a mathematician, Alex Campbell, who sent me the proof he extracted from the paper by G. M. Kelly (see Bibliography).

I worked my way through this proof, filling some of the steps that might have been obvious to a mathematician, but not to an outsider. I even learned how to draw diagrams using the TikZ package for LaTeX.

What I realized was that category theorists have developed a unique language to manipulate mathematical formulas: the language of 2-dimensional diagrams. For a programmer interested in languages this is a fascinating topic. We are used to working with grammars that essentially deal with string substitutions. Although diagrams can be serialized—TikZ lets you do it—you can’t really operate on diagrams without laying them out on a page. The most common operation—diagram pasting—involves combining two or more diagrams along common edges. I am amazed that, to my knowledge, there is no tool to mechanize this process.

In this post you’ll also see lots of examples of using the same diagram shape (the free-construction diagram, or the algebra-morphism diagram), substituting new expressions for some of the nodes and edges. Not all substitutions are valid and I’m sure one could develop some kind of type system to verify their correctness.

Because of proliferation of diagrams, this blog post started growing out of proportion, so I decided to split it into two parts. If you are serious about studying this proof, I strongly suggest you download (or even print out) the PDF version of this blog.

# Part I: Free Algebras

## Introduction

Here’s the broad idea: The initial algebra that defines a free monoid is a fixed point of the functor $I + h \otimes -$, which I will call the list functor. Roughly speaking, it’s the result of replacing the dash in the definition of the functor with the result of the replacement. For instance, in the first iteration we would get:

$I + h \otimes (I + h \otimes -) \cong I + h + h \otimes h \otimes -$

I used the fact that $I$ is the unit of the tensor product, the associativity of $\otimes$ (all up to isomorphism), and the distributivity of tensor product over coproduct.

Continuing this process, we would arrive at an infinite sum of powers of $h$:

$m = I + h + h \otimes h + h \otimes h \otimes h + ...$

Intuitively, a free monoid is a list of $h$s, and this representation expresses the fact that a list is either trivial (corresponding to the unit $I$), or a single $h$, or a product of two $h$s, and so on…

Let’s have a look at the structure map of the initial algebra of the list functor:

$I + h \otimes m \to m$

Mapping out of a coproduct (sum) is equivalent to defining a pair of morphisms $\langle \pi, \sigma \rangle$:

$\pi \colon I \to m$

$\sigma \colon h \otimes m \to m$

the second of which may, in itself, be considered an algebra for the product functor $h \otimes -$.

Our goal is to show that the initial algebra of the list functor is a monoid so, in particular, it supports multiplication:

$\mu \colon m \otimes m \to m$

Following our intuition about lists, this multiplication corresponds to list concatenation. One way of concatenating two lists is to keep multiplying the second list by elements taken from the first list. This operation is described by the application of our product functor $h \otimes -$. Such repetitive application of a functor is described by a free algebra.

There is just one tricky part: when concatenating two lists, we have to disassemble the left list starting from the tail (alternatively, we could disassemble the right list from the head, but then we’d have to append elements to the tail of the left list, which is the same problem). And here’s the ingenious idea: you disassemble the left list from the head, but instead of applying the elements directly to the right list, you turn them into functions that prepend an element. In other words you convert a list of elements into a (reversed) list of functions. Then you apply this list of functions to the right list one by one.

This conversion is only possible if you can trade product for function — the process we know as currying. Currying is possible if there is an adjunction between the product and the exponential, a.k.a, the internal hom, $[k, n]$ (which generalizes the set of functions from $k$ to $n$):

$C(m \otimes k, n) \cong C(m, [k, n])$

We’ll assume that the underlying category $C$ is monoidal closed, so that we can curry morphisms that map out from the tensor product:

$g \colon m \otimes k \to n$

$\bar{g} \colon m \to [k, n]$

(In what follows I’ll be using the overbar to denote the curried version of a morphism.)

The internal hom can also be defined using a universal construction, see Fig. 1. The morphism $eval$ corresponds to the counit of the adjunction (although the universal construction is more general than the adjunction).

Fig. 1. Universal construction of the internal hom $[k, n]$. For any object $m$ and a morphism $g \colon m \otimes k \to n$ there is a unique morphism $\bar{g}$ (the curried version of $g$) which makes the triangle commute.

The hard part of the proof is to show that the initial algebra produces a free monoid, which is a free object in the category of monoids. I’ll start by defining the notion of a free object.

## Free Objects

You might be familiar with the definition of a free construction as the left adjoint to the forgetful functor. Fig 2 illustrates the essence of such an adjunction.

The left hand side is in some category $D$ of structured objects: algebras, monoids, monads, etc. The right hand side is in the underlying category $C$, often the category of sets. The adjunction establishes a one-to-one correspondence between sets of morphisms, of which $g$ and $f$ are examples. If $U$ is the forgetful functor, then $F$ is the free functor, and the object $F x$ is called the free object generated by $x$. The adjunction is an isomorphism of hom-sets, natural in both $x$ and $z$:

$D(F x, z) \cong C(x, U z)$

Unfortunately, this description doesn’t quite work for some important cases, like free monads. In the case of free monads, the right category is the category of endofunctors, and the left category is the category of monads. Because of size issues, not every endofunctor on the right generates a free monad on the left.

It turns out that there is a weaker definition of a free object that doesn’t require a full blown adjunction; and which reduces to it, when the adjunction can be defined globally.

Let’s start with the object $x$ on the right, and try to define the corresponding free object $F x$ on the left (by abuse of notation I will call this object $F x$, even if there is no functor $F$). For our definition, we need a condition that would work universally for any object $z$, and any morphism $f$ from $x$ to $U z$.

We are still missing one important ingredient: something that would tell us that $x$ acts as a set of generators for $F x$. This property can be expressed by providing a morphism that inserts $x$ into $U (F x)$—the object underlying the free object. In the case of an adjunction, this morphism happens to be the component of the unit natural transformation:

$\eta \colon Id \to U \circ F$

where $Id$ is the identity functor (see Fig. 3).

The important property of the unit of adjunction $\eta$ is that it can be used to recover the mapping from the left hom-set to the right hom-set in Fig. 2. Take a morphism $g \colon F x \to z$, lift it using $U$, and compose it with $\eta_x$. You get a morphism $f \colon x \to U z$:

$f = U g \circ \eta_x$

In the absence of an adjunction, we’ll make the existence of the morphism $\eta_x$ part of the definition of the free object.

Definition. {Free object.}
A free object on $x$ consists of an object $F x$ and a morphism $\eta_x \colon x \to U (F x)$ such that, for every object $z$ and a morphism $f \colon x \to U z$, there is a unique morphism $g \colon F x \to z$ such that:

$U g \circ \eta_x = f$

The diagram in Fig. 4 illustrates this definition. It is essentially a composition of the two previous diagrams, except that we don’t require the existence of a global mapping, let alone a functor, $F$.

Fig. 4. Definition of a free object $F x$

The morphism $\eta_x$ is called the universal mapping, because it’s independent of $z$ and $f$. The equation:

$U g \circ \eta_x = f$

is called the universal condition, because it works universally for every $z$ and $f$. We say that $g$ is induced by the morphism $f$.

There is a standard trick that uses the universal condition: Suppose you have two morphisms $g$ and $g'$ from the universal object to some $z$. To prove that they are equal, it’s enough to show that they are both induced by the same $f$. Showing the equality:

$U g \circ \eta_x = U g' \circ \eta_x$

is often easier because it lets you work in the simpler, underlying category.

## Free Algebras

As I mentioned in the introduction, we are interested in algebras for the product functor: $h \otimes -$, which we’ll call $h$-algebras. Such an algebra consists of a carrier $n$ and a structure map:

$\nu \colon h \otimes n \to n$

For every $h$, $h$-algebras form a category; and there is a forgetful functor $U$ that maps an $h$-algebra to its carrier object $n$. We can therefore define a free algebra as a free object in the category of $h$-algebras, which may or may not be generalizable to a full-blown free/forgetful adjunction. Fig. 5 shows the universal condition that defines such a free algebra $(m_k, \sigma)$ generated by an object $k$.

Fig. 5. Free $h$-algebra $(m_k, \sigma)$ generated by $k$

In particular, we can define an important free $h$-algebra generated by the identity object $I$. This algebra $(m, \sigma)$ has the structure map:

$\sigma \colon h \otimes m \to m$

and is described by the diagram in Fig. 6:

Fig. 6. Free $h$-algebra $(m, \sigma)$ generated by $I$

$g \circ \pi = f$

By definition, since $g$ is an algebra morphism, it makes the diagram in Fig. 7 commute:

Fig. 7. $g$ is an $h$-algebra morphism

We use a notational convenience: $h \otimes g$ is the lifting of the morphism $g$ by the product functor $h \otimes -$. This might be confusing at first, because it looks like we are multiplying an object $h$ by a morphism $g$. One way to parse it is to consider that, if we keep the first argument to the tensor product constant, then it’s a functor in the second component, symbolically $h \otimes -$. Since it’s a functor, we can use it to lift a morphism $g$, which can be notated as $h \otimes g$.

Alternatively, we can exploit the fact that tensor product is a bifunctor, and therefore it may lift a pair of morphism, as in $id_h \otimes g$; and $h \otimes g$ is just a shorthand notation for this.

Bifunctoriality also means that the tensor product preserves composition and identity in both arguments. We’ll use these facts extensively later, especially as the basis for string diagrams.

The important property of $m$ is that it also happens to be the initial algebra for the list functor $I + h \otimes -$. Indeed, for any other algebra with the carrier $n$ and the structure map a pair $\langle f, \nu \rangle$, there exists a unique $g$ given by Fig 6, such that the diagram in Fig. 8 commutes:

Fig. 8. Initiality of the algebra $(m, \langle \pi, \sigma \rangle)$ for the functor $I + h \otimes -$.

Here, $inl$ and $inr$ are the two injections into the coproduct.

If you visualize $m$ as the sum of all powers of $h$, $\pi$ inserts the unit $I$ (zeroth power) into it, and $\sigma$ inserts the sum of non-zero powers.

$\langle \pi, \sigma \rangle \colon I + h \otimes m \to m$

The advantage of this result is that we can concentrate on studying the simpler problem of free $h$-algebras rather than the problem of initial algebras for the more complex list functor.

## Example

Here’s some useful intuition behind $h$-algebras. Think of the functor $(h \otimes -)$ as a means of forming formal expressions. These are very simple expressions: you take a variable and pair it (using the tensor product) with $h$. To define an algebra for this functor you pick a particular type $n$ for your variable (i.e., the carrier object) and a recipe for evaluating any expression of type $h \otimes n$ (i.e., the structure map).

Let’s try a simple example in Haskell. We’ll use pairs (and tuples) as our tensor product, with the empty tuple () as unit (up to isomorphism). An algebra for a functor f is defined as:

type Algebra f a = f a -> a


Consider $h$-algebras for a particular choice of $h$: the type of real numbers Double. In other words, these are algebras for the functor ((,) Double). Pick, as your carrier, the type of vectors:

data Vector = Vector { x :: Double
, y :: Double
} deriving Show


and the structure map that scales a vector by multiplying it by a number:

vecAlg :: Algebra ((,) Double) Vector
vecAlg (a, v) = Vector { x = a * x v
, y = a * y v }


Define another algebra with the carrier the set of real numbers, and the structure map multiplication by a number.

mulAlg :: Algebra ((,) Double) Double
mulAlg (a, x) = a * x


There is an algebra morphism from vecAlg to mulAlg, which takes a vector and maps it to its length.

algMorph :: Vector -> Double
algMorph v = sqrt((x v)^2 + (y v)^2)


This is an algebra morphism, because it doesn’t matter if you first calculate the length of a vector and then multiply it by a, or first multiply the vector by a and then calculate its length (modulo floating-point rounding).

A free algebra has, as its carrier, the set of all possible expressions, and an evaluator that tells you how to convert a functor-ful of such expressions to another valid expression. A good analogy is to think of the functor as defining a grammar (as in BNF grammar) and a free algebra as a language generated by this grammar.

You can generate free $h$-expressions recursively. The starting point is the set of generators $k$ as the source of variables. Applying the functor to it produces $h \otimes k$. Applying it again, produces $h \otimes h \otimes k$, and so on. The whole set of expressions is the infinite coproduct (sum):

$k + h \otimes k + h \otimes h \otimes k + ...$

An element of this coproduct is either an element of $k$, or an element of the product $h \otimes k$, and so on…

The universal mapping injects the set of generators $k$ into the set of expressions (here, it would be the leftmost injection into the coproduct).

In the special case of an algebra generated by the unit $I$, the free algebra simplifies to the power series:

$I + h + h \otimes h + ...$

and $\pi$ injects $I$ into it.

Continuing with our example, let’s consider the free algebra for the functor ((,) Double) generated by the unit (). The free algebra is, symbolically, an infinite sum (coproduct) of tuples:

data Expr =
()
| Double
| (Double, Double)
| (Double, Double, Double)
| ...


Here’s an example of an expression:

e = (1.1, 2.2, 3.3)


As you can see, free expressions are just lists of numbers. There is a function that inserts the unit into the set of expressions:

pi :: () -> Expr
pi () = ()


The free evaluator is a function (not actual Haskell):

sigma (a, ()) = a
sigma (a, x)  = (a, x)
sigma (a, (x, y)) = (a, x, y)
sigma (a, (x, y, z)) = (a, x, y, z)
...


Let’s see how the universal property works in our example. Let’s try, as the target $(n, \nu)$, our earlier vector algebra:

vecAlg :: Algebra ((,) Double) Vector
vecAlg (a, v) = Vector { x = a * x v
, y = a * y v }


The morphism f picks some concrete vector, for instance:

f :: () -> Vector
f () = Vector 3 4


There should be a unique morphism of algebras g that takes an arbitrary expression (a list of Doubles) to a vector, such that g . pi = f picks our special vector:

Vector 3 4


In other words, g must take the empty tuple (the result of pi) to Vector 3 4. The question is, how is g defined for an arbitrary expression? Look at the diagram in Fig. 7 and the commuting condition it implies:

$g \circ \sigma = \nu \circ (h \otimes g)$

Let’s apply it to an expression (a, ()) (the action of the functor (h, -) on ()). Applying sigma to it produces a, followed by the action of g resulting in g a. This should be the same as first applying the lifted (id, g) acting on (a, ()), which gives us (a, Vector 3 4); followed by vecAlg, which produces Vector (a * 3) (a * 4). Together, we get:

g a = Vector (a * 3) (a * 4)


Repeating this process gives us:

g :: Expr -> Vector
g () = Vector 3 4
g a  = Vector (a * 3) (a * 4)
g (a, b) = Vector (a * b * 3) (a * b * 4)
...


This is the unique g induced by our f.

## Properties of Free Algebras

Here are some interesting properties that will help us later: $h$-algebras are closed under multiplication and exponentiation. If $(n, \nu)$ is an $h$-algebra, then there are also $h$-algebras with the carriers $n \otimes k$ and $[k, n]$, for an arbitrary object $k$. Let’s find their structure maps.

The first one should be a mapping:

$h \otimes n \otimes k \to n \otimes k$

which we can simply take as the lifting of $\nu$ by the tensor product: $\nu \otimes k$.

The second one is:

$\tau_{k} \colon h \otimes [k, n] \to [k, n]$

which can be uncurried to:

$h \otimes [k, n] \otimes k \to n$

We have at our disposal the counit of the hom-adjunction:

$eval \colon [k, n] \otimes k \to n$

which we can use in combination with $\nu$:

$\nu \circ (h \otimes eval)$

to implement the (uncurried) structure map.

Here’s the same construction for Haskell programmers. Given an algebra:

nu :: Algebra ((,) h) n


we can create two algebras:

alpha :: Algebra ((,) h) (n, k)
alpha (a, (n, k)) = (nu (a, n), k)


and:

tau :: Algebra ((,) h) (k -> n)
tau (a, kton) = k -> nu (a, kton k)


These two algebras are related through an adjunction in the category of $h$-algebras:

$Alg\big((n \otimes k, \nu \otimes k), (l, \lambda)\big) \cong Alg\big((n, \nu), ([k, l], \tau_{k})\big)$

which follows directly from hom-adjunction acting on carriers.

$C(n \otimes k, l) \cong C(n, [k, l])$

Finally, we can show how to generate free algebras from arbitrary generators. Intuitively, this is obvious, since it can be described as the application of distributivity of tensor product over coproduct:

$k + h \otimes k + h \otimes h \otimes k + ... = (I + h + h \otimes h + ...) \otimes k$

More formally, we have:

Proposition.
If $(m, \sigma)$ is is the free $h$-algebra generated by $I$, then $(m \otimes k, \sigma \otimes k)$ is the free $h$-algebra generated by $k$ with the universal map given by $\pi \otimes k$.

Proof.
Let’s show the universal property of $(m \otimes k, \sigma \otimes k)$. Take any $h$-algebra $(n, \nu)$ and a morphism:

$f \colon k \to n$

We want to show that there is a unique $g \colon m \otimes k \to n$ that closes the diagram in Fig. 9.

Fig. 9. Free $h$-algebra generated by $k \cong I \otimes k$

I have rewritten $f$ (taking advantage of the isomorphism $I \otimes k \cong k$), as:

$f \colon I \otimes k \to n$

We can curry it to get:

$\bar{f} \colon I \to [k, n]$

The intuition is that the morphism $\bar{f}$ selects an element of the internal hom $[k, n]$ that corresponds to the original morphism $f \colon k \to n$.

We’ve seen earlier that there exists an $h$-algebra with the carrier $[k, n]$ and the structure map $\tau_k$. But since $m$ is the free $h$-algebra, there must be a unique algebra morphism $\bar{g}$ (see Fig. 10):

$\bar{g} \colon (m, \sigma) \to ([k, n], \tau_k)$

such that:

$\bar{g} \circ \pi = \bar{f}$

Fig. 10. The construction of the unique morphism $\bar{g}$

Uncurying this $\bar{g}$ gives us the sought after $g$.

The universal condition in Fig. 9:

$g \circ (\pi \otimes k) = f$

follows from pasting together two diagrams that define the relevant hom-adjunctions in Fig 11 (c.f., Fig. 1).

Fig. 11. The diagram defining the currying of both $g$ and $g \circ (\pi \otimes k)$. This is the pasting together of two diagrams that define the universal property of the internal hom $[k, n]$, one for the object $I$ and one for the object $m$.

$\square$

The immediate consequence of this proposition is that, in order to show that two $h$-algebra morphisms $g, g' \colon m \otimes k \to n$ are equal, it’s enough to show the equality of two regular morphisms:

$g \circ (\pi \otimes k) = g' \circ (\pi \otimes k) \colon k \to n$

(modulo isomorphism between $k$ and $I \otimes k$). We’ll use this property later.

It’s instructive to pick apart this proof in the simpler Haskell setting. We have the target algebra:

nu :: Algebra ((,) h) n


There is a related algebra $[k, n]$ of the form:

tau :: Algebra ((,) h) (k -> n)
tau (a, kton) = k -> nu (a, kton k)


We’ve analyzed, previously, a version of the universal construction of $g$, which we can now generalize to Fig. 10. We can build up the definition of $\bar{g}$, starting with the condition $\bar{g} \circ \pi = \bar{f}$. Here, $\bar{f}$ selects a function from the hom-set: this function is our $f$. That gives us the action of $g$ on the unit:

g :: Expr -> (k -> n)
g () = f


Next, we make use of the fact that $\bar{g}$ is an algebra morphism that satisfies the commuting condition:

$\bar{g} \circ \sigma = \tau \circ (h \otimes \bar{g})$

As before, we apply this equation to the expression (a, ()). The left hand side produces g a, while the right hand side produces tau (a, f). Next, we
apply the same equation to (a, (b, ())). The left hand side produces g (a, b). The right hand produces tau (a, tau (b, f)), and so on. Applying the definition of tau, we get:

g :: Expr -> (k -> n)
g () = f
g a  = k -> nu (a, f k)
g (a, b) = k -> nu (a, nu (b, f k))
...


Notice the order reversal in function application. The list that is the argument to g is converted to a series of applications of nu, with list elements in reverse order. We first apply nu (b, -) and then nu (a, -). This reversal is crucial to implementing list concatenation, where nu will prepend elements of the first list to the second list. We’ll see this in the second installment of this blog post.

## Bibliography

1. G.M. Kelly, A Unified Treatment of Transfinite Constructions for Free Algebras, Free Monoids, Colimits, Associated Sheaves, and so On. Bulletin of the Australian Mathematical Society, vol. 22, no. 01, 1980, p. 1.