In the previous installment of Category Theory for Programmers we talked about the category of types and functions. If you’re new to the series, here’s the link to the initial post.

You can get real appreciation for categories by studying a variety of examples. Categories come in all shapes and sizes and often pop up in unexpected places. We’ll start with something really simple.

No Objects

The most trivial category is one with zero objects and, consequently, zero morphisms. It’s a very sad category by itself, but it may be important in the context of other categories, for instance, in the category of all categories (yes, there is one). If you think that an empty set makes sense, then why not an empty category?

Simple Graphs

You can build categories just by connecting objects with arrows. You can imagine starting with any directed graph and making it into a category by simply adding more arrows. First, add an identity arrow at each node. Then, for any two arrows such that the end of one coincides with the beginning of the other (in other words, any two composable arrows), add a new arrow to serve as their composition. Every time you add a new arrow, you have to also consider its composition with any other arrow (except for the identity arrows) and itself. You usually end up with infinitely many arrows, but that’s okay.

Another way of looking at this process is that you’re creating a category, which has an object for every node in the graph, and all possible chains of composable graph edges as morphisms. (You may even consider identity morphisms as special cases of chains of length zero.)

Such a category is called a free category generated by a given graph. It’s an example of a free construction, a process of completing a given structure by extending it with a minimum number of items to satisfy its laws (here, the laws of a category). We’ll see more examples of it in the future.


And now for something completely different! A category where a morphism is a relation between objects: the relation of being less than or equal. Let’s check if it indeed is a category. Do we have identity morphisms? Every object is less than or equal to itself: check! Do we have composition? If a <= b and b <= c then a <= c: check! Is composition associative? Check! A set with a relation like this is called a preorder, so a preorder is indeed a category.

You can also have a stronger relation, that satisfies an additional condition that, if a <= b and b <= a then a must be the same as b. That’s called a partial order.

Finally, you can impose the condition that any two objects are in a relation with each other, one way or another; and that gives you a linear order or total order.

Let’s characterize these ordered sets as categories. A preorder is a category where there is at most one morphism going from any object a to any object b. Another name for such a category is “thin.” A preorder is a thin category.

A set of morphisms from object a to object b in a category C is called a hom-set and is written as C(a, b) (or, sometimes, HomC(a, b)). So every hom-set in a preorder is either empty or a singleton. That includes the hom-set C(a, a), the set of morphisms from a to a, which must be a singleton, containing only the identity, in any preorder. You may, however, have cycles in a preorder. Cycles are forbidden in a partial order.

You can easily convince yourself that any directed acyclic graph generates a partial order as its free category.

It’s very important to be able to recognize preorders, partial orders, and total orders because of sorting. Sorting algorithms, such as quicksort, bubble sort, merge sort, etc., can only work correctly on total orders. Partial orders can be sorted using topological sort.

Monoid as Set

Monoid is an embarrassingly simple but amazingly powerful concept. It’s the concept behind basic arithmetics: Both addition and multiplication form a monoid. Monoids are ubiquitous in programming. They show up as strings, lists, foldable data structures, futures in concurrent programming, events in functional reactive programming, and so on.

Traditionally, a monoid is defined as a set with a binary operation. All that’s required from this operation is that it’s associative, and that there is one special element that behaves like a unit with respect to it.

For instance, natural numbers with zero form a monoid under addition. Associativity means that:

(a + b) + c = a + (b + c)

(In other words, we can skip parentheses when adding numbers.)

The neutral element is zero, because:

0 + a = a


a + 0 = a

The second equation is redundant, because addition is commutative (a + b = b + a), but commutativity is not part of the definition of a monoid. For instance, string concatenation is not commutative and yet it forms a monoid. The neutral element for string concatenation, by the way, is an empty string, which can be attached to either side of a string without changing it.

In Haskell we can define a type class for monoids — a type for which there is a neutral element called mempty and a binary operation called mappend:

class Monoid m where
    mempty  :: m
    mappend :: m -> m -> m

The type signature for a two-argument function, m->m->m, might look strange at first, but it will make perfect sense after we talk about currying. You may interpret a signature with multiple arrows in two basic ways: as a function of multiple arguments, with the rightmost type being the return type; or as a function of one argument (the leftmost one), returning a function. The latter interpretation may be emphasized by adding parentheses (which are redundant, because the arrow is right-associative), as in: m->(m->m). We’ll come back to this interpretation in a moment.

Notice that, in Haskell, there is no way to express the monoidal properties of mempty and mappend (i.e., the fact that mempty is neutral and that mappend is associative). It’s the responsibility of the programmer to make sure they are satisfied.

Haskell classes are not as intrusive as C++ classes. When you’re defining a new type, you don’t have to specify its class up front. You are free to procrastinate and declare a given type to be an instance of some class much later. As an example, let’s declare String to be a monoid by providing the implementation of mempty and mappend (this is, in fact, done for you in the standard Prelude):

instance Monoid String where
    mempty = ""
    mappend = (++)

Here, we have reused the list concatenation operator (++), because a String is just a list of characters.

A word about Haskell syntax: Any infix operator can be turned into a two-argument function by surrounding it with parentheses. Given two strings, you can concatenate them by inserting ++ between them:

"Hello " ++ "world!"

or by passing them as two arguments to the parenthesized (++):

(++) "Hello " "world!"

Notice that arguments to a function are not separated by commas or surrounded by parentheses. (This is probably the hardest thing to get used to when learning Haskell.)

It’s worth emphasizing that Haskell lets you express equality of functions, as in:

mappend = (++)

Conceptually, this is different than expressing the equality of values produced by functions, as in:

mappend s1 s2 = (++) s1 s2

The former translates into equality of morphisms in the category Hask (or Set, if we ignore bottoms). Such equations are not only more succinct, but can often be generalized to other categories. The latter is called extensional equality, and states the fact that for any two input strings, the outputs of mappend and (++) are the same. Since the values of arguments are sometimes called points (as in: the value of f at point x), this is called point-wise equality. Function equality without specifying the arguments is described as point-free. (Incidentally, point-free equations often involve composition of functions, which is symbolized by a point, so this might be a little confusing to the beginner.)

The closest one can get to declaring a monoid in C++ would be to use the (proposed) syntax for concepts.

template<class T>
  T mempty = delete;

template<class T>
  T mappend(T, T) = delete;

template<class M>
  concept bool Monoid = requires (M m) {
    { mempty<M> } -> M;
    { mappend(m, m); } -> M;

The first definition uses a value template (also proposed). A polymorphic value is a family of values — a different value for every type.

The keyword delete means that there is no default value defined: It will have to be specified on a case-by-case basis. Similarly, there is no default for mappend.

The concept Monoid is a predicate (hence the bool type) that tests whether there exist appropriate definitions of mempty and mappend for a given type M.

An instantiation of the Monoid concept can be accomplished by providing appropriate specializations and overloads:

std::string mempty<std::string> = {""};

std::string mappend(std::string s1, std::string s2) {
    return s1 + s2;

Monoid as Category

That was the “familiar” definition of the monoid in terms of elements of a set. But as you know, in category theory we try to get away from sets and their elements, and instead talk about objects and morphisms. So let’s change our perspective a bit and think of the application of the binary operator as “moving” or “shifting” things around the set.

For instance, there is the operation of adding 5 to every natural number. It maps 0 to 5, 1 to 6, 2 to 7, and so on. That’s a function defined on the set of natural numbers. That’s good: we have a function and a set. In general, for any number n there is a function of adding n — the “adder” of n.

How do adders compose? The composition of the function that adds 5 with the function that adds 7 is a function that adds 12. So the composition of adders can be made equivalent to the rules of addition. That’s good too: we can replace addition with function composition.

But wait, there’s more: There is also the adder for the neutral element, zero. Adding zero doesn’t move things around, so it’s the identity function in the set of natural numbers.

Instead of giving you the traditional rules of addition, I could as well give you the rules of composing adders, without any loss of information. Notice that the composition of adders is associative, because the composition of functions is associative; and we have the zero adder corresponding to the identity function.

An astute reader might have noticed that the mapping from integers to adders follows from the second interpretation of the type signature of mappend as m->(m->m). It tells us that mappend maps an element of a monoid set to a function acting on that set.

Now I want you to forget that you are dealing with the set of natural numbers and just think of it as a single object, a blob with a bunch of morphisms — the adders. A monoid is a single object category. In fact the name monoid comes from Greek mono, which means single. Every monoid can be described as a single object category with a set of morphisms that follow appropriate rules of composition.


String concatenation is an interesting case, because we have a choice of defining right appenders and left appenders (or prependers, if you will). The two models are a mirror inverse of each other. You can easily convince yourself that appending “bar” to “foo” corresponds to prepending “rab” to “oof”.

You might ask the question whether every categorical monoid — a one-object category — defines a unique set-with-binary-operator monoid. It turns out that we can always extract a set from a single-object category. This set is the set of morphisms — the adders in our example. In other words, we have the hom-set M(m, m) of the single object m in the category M. We can easily define a binary operator in this set: The monoidal product of two set-elements is the element corresponding to the composition of the corresponding morphisms. If you give me two elements of M(m, m) corresponding to f and g, their product will correspond to the composition g∘f. The composition always exists, because the source and the target for these morphisms are the same object. And it’s associative by the rules of category. The identity morphism is the neutral element of this product. So we can always recover a set monoid from a category monoid. For all intents and purposes they are one and the same.

Monoid hom-set seen as morphisms and as points in a set

Monoid hom-set seen as morphisms and as points in a set

There is just one little nit for mathematicians to pick: morphisms don’t have to form a set. In the world of categories there are things larger than sets. A category in which morphisms between any two objects form a set is called locally small. As promised, I will be mostly ignoring such subtleties, but I thought I should mention them for the record.

A lot of interesting phenomena in category theory have their root in the fact that elements of a hom-set can be seen both as morphisms, which follow the rules of composition, and as points in a set. Here, composition of morphisms in M translates into monoidal product in the set M(m, m).


I’d like to thank Andrew Sutton for rewriting my C++ monoid concept code according to his and Bjarne Stroustrup’s latest proposal.


  1. Generate a free category from:
    1. A graph with one node and no edges
    2. A graph with one node and one (directed) edge (hint: this edge can be composed with itself)
    3. A graph with two nodes and a single arrow between them
    4. A graph with a single node and 26 arrows marked with the letters of the alphabet: a, b, c … z.
  2. What kind of order is this?
    1. A set of sets with the inclusion relation: A is included in B if every element of A is also an element of B.
    2. C++ types with the following subtyping relation: T1 is a subtype of T2 if a pointer to T1 can be passed to a function that expects a pointer to T2 without triggering a compilation error.
  3. Considering that Bool is a set of two values True and False, show that it forms two (set-theoretical) monoids with respect to, respectively, operator && (AND) and || (OR).
  4. Represent the Bool monoid with the AND operator as a category: List the morphisms and their rules of composition.
  5. Represent addition modulo 3 as a monoid category.

This is part of the book Category Theory for Programmers. The previous instalment was Category: The Essence of Composition.

The category of types and functions plays an important role in programming, so let’s talk about what types are and why we need them.

Who Needs Types?

There seems to be some controversy about the advantages of static vs. dynamic and strong vs. weak typing. Let me illustrate these choices with a thought experiment. Imagine millions of monkeys at computer keyboards happily hitting random keys, producing programs, compiling, and running them.


With machine language, any combination of bytes produced by monkeys would be accepted and run. But with higher level languages, we do appreciate the fact that a compiler is able to detect lexical and grammatical errors. Lots of monkeys will go without bananas, but the remaining programs will have a better chance of being useful. Type checking provides yet another barrier against nonsensical programs. Moreover, whereas in a dynamically typed language, type mismatches would be discovered at runtime, in strongly typed statically checked languages type mismatches are discovered at compile time, eliminating lots of incorrect programs before they have a chance to run.

So the question is, do we want to make monkeys happy, or do we want to produce correct programs?

The usual goal in the typing monkeys thought experiment is the production of the complete works of Shakespeare. Having a spell checker and a grammar checker in the loop would drastically increase the odds. The analog of a type checker would go even further by making sure that, once Romeo is declared a human being, he doesn’t sprout leaves or trap photons in his powerful gravitational field.

Types Are About Composability

Category theory is about composing arrows. But not any two arrows can be composed. The target object of one arrow must be the same as the source source object of the next arrow. In programming we pass the results on one function to another. The program will not work if the target function is not able to correctly interpret the data produced by the source function. The two ends must fit for the composition to work. The stronger the type system of the language, the better this match can be described and mechanically verified.

The only serious argument I hear against strong static type checking is that it might eliminate some programs that are semantically correct. In practice, this happens extremely rarely and, in any case, every language provides some kind of a backdoor to bypass the type system when that’s really necessary. Even Haskell has unsafeCoerce. But such devices should by used judiciously. Franz Kafka’s character, Gregor Samsa, breaks the type system when he metamorphoses into a giant bug, and we all know how it ends.

Another argument I hear a lot is that dealing with types imposes too much burden on the programmer. I could sympathize with this sentiment after having to write a few declarations of iterators in C++ myself, except that there is a technology called type inference that lets the compiler deduce most of the types from the context in which they are used. In C++, you can now declare a variable auto and let the compiler figure out its type.

In Haskell, except on rare occasions, type annotations are purely optional. Programmers tend to use them anyway, because they can tell a lot about the semantics of code, and they make compilation errors easier to understand. It’s a common practice in Haskell to start a project by designing the types. Later, type annotations drive the implementation and become compiler-enforced comments.

Strong static typing is often used as an excuse for not testing the code. You may sometimes hear Haskell programmers saying, “If it compiles, it must be correct.” Of course, there is no guarantee that a type-correct program is correct in the sense of producing the right ouput. The result of this cavalier attitude is that in several studies Haskell didn’t come as strongly ahead of the pack in code quality as one would expect. It seems that, in the commercial setting, the pressure to fix bugs is applied only up to a certain quality level, which has everything to do with the economics of software development and the tolerance of the end user, and very little to do with the programming language or methodology. A better criterion would be to measure how many projects fall behind schedule or are delivered with drastically reduced functionality.

As for the argument that unit testing can replace strong typing, consider the common refactoring practice in strongly typed languages: changing the type of an argument of a particular function. In a strongly typed language, it’s enough to modify the declaration of that function and then fix all the build breaks. In a weakly typed language, the fact that a function now expects different data cannot be propagated to call sites. Unit testing may catch some of the mismatches, but testing is almost always a probabilistic rather than a deterministic process. Testing is a poor substitute for proof.

What Are Types?

The simplest intuition for types is that they are sets of values. The type Bool (remember, concrete types start with a capital letter in Haskell) is a two-element set of True and False. Type Char is a set of all Unicode characters like 'a' or 'ą'.

Sets can be finite or infinite. The type of String, which is a synonym for a list of Char, is an example of an infinite set.

When we declare x to be an Integer:

x :: Integer

we are saying that it’s an element of the set of integers. Integer in Haskell is an infinite set, and it can be used to do arbitrary precision arithmetic. There is also a finite-set Int that corresponds to machine type, just like the C++ int.

There are some subtleties that make this identification of types and sets tricky. There are problems with polymorphic functions that involve circular definitions, and with the fact that you can’t have a set of all sets; but as I promised, I won’t be a stickler for math. The great thing is that there is a category of sets, which is called Set, and we’ll just work with it. In Set, objects are sets and morphisms (arrows) are functions.

Set is a very special category, because we can actually peek inside its objects and get a lot of intuitions from doing that. For instance, we know that an empty set has no elements. We know that there are special one-element sets. We know that functions map elements of one set to elements of another set. They can map two elements to one, but not one element to two. We know that an identity function maps each element of a set to itself, and so on. The plan is to gradually forget all this information and instead express all those notions in purely categorical terms, that is in terms of objects and arrows.

In the ideal world we would just say that Haskell types are sets and Haskell functions are mathematical functions between sets. There is just one little problem: A mathematical function does not execute any code — it just knows the answer. A Haskell function has to calculate the answer. It’s not a problem if the answer can be obtained in a finite number of steps — however big that number might be. But there are some calculations that involve recursion, and those might never terminate. We can’t just ban non-terminating functions from Haskell because distinguishing between terminating and non-terminating functions is undecidable — the famous halting problem. That’s why computer scientists came up with a brilliant idea, or a major hack, depending on your point of view, to extend every type by one more special value called the bottom and denoted by _|_, or Unicode ⊥. This “value” corresponds to a non-terminating computation. So a function declared as:

f :: Bool -> Bool

may return True, False, or _|_; the latter meaning that it would never terminate.

Interestingly, once you accept the bottom as part of the type system, it is convenient to treat every runtime error as a bottom, and even allow functions to return the bottom explicitly. The latter is usually done using the expression undefined, as in:

f :: Bool -> Bool
f x = undefined

This definition type checks because undefined evaluates to bottom, which is a member of any type, including Bool. You can even write:

f :: Bool -> Bool
f = undefined

(without the x) because the bottom is also a member of the type Bool->Bool.

Functions that may return bottom are called partial, as opposed to total functions, which return valid results for every possible argument.

Because of the bottom, you’ll see the category of Haskell types and functions referred to as Hask rather than Set. From the theoretical point of view, this is the source of never-ending complications, so at this point I will use my butcher’s knife and terminate this line of reasoning. From the pragmatic point of view, it’s okay to ignore non-terminating functions and bottoms, and treat Hask as bona fide Set (see Bibliography at the end).

Why Do We Need a Mathematical Model?

As a programmer you are intimately familiar with the syntax and grammar of your programming language. These aspects of the language are usually described using formal notation at the very beginning of the language spec. But the meaning, or semantics, of the language is much harder to describe; it takes many more pages, is rarely formal enough, and almost never complete. Hence the never ending discussions among language lawyers, and a whole cottage industry of books dedicated to the exegesis of the finer points of language standards.

There are formal tools for describing the semantics of a language but, because of their complexity, they are mostly used with simplified academic languages, not real-life programming behemoths. One such tool called operational semantics describes the mechanics of program execution. It defines a formalized idealized interpreter. The semantics of industrial languages, such as C++, is usually described using informal operational reasoning, often in terms of an “abstract machine.”

The problem is that it’s very hard to prove things about programs using operational semantics. To show a property of a program you essentially have to “run it” through the idealized interpreter.

It doesn’t matter that programmers never perform formal proofs of correctness. We always “think” that we write correct programs. Nobody sits at the keyboard saying, “Oh, I’ll just throw a few lines of code and see what happens.” We think that the code we write will perform certain actions that will produce desired results. We are usually quite surprised when it doesn’t. That means we do reason about programs we write, and we usually do it by running an interpreter in our heads. It’s just really hard to keep track of all the variables. Computers are good at running programs — humans are not! If we were, we wouldn’t need computers.

But there is an alternative. It’s called denotational semantics and it’s based on math. In denotational semantics every programing construct is given its mathematical interpretation. With that, if you want to prove a property of a program, you just prove a mathematical theorem. You might think that theorem proving is hard, but the fact is that we humans have been building up mathematical methods for thousands of years, so there is a wealth of accumulated knowledge to tap into. Also, as compared to the kind of theorems that professional mathematicians prove, the problems that we encounter in programming are usually quite simple, if not trivial.

Consider the definition of a factorial function in Haskell, which is a language quite amenable to denotational semantics:

fact n = product [1..n]

The expression [1..n] is a list of integers from 1 to n. The function product multiplies all elements of a list. That’s just like a definition of factorial taken from a math text. Compare this with C:

int fact(int n) {
    int i;
    int result = 1;
    for (i = 2; i <= n; ++i)
        result *= i;
    return result;

Need I say more?

Okay, I’ll be the first to admit that this was a cheap shot! A factorial function has an obvious mathematical denotation. An astute reader might ask: What’s the mathematical model for reading a character from the keyboard or sending a packet across the network? For the longest time that would have been an awkward question leading to a rather convoluted explanation. It seemed like denotational semantics wasn’t the best fit for a considerable number of important tasks that were essential for writing useful programs, and which could be easily tackled by operational semantics. The breakthrough came from category theory. Eugenio Moggi discovered that computational effect can be mapped to monads. This turned out to be an important observation that not only gave denotational semantics a new lease on life and made pure functional programs more usable, but also shed new light on traditional programming. I’ll talk about monads later, when we develop more categorical tools.

One of the important advantages of having a mathematical model for programming is that it’s possible to perform formal proofs of correctness of software. This might not seem so important when you’re writing consumer software, but there are areas of programming where the price of failure may be exorbitant, or where human life is at stake. But even when writing web applications for the health system, you may appreciate the thought that functions and algorithms from the Haskell standard library come with proofs of correctness.

Pure and Dirty Functions

The things we call functions in C++ or any other imperative language, are not the same things mathematicians call functions. A mathematical function is just a mapping of values to values.

We can implement a mathematical function in a programming language: Such a function, given an input value will calculate the output value. A function to produce a square of a number will probably multiply the input value by itself. It will do it every time it’s called, and it’s guaranteed to produce the same output every time it’s called with the same input. The square of a number doesn’t change with the phases of the Moon.

Also, calculating the square of a number should not have a side effect of dispensing a tasty treat for your dog. A “function” that does that cannot be easily modelled as a mathematical function.

In programming languages, functions that always produce the same result given the same input and have no side effects are called pure functions. In a pure functional language like Haskell all functions are pure. Because of that, it’s easier to give these languages denotational semantics and model them using category theory. As for other languages, it’s always possible to restrict yourself to a pure subset, or reason about side effects separately. Later we’ll see how monads let us model all kinds of effects using only pure functions. So we really don’t lose anything by restricting ourselves to mathematical functions.

Examples of Types

Once you realize that types are sets, you can think of some rather exotic types. For instance, what’s the type corresponding to an empty set? No, it’s not C++ void, although this type is called Void in Haskell. It’s a type that’s not inhabited by any values. You can define a function that takes Void, but you can never call it. To call it, you would have to provide a value of the type Void, and there just aren’t any. As for what this function can return, there are no restrictions whatsoever. It can return any type (although it never will, because it can’t be called). In other words it’s a function that’s polymorphic in the return type. Haskellers have a name for it:

absurd :: Void -> a

(Remember, a is a type variable that can stand for any type.) The name is not coincidental. There is deeper interpretation of types and functions in terms of logic called the Curry-Howard isomorphism. The type Void represents falsity, and the function absurd is the statement that from falsity follows anything, as in the Latin adage “ex falso sequitur quodlibet.”

Next is the type that corresponds to a singleton set. It’s a type that has only one possible value. This value just “is.” You might not immediately recognise it as such, but that is the C++ void. Think of functions from and to this type. A function from void can always be called. If it’s a pure function, it will always return the same result. Here’s an example of such a function:

int f44() { return 44; }

You might think of this function as taking “nothing”, but as we’ve just seen, a function that takes “nothing” can never be called because there is no value representing “nothing.” So what does this function take? Conceptually, it takes a dummy value of which there is only one instance ever, so we don’t have to mention it explicitly. In Haskell, however, there is a symbol for this value: an empty pair of parentheses, (). So, by a funny coincidence (or is it a coincidence?), the call to a function of void looks the same in C++ and in Haskell. Also, because of the Haskell’s love of terseness, the same symbol () is used for the type, the constructor, and the only value corresponding to a singleton set. So here’s this function in Haskell:

f44 :: () -> Integer
f44 () = 44

The first line declares that f44 takes the type (), pronounced “unit,” into the type Integer. The second line defines f44 by pattern matching the only constructor for unit, namely (), and producing the number 44. You call this function by providing the unit value ():

f44 ()

Notice that every function of unit is equivalent to picking a single element from the target type (here, picking the Integer 44). In fact you could think of f44 as a different representation for the number 44. This is an example of how we can replace explicit mention of elements of a set by talking about functions (arrows) instead. Functions from unit to any type A are in one-to-one correspondence with the elements of that set A.

What about functions with the void return type, or, in Haskell, with the unit return type? In C++ such functions are used for side effects, but we know that these are not real functions in the mathematical sense of the word. A pure function that returns unit does nothing: it discards its argument.

Mathematically, a function from a set A to a singleton set maps every element of A to the single element of that singleton set. For every A there is exactly one such function. Here’s this function for Integer:

fInt :: Integer -> ()
fInt x = ()

You give it any integer, and it gives you back a unit. In the spirit of terseness, Haskell lets you use the wildcard pattern, the underscore, for an argument that is discarded. This way you don’t have to invent a name for it. So the above can be rewritten as:

fInt :: Integer -> ()
fInt _ = ()

Notice that the implementation of this function not only doesn’t depend on the value passed to it, but it doesn’t even depend on the type of the argument.

Functions that can be implemented with the same formula for any type are called parametrically polymorphic. You can implement a whole family of such functions with one equation using a type parameter instead of a concrete type. What should we call a polymorphic function from any type to unit type? Of course we’ll call it unit:

unit :: a -> ()
unit _ = ()

In C++ you would write this function as:

template<class T>
void unit(T) {}

Next in the typology of types is a two-element set. In C++ it’s called bool and in Haskell, predictably, Bool. The difference is that in C++ bool is a built-in type, whereas in Haskell it can be defined as follows:

data Bool = True | False

(The way to read this definition is that Bool is either True or False.) In principle, one should also be able to define a Boolean type in C++ as an enumeration:

enum bool {

but C++ enum is secretly an integer. The C++11 “enum class” could have been used instead, but then you would have to qualify its values with the class name, as in bool::true and bool::false, not to mention having to include the appropriate header in every file that uses it.

Pure functions from Bool just pick two values from the target type, one corresponding to True and another to False.

Functions to Bool are called predicates. For instance, the Haskell library Data.Char is full of predicates like isAlpha or isDigit. In C++ there is a similar library <cctype> that defines, among others, isalpha and isdigit, but these return an int rather than a Boolean. The actual predicates are defined in <locale> and have the form ctype::is(alpha, c), ctype::is(digit, c), etc.


  1. Define a higher-order function (or a function object) memoize in your favorite language. This function takes a pure function f as an argument and returns a function that behaves almost exactly like f, except that it only calls the original function once for every argument, stores the result internally, and subsequently returns this stored result every time it’s called with the same argument. You can tell the memoized function from the original by watching its performance. For instance, try to memoize a function that takes a long time to evaluate. You’ll have to wait for the result the first time you call it, but on subsequent calls, with the same argument, you should get the result immediately.
  2. Try to memoize a function from your standard library that you normally use to produce random numbers. Does it work?
  3. Most random number generators can be initialized with a seed. Implement a function that takes a seed, calls the random number generator with that seed, and returns the result. Memoize that function. Does it work?
  4. Which of these C++ functions are pure? Try to memoize them and observe what happens when you call them multiple times: memoized and not.
    1. The factorial function from the example in the text.
    2. std::getchar()
    3. bool f() { 
          std::cout << "Hello!" << std::endl;
          return true; 
    4. int f(int x)
          static int y = 0;
          y += x;
          return y;
  5. How many different functions are there from Bool to Bool? Can you implement them all?
  6. Draw a picture of a category whose only objects are the types Void, () (unit), and Bool; with arrows corresponding to all possible functions between these types. Label the arrows with the names of the functions.


  1. Nils Anders Danielsson, John Hughes, Patrik Jansson, Jeremy Gibbons, Fast and Loose Reasoning is Morally Correct. This paper provides justification for ignoring bottoms in most contexts.

Next: Categories Great and Small

Ferrari museum in Maranello

Ferrari museum in Maranello

I was recently visiting the Ferrari museum in Maranello, Italy, where I saw this display of telemetry data from racing cars.

Telementry data from a racing car. The contour of the racing track is shown in the upper left corner and various data channels are displayed below.

Telementry data from a racing car. The racing track is displayed in the upper left corner and various data channels are displayed below.

The processing and the display of telemetry data is an interesting programming challenge. It has application in space exploration (as in, when you land a probe on a surface of a comet), medicine, and the military. The same techniques are used in financial systems where streams carry information about stock prices, commodity prices, and currency exchange rates.

It’s also a problem that lends itself particularly well to functional programming. If you are one of these shops working with telemetry, and you have to maintain legacy code written in imperative style, you might be interested in an alternative approach, especially if you are facing constant pressure to provide more sophisticated analysis tools and introduce concurrency to make the system faster and more responsive.

What all these applications have in common is that they deal with multiple channels generating streams of data. The data has to be either displayed in real time or stored for later analysis and processing. It’s pretty obvious to a functional programmer that channels are functors, and that they should be composed using combinators. In fact this observation can drive the whole architecture. The clincher is the issue of concurrency: retrofitting non-functional code to run in parallel is a lost battle — it’s almost easier to start from scratch. But treating channels as immutable entities makes concurrency almost an after-thought.

Everything is a Number

The most basic (and totally wrong) approach is to look at telemetry as streams of numbers. This is the assembly language of data processing. When everything is a number and you can apply your math any way you wish. The problem is that you are throwing away a lot of useful information. You want to use types as soon as possible to encode additional information and to prevent nonsensical operations like adding temperature to velocity.

In an engineering application, the least you can do is to keep track of units of measurement. You also want to distinguish between channels that produce floating-point numbers and ones that produce integers, or Booleans, or strings. This immediately tells you that a channel should be a polymorphic data structure. You should be able to stream any type of data, be it bytes, complex numbers, or vectors.

Everything is an Object

To an object-oriented mind it looks very much like a channel should be an object that is parameterized by the type of data it carries. And as an object it should have some methods. We need the get method to access the current value, and the next method to increment the position in the stream. As an imperative programmer you might also be tempted to provide a mutator, set. If you ever want your program to be concurrent, don’t even think about it!

If you’re a C++ programmer, you may overload some operators, and use * and ++ instead. That would make a channel look more like a forward iterator. But whatever you call it, a functional programmer will recognize it as a list, with the head and tail functionality.

Everything is a List

Let’s talk about lists, because there is a lot of misunderstanding around them. When people think of lists in imperative languages they think about storage. A list is probably the worst data type for storing data. Imperative programmers naturally assume that functional programmers, who use lists a lot, must be crazy. They are not! A Haskell list is almost never used for storing bulk data. A list is either an interface to data that is stored elsewhere, or a generator of data. Haskell is a lazy functional language, so what looks like a data structure is really a bunch of functions that provide data on demand.

That’s why I wouldn’t hesitate to implement channels as lists in Haskell. As an added bonus, lists can provide a pull interface to data that is being pushed. Reactive programs that process streams of data may be written as if all the data were already there — the event handler logic can be hidden inside the objects that generate the data. And this is just what’s needed for live telemetry data.

Obviously, functional programming is easier in Haskell than in C++, C#, or Java. But given how much legacy software there is, it could be a lost cause to ask management to (a) throw away existing code and start from scratch, (b) retrain the team to learn a new language, and (c) deal with completely new performance characteristics, e.g., lazy evaluation and garbage collection. So, realistically, the best we can do is to keep introducing functional methods into imperative languages, at least for the time being. It doesn’t mean that Haskell should’t play an important role in it. Over and over again I find myself prototyping solutions in Haskell before translating them into C++. The added effort pays back handsomely through faster prototyping, better code quality, and fewer bugs to chase. So I would highly recommend to every imperative programmer to spend, say, an hour a day learning and playing with Haskell. You’d be amazed how it helps in developing your programming skills.

Everything is a Functor

So, if you’re an object oriented programmer, you’ll probably implement a channel as something like this:

template <class T> Channel {
    virtual T get();
    virtual bool next();

and then get stuck. With this kind of interface, the rest of your program is bound to degenerate into a complex system of loops that extract data from streams and process them, possibly stuffing it back into other streams.

Instead, I propose to try the functional way. I will show you some prototype code in Haskell, but mostly explain how things work, so a non-Haskell programmer can gain some insight.

Here’s the definition of a polymorphic channel type, Chan:

data Chan a = Chan [a]

where a plays the role of a type variable, analogous to T in the C++ code above. The right hand side of the equal sign defines the constructor Chan that takes a list as an argument. Constructors are used both for constructing and for pattern matching. The notation [a] means a list of a.

The details don’t really matter, as long as you understand that the channel is implemented as a list. Also, I’m making things a little bit more explicit for didactic purposes. A Haskell programmer would implement the channel as a type alias, type, rather than a separate type.

Rule number one of dealing with lists is: try not to access their elements in a loop (or, using the functional equivalent of a loop — recursively). Operate on lists holistically. For instance, one of the most common operations on lists is to apply a function to every element. That means we want our Chan to be a functor.

A functor is a polymorphic data type that supports operating on its contents with a function. In the case of Chan that’s easy, since a list itself is a functor. I’ll be explicit here, again for didactic reasons. This is how you make Chan an instance of the Functor class by defining how to fmap a function f over it:

instance Functor Chan where
    fmap f (Chan xs) = Chan (map f xs)

Here, map is a library function that applies f to every element of the list. This is very much like applying C++ std::transform to a container, except that in Haskell everything is evaluated lazily, so you can apply fmap to an infinite list, or to a list that is not there yet because, for instance, it’s being generated in real time from incoming telemetry.

Everything is a Combinator

Let’s see how far we can get with this channel idea. The next step is to be able to combine multiple channels to generate streams of derived data. For instance, suppose that you have one channel from a pressure gauge, and another providing volume data, and you want to calculate instantaneous temperature using the ideal gas equation.

Let’s start with defining some types. We want separate types for quantities that are measured using different units. Once more, I’m being didactic here, because there are ready-made Haskell libraries that use so called phantom types to encode measurement units. Here I’ll do it naively:

data Pressure = Pascal Float
data Volume   = Meter3 Float
data Temp     = Kelvin Float

I’ll also define the ideal gas constant:

constR = 8.314472 -- J/(mol·K)

Here’s the function that calculates the temperature of ideal gas:

getT :: Float -> Pressure -> Volume -> Temp
getT n (Pascal p) (Meter3 v) = Kelvin (p * v / (n * constR))

The question is, how can we apply this function to the pressure and volume channels to get the temperature channel? We know how to apply a function to a single channel using fmap, but here we have to work with two channels. Fortunately, a channel is not just a functor — it’s an applicative functor. It defines the action of multi-argument functions on multiple channels. I’ll give you a Haskell implementation, but you should be able to do the same in C++ by overloading fmap or transform.

instance Applicative Chan where
    pure x = Chan (repeat x)
    (Chan fs) <*> (Chan xs) = Chan (zipWith ($) fs xs)

The Applicative class defines two functions. One is called pure, and it creates a constant channel from a value by repeatedly returning the same value. The other is a binary operator <*> that applies a channel of functions (yes, you can treat functions the same way you treat any other data) to a channel of values. The function zipWith applies, pairwise, functions to arguments using the function application operator ($).

Again, the details are not essential. The bottom line is that this allows us to apply our function getT to two channels (actually, three channels, since we also need to provide the amount of gas in moles — here I’m assuming 0.1 moles).

chT :: Chan Pressure -> Chan Volume -> Chan Temp
chT chP chV = getT <$> pure 0.1 <*> chP <*> chV

Such functions that combine channels into new channels are called combinators, and an applicative functor makes the creation of new combinators very easy.

The combinators are not limited to producing physical quantities. They may as well produce channels of alerts, channels of pixels for display, or channels of visual widgets. You can construct the whole architecture around channels. And since we’ve been only considering functional data structures, the resulting architecture can be easily subject to parallelization.

Moving Average

But don’t some computations require mutable state? For instance, don’t you need some kind of accumulators in order to calculate, let’s say, moving averages? Let’s see how this can be done functionally.

The idea is to keep a running sum of list elements within a fixed window of size n. When advancing through the list, we will add the new incoming element to the running sum and subtract the old outgoing element. The average is just this sum divided by n.

We can use the old trick of delaying the list by n positions. We’ll pad the beginning of the delayed list with n zeros. Here’s the Haskell code:

delay :: Num a => Int -> [a] -> [a]
delay n lst = replicate n 0 ++ lst

The first line is the (optional, but very useful) type signature. The second line defines the function delay that takes the delay counter n and the list. The function returns a list that is obtained by concatenating (operator ++) the zero-filled list (replicate n 0) in front of the original list. For instance, if you start with the list [1, 2, 3, 4] and delay it by 2, you’ll get [0, 0, 1, 2, 3, 4].

The next step is to create a stream of deltas — the differences between elements separated by n positions. We do it by zipping two lists: the original and the delayed one.

zip lst (delay n lst)

The function zip pairs elements from the first list with the elements from the second list.

Continuing with our example, the zipping will produce the pairs [(1, 0), (2, 0), (3, 1), (4, 2)]. Notice that the left number in each pair is the incoming element that is to be added to the running sum, while the right number is the outgoing one, to be subtracted from the running sum.

Now if we subtract the two numbers in each pair we’ll get exactly the delta that has to be added to the running sum at each step. We do the subtraction by mapping the operator (-) over the list. (To make the subtraction operator (-) operate on pairs we have to uncurry it. (If you don’t know what currying is, don’t worry.)

deltas :: Num a => Int -> [a] -> [a]
deltas n lst = map (uncurry (-)) (zip lst (delay n lst))

Continuing with the example, we will get [1, 2, 2, 2]. These are the amounts by which the running sum should change at every step. (Incidentally, for n equal to one, the deltas are proportional to the derivative of the sampled stream.)

Finally, we have to start accumulating the deltas. There is a library function scanl1 that can be used to produce a list of partial sums when called with the summation operator (+).

slidingSums :: Num a => Int -> [a] -> [a]
slidingSums n lst =  scanl1 (+) (deltas n lst)

At each step, scanl1 will add the delta to the previous running sum. The “1” in its name means that it will start with the first element of the list as the accumulator. The result, in our little example, is [1, 3, 5, 7]. What remains is to divide each sum by n and we’re done:

movingAverage :: Fractional a => Int -> [a] -> [a]
movingAverage n list = map (/ (fromIntegral n)) (slidingSums n list)

Since n is an integer, it has to be explicitly converted to a fractional number before being passed to the division operator. This is done using fromIntegral. The slightly cryptic notation (/ (fromIntegral n)) is called operator section. It just means “divide by n.”

As expected, the final result for the two-element running average of [1, 2, 3, 4] is [0.5, 1.5, 2.5, 3.5]. Notice that we haven’t used any mutable state to achieve this result, which makes this code automatically thread safe. Also, because the calculation is lazy, we can calculate the moving average of an infinite list as long as we only extract a finite number of data points. Here, we are printing the first 10 points of the 5-element moving average of the list of integers from 1 to infinity.

print (take 10 (movingAverage 5 [1..]))

The result is:

[0.2, 0.6, 1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]


The functional approach is applicable to designing software not only in the small but, more importantly, in the large. It captures the patterns of interaction between components and the ways they compose. The patterns I mentioned in this post, the functor and the applicative functor, are probably the most common, but functional programmers have at their disposal a large variety of patterns borrowed from various branches of mathematics. These patterns can be used by imperative programmers as well, resulting in cleaner and more maintainable software that is, by construction, multithread-ready.

I was overwhelmed by the positive response to my previous post, the Preface to Category Theory for Programmers. At the same time, it scared the heck out of me because I realized what high expectations people were placing in me. I’m afraid that no matter what I’ll write, a lot of readers will be disappointed. Some readers would like the book to be more practical, others more abstract. Some hate C++ and would like all examples in Haskell, others hate Haskell and demand examples in Java. And I know that the pace of exposition will be too slow for some and too fast for others. This will not be the perfect book. It will be a compromise. All I can hope is that I’ll be able to share some of my aha! moments with my readers. Let’s start with the basics.

A category is an embarrassingly simple concept. A category consists of objects and arrows that go between them. That’s why categories are so easy to represent pictorially. An object can be drawn as a circle or a point, and an arrow… is an arrow. (Just for variety, I will occasionally draw objects as piggies and arrows as fireworks.) But the essence of a category is composition. Or, if you prefer, the essence of composition is a category. Arrows compose, so if you have an arrow from object A to object B, and another arrow from object B to object C, then there must be an arrow — their composition — that goes from A to C.


In a category, if there is an arrow going from A to B and an arrow going from B to C then there must also be a direct arrow from A to C that is their composition. This diagram is not a full category because it’s missing identity morphisms (see later).


Arrows as Functions

Is this already too much abstract nonsense? Do not despair. Let’s talk concretes. Think of arrows, which are also called morphisms, as functions. You have a function f that takes an argument of type A and returns a B. You have another function g that takes a B and returns a C. You can compose them by passing the result of f to g. You have just defined a new function that takes an A and returns a C.

In math, such composition is denoted by a small circle between functions: g∘f. Notice the right to left order of composition. For some people this is confusing. You may be familiar with the pipe notation in Unix, as in:

lsof | grep Chrome

or the chevron >> in F#, which both go from left to right. But in mathematics and in Haskell functions compose right to left. It helps if you read g∘f as “g after f.”

Let’s make this even more explicit by writing some C code. We have one function f that takes an argument of type A and returns a value of type B:

B f(A a);

and another:

C g(B b);

Their composition is:

C g_after_f(A a)
    return g(f(a));

Here, again, you see right-to-left composition: g(f(a)); this time in C.

I wish I could tell you that there is a template in the C++ Standard Library that takes two functions and returns their composition, but there isn’t one. So let’s try some Haskell for a change. Here’s the declaration of a function from A to B:

f :: A -> B


g :: B -> C

Their composition is:

g . f

Once you see how simple things are in Haskell, the inability to express straightforward functional concepts in C++ is a little embarrassing. In fact, Haskell will let you use Unicode characters so you can write composition as:

g ∘ f

You can even use Unicode double colons and arrows:

f ∷ A → B

So here’s the first Haskell lesson: Double colon means “has the type of…” A function type is created by inserting an arrow between two types. You compose two functions by inserting a period between them (or a Unicode circle).

Properties of Composition

There are two extremely important properties that the composition in any category must satisfy.

1. Composition is associative. If you have three morphisms, f, g, and h, that can be composed (that is, their objects match end-to-end), you don’t need parentheses to compose them. In math notation this is expressed as:

h∘(g∘f) = (h∘g)∘f = h∘g∘f

In (pseudo) Haskell:

f :: A -> B
g :: B -> C
h :: C -> D
h . (g . f) == (h . g) . f == h . g . f

(I said “pseudo,” because equality is not defined for functions.)

Associativity is pretty obvious when dealing with functions, but it may be not as obvious in other categories.

2. For every object A there is an arrow which is a unit of composition. This arrow loops from the object to itself. Being a unit of composition means that, when composed with any arrow that either starts at A or ends at A, respectively, it gives back the same arrow. The unit arrow for object A is called idA (identity on A). In math notation, if f goes from A to B then

f∘idA = f


idB∘f = f

When dealing with functions, the identity arrow is implemented as the identity function that just returns back its argument. The implementation is the same for every type, which means this function is universally polymorphic. In C++ we could define it as a template:

template<class T> T id(T x) { return x; }

Of course, in C++ nothing is that simple, because you have to take into account not only what you’re passing but also how (that is, by value, by reference, by const reference, by move, and so on).

In Haskell, the identity function is part of the standard library (called Prelude). Here’s its declaration and definition:

id :: a -> a
id x = x

As you can see, polymorphic functions in Haskell are a piece of cake. In the declaration, you just replace the type with a type variable. Here’s the trick: names of concrete types always start with a capital letter, names of type variables start with a lowercase letter. So here a stands for all types.

Haskell function definition consists of the name of the function followed by formal parameters — here just one, x. The body of the function follows the equal sign. This terseness is often shocking to newcomers but you will quickly see that it makes perfect sense. Function definition and function call are the bread and butter of functional programming so their syntax is reduced to the bare minimum. Not only there are no parentheses around the argument list but there are no commas between arguments (you’ll see that later, when we define functions of multiple arguments).

The body of a function is always an expression — there are no statements in functions. The result of a function is this expression — here, just x.

This concludes our second Haskell lesson.

The identity conditions can be written (again, in pseudo-Haskell) as:

f . id == f
id . f == f

You might be asking yourself the question: Why would anyone bother with the identity function — a function that does nothing? Then again, why do we bother with the number zero? Zero is a symbol for nothing. Ancient Romans had a number system without a zero and they were able to build excellent roads and aqueducts, some of which survive to this day.

Neutral values like zero or id are extremely useful when working with symbolic variables. That’s why Romans were not very good at algebra, whereas the Arabs and the Persians, who were familiar with the concept of zero, were. So the identity function becomes very handy as an argument to, or a return from, a higher-order function. Higher order functions are what makes symbolic manipulation of functions possible. They are the algebra of functions.

To summarize: A category consists of objects and arrows (morphisms). Arrows can be composed, and the composition is associative. Every object has an identity arrow that serves as a unit under composition.

Composition is the Essence of Programming

Functional programmers have a peculiar way of approaching problems. They start by asking very Zen-like questions. For instance, when designing an interactive program, they would ask: What is interaction? When implementing Conway’s Game of Life, they would probably ponder about the meaning of life. In this spirit, I’m going to ask: What is programming? At the most basic level, programming is about telling the computer what to do. “Take the contents of memory address x and add it to the contents of the register EAX.” But even when we program in assembly, the instructions we give the computer are an expression of something more meaningful. We are solving a non-trivial problem (if it were trivial, we wouldn’t need the help of the computer). And how do we solve problems? We decompose bigger problems into smaller problems. If the smaller problems are still too big, we decompose them further, and so on. Finally, we write code that solves all the small problems. And then comes the essence of programming: we compose those pieces of code to create solutions to larger problems. Decomposition wouldn’t make sense if we weren’t able to put the pieces back together.

This process of hierarchical decomposition and recomposition is not imposed on us by computers. It reflects the limitations of the human mind. Our brains can only deal with a small number of concepts at a time. One of the most cited papers in psychology, The Magical Number Seven, Plus or Minus Two, postulated that we can only keep 7 ± 2 “chunks” of information in our minds. The details of our understanding of the human short-term memory might be changing, but we know for sure that it’s limited. The bottom line is that we are unable to deal with the soup of objects or the spaghetti of code. We need structure not because well-structured programs are pleasant to look at, but because otherwise our brains can’t process them efficiently. We often describe some piece of code as elegant or beautiful, but what we really mean is that it’s easy to process by our limited human minds. Elegant code creates chunks that are just the right size and come in just the right number for our mental digestive system to assimilate them.

So what are the right chunks for the composition of programs? Their surface area has to be smaller than their volume. (I like this analogy because of the intuition that the surface area of a geometric object grows with the square of its size — slower than the volume, which grows with the cube of its size.) The surface area is the information we need in order to compose chunks. The volume is the information we need in order to implement them. The idea is that, once a chunk is implemented, we can forget about the details of its implementation and concentrate on how it interacts with other chunks. In object-oriented programming, the surface is the class declaration of the object, or its abstract interface. In functional programming, it’s the declaration of a function. (I’m simplifying things a bit, but that’s the gist of it.)

Category theory is extreme in the sense that it actively discourages us from looking inside the objects. An object in category theory is an abstract nebulous entity. All you can ever know about it is how it relates to other object — how it connects with them using arrows. This is how internet search engines rank web sites by analyzing incoming and outgoing links (except when they cheat). In object-oriented programming, an idealized object is only visible through its abstract interface (pure surface, no volume), with methods playing the role of arrows. The moment you have to dig into the implementation of the object in order to understand how to compose it with other objects, you lost the advantages of your programming paradigm.


  1. Implement, as best as you can, the identity function in your favorite language (or the second favorite, if your favorite language happens to be Haskell).
  2. Implement the composition function in your favorite language. It takes two functions as arguments and returns a function that is their composition.
  3. Write a program that tries to test that your composition function respects identity.
  4. Is the world-wide web a category in any sense? Are links morphisms?
  5. Is Facebook a category, with people as objects and friendships as morphisms?
  6. When is a directed graph a category?

Next: Types and Functions.

For some time now I’ve been floating the idea of writing a book about category theory that would be targeted at programmers. Mind you, not computer scientists but programmers — engineers rather than scientists. I know this sounds crazy and I am properly scared. I can’t deny that there is a huge gap between science and engineering because I have worked on both sides of the divide. But I’ve always felt a very strong compulsion to explain things. I have tremendous admiration for Richard Feynman who was the master of simple explanations. I know I’m no Feynman, but I will try my best. I’m starting by publishing this preface — which is supposed to motivate the reader to learn category theory — in hopes of starting a discussion and soliciting feedback.

I will attempt, in the space of a few paragraphs, to convince you that this book is written for you, and whatever objections you might have to learning one of the most abstract branches of mathematics in your “copious spare time” are totally unfounded.

My optimism is based on several observations. First, category theory is a treasure trove of extremely useful programming ideas. Haskell programmers have been tapping this resource for a long time, and the ideas are slowly percolating into other languages, but this process is too slow. We need to speed it up.

Second, there are many different kinds of math, and they appeal to different audiences. You might be allergic to calculus or algebra, but it doesn’t mean you won’t enjoy category theory. I would go as far as to argue that category theory is the kind of math that is particularly well suited for the minds of programmers. That’s because category theory — rather than dealing with particulars — deals with structure. It deals with the kind of structure that makes programs composable.

Composition is at the very root of category theory — it’s part of the definition of the category itself. And I will argue strongly that composition is the essence of programming. We’ve been composing things forever, long before some great engineer came up with the idea of a subroutine. Some time ago the principles of structural programming revolutionized programming because they made blocks of code composable. Then came object oriented programming, which is all about composing objects. Functional programming is not only about composing functions and algebraic data structures — it makes concurrency composable — something that’s virtually impossible with other programming paradigms.

Third, I have a secret weapon, a butcher’s knife, with which I will butcher math to make it more palatable to programmers. When you’re a professional mathematician, you have to be very careful to get all your assumptions straight, qualify every statement properly, and construct all your proofs rigorously. This makes mathematical papers and books extremely hard to read for an outsider. I’m a physicist by training, and in physics we made amazing advances using informal reasoning. Mathematicians laughed at the Dirac delta function, which was made up on the spot by the great physicist P. A. M. Dirac to solve some differential equations. They stopped laughing when they discovered a completely new branch of calculus called distribution theory that formalized Dirac’s insights.

Of course when using hand-waving arguments you run the risk of saying something blatantly wrong, so I will try to make sure that there is solid mathematical theory behind informal arguments in this book. I do have a worn-out copy of Saunders Mac Lane’s Category Theory for the Working Mathematician on my nightstand.

Since this is category theory for programmers I will illustrate all major concepts using computer code. You are probably aware that functional languages are closer to math than the more popular imperative languages. They also offer more abstracting power. So a natural temptation would be to say: You must learn Haskell before the bounty of category theory becomes available to you. But that would imply that category theory has no application outside of functional programming and that’s simply not true. So I will provide a lot of C++ examples. Granted, you’ll have to overcome some ugly syntax, the patterns might not stand out from the background of verbosity, and you might be forced to do some copy and paste in lieu of higher abstraction, but that’s just the lot of a C++ programmer.

But you’re not off the hook as far as Haskell is concerned. You don’t have to become a Haskell programmer, but you need it as a language for sketching and documenting ideas to be implemented in C++. That’s exactly how I got started with Haskell. I found its terse syntax and powerful type system a great help in understanding and implementing C++ templates, data structures, and algorithms. But since I can’t expect the readers to already know Haskell, I will introduce it slowly and explain everything as I go.

If you’re an experienced programmer, you might be asking yourself: I’ve been coding for so long without worrying about category theory or functional methods, so what’s changed? Surely you can’t help but notice that there’s been a steady stream of new functional features invading imperative languages. Even Java, the bastion of object-oriented programming, let the lambdas in. C++ has recently been evolving at a frantic pace — a new standard every few years — trying to catch up with the changing world. All this activity is in preparation for a disruptive change or, as we physicist call it, a phase transition. If you keep heating water, it will eventually start boiling. We are now in the position of a frog that must decide if it should continue swimming in increasingly hot water, or start looking for some alternatives.


One of the forces that are driving the big change is the multicore revolution. The prevailing programming paradigm, object oriented programming, doesn’t buy you anything in the realm of concurrency and parallelism, and instead encourages dangerous and buggy design. Data hiding, the basic premise of object orientation, when combined with sharing and mutation, becomes a recipe for data races. The idea of combining a mutex with the data it protects is nice but, unfortunately, locks don’t compose, and lock hiding makes deadlocks more likely and harder to debug.

But even in the absence of concurrency, the growing complexity of software systems is testing the limits of scalability of the imperative paradigm. To put it simply, side effects are getting out of hand. Granted, functions that have side effects are often convenient and easy to write. Their effects can in principle be encoded in their names and in the comments. A function called SetPassword or WriteFile is obviously mutating some state and generating side effects, and we are used to dealing with that. It’s only when we start composing functions that have side effects on top of other functions that have side effects, and so on, that things start getting hairy. It’s not that side effects are inherently bad — it’s the fact that they are hidden from view that makes them impossible to manage at larger scales. Side effects don’t scale, and imperative programming is all about side effects.

Changes in hardware and the growing complexity of software are forcing us to rethink the foundations of programming. Just like the builders of Europe’s great gothic cathedrals we’ve been honing our craft to the limits of material and structure. There is an unfinished gothic cathedral in Bauvais, France, that stands witness to this deeply human struggle with limitations. It was intended to beat all previous records of height and lightness, but it suffered a series of collapses. Ad hoc measures like iron rods and wooden supports keep if from disintegrating, but obviously a lot of things went wrong. From a modern perspective, it’s a miracle that so many gothic structures had been successfully completed without the help of modern material science, computer modelling, finite element analysis, and general math and physics. I hope future generations will be as admiring of the programming skills we’ve been displaying in building complex operating systems, web servers, and the internet infrastructure. And, frankly, they should, because we’ve done all this based on very flimsy theoretical foundations. We have to fix those foundations if we want to move forward.

Ad hoc measures preventing the Beauvais cathedral from collapsing

Ad hoc measures preventing the Beauvais cathedral from collapsing

Next: Category: The Essence of Composition.

Data races lead to undefined behavior; but how bad can they really be? In my previous post I talked about benign data races and I gave several examples taken from the Windows kernel. Those examples worked because the kernel was compiled with a specific compiler for a specific processor. But in general, if you want your code to be portable, you can’t have data races, period.

You just cannot reason about something that is specifically defined to be “undefined.” So, obviously, you cannot prove the correctness of a program that has data races. However, very few people engage in proofs of correctness. In most cases the argument goes, “I can’t see how this can fail, therefore it must be correct” (maybe not in so many words). I call this “proof by lack of imagination.” If you want to become a concurrency expert, you have to constantly stretch your imagination. So let’s do a few stretches.

One of the readers of my previous post, Duarte Nunes, posted a very interesting example of a benign data race. Here’s the code:

int owner = 0;
int count = 0;
std::mutex mtx;

bool TryEnter() {
    if (owner == std::this_thread::get_id()) {
        count += 1;
        return true;

    if (mtx.try_lock()) {
        owner = std::this_thread::get_id();
        return true;
    return false;

void Exit() {
    if (count != 0) {
        count -= 1;
    owner = 0;

I highlighted in blue the parts that are executed under the lock (in a correct program, Exit will always be called after the lock has been acquired). Notice that the variable count is always accessed under the lock, so no data races there. However, the variable owner may be read outside of the lock– I highlighted that part of code in red. That’s the data race we are talking about.

Try to analyze this code and imagine what could go wrong. Notice that the compiler or the processor can’t be too malicious. The code still has to work correctly if the data race is removed, for instance if the racy read is put under the lock.

Here’s an attempt at the “proof” of correctness. First, Duarte observed that “The valid values for the owner variable are zero and the id of any thread in the process.” That sort of makes sense, doesn’t it? Now, the only way the racy read can have any effect is if the value of owner is equal to the current thread’s ID. But that’s exactly the value that could have been written only by the current thread– and under the lock.

There are two possibilities when reading owner: either we are still under that lock, in which case the read is not at all racy; or we have already released the lock. But the release of the lock happens only after the current thread zeroes owner.

Notice that this is a single-thread analysis and, within a single thread, events must be ordered (no need to discuss memory fences or acquire/release semantics at this point). A read following a write in the same thread cannot see the value that was there before the write. That would break regular single-threaded programs. Of course, other threads may have overwritten this zero with their own thread IDs, but never with the current thread ID. Or so the story goes…

Brace yourself: Here’s an example how a compiler or the processor may “optimize” the program:

void Exit() {
    if (count != 0) {
        count -= 1;
    owner = 42;
    owner = 0;

You might argue that this is insane and no compiler in its right mind would do a thing like this; but the truth is: It’s a legal program transformation. The effect of this modification is definitely not observable in the current thread. Neither is it observable by other threads in the absence of data races. Now, the unfortunate thread whose ID just happens to be 42 might observe this value and take the wrong turn. But it can only observe it through a racy read. For instance, it would never see this value if it read owner under the lock. Moreover, it would never see it if the variable owner were defined as ‘atomic':

std::atomic<int> owner = 0

Stores and loads of atomic variables are, by default, sequentially consistent. Unfortunately sequential consistency, even on an x86, requires memory fences, which can be quite costly. It would definitely be an overkill in our example. So here’s the trick: Tell the compiler to forgo sequential consistency on a per read/write basis. For instance, here’s how you read an atomic variable without imposing any ordering constraints:


Such ‘relaxed’ operations will not introduce any memory fences– at least not on any processor I know about.

Here’s the version of the code that is exactly equivalent to the original, except that it’s correct and portable:

std::atomic<int> owner = 0;
int count = 0;
std::mutex mtx;

bool TryEnter() {
    if (owner.load(memory_order_relaxed) == std::this_thread::get_id()) {
        count += 1;
        return true;

    if (mtx.try_lock()) {, memory_order_relaxed);
        return true;
    return false;

void Exit() {
    if (count != 0) {
        count -= 1;
    }, memory_order_relaxed);

So what has changed? Can’t the compiler still do the same dirty trick, and momentarily store 42 in the owner variable? No, it can’t! Since the variable is declared ‘atomic,’ the compiler can no longer assume that the write can’t be observed by other threads.

The new version has no data races: The Standard specifically states that ‘atomic’ variables don’t contribute to data races, even in their most relaxed form.

C++ Standard, (1.10.5):
[…] “Relaxed” atomic operations are not synchronization operations even though, like synchronization operations, they cannot contribute to data races.

With those changes, I believe that our “proof” of correctness may actually be turned into a more rigorous proof using the axioms of the C++ memory model (although I’m not prepared to present one). We can have our cake (correct, portable code) and eat it too (no loss of performance). And, by the way, the same trick may be used in the case of lossy counters from my previous post.

Warning: I do not recommend this style of coding, or the use of weak atomics, to anybody who is not implementing operating system kernels or concurrency libraries.


I’d like to thank Luis Ceze, Hans Boehm, and Anthony Williams for useful remarks and for verifying my assumptions about the C++ memory model.


  1. C++ Draft Standard

We have this friendly competition going on between Eric Niebler and myself. He writes some clever C++ template code, and I feel the compulsion to explain it to him in functional terms. Then I write a blog about Haskell or category theory and Eric feels a compulsion to translate it into C++.

Eric is now working on his proposal to rewrite the C++ STL in terms of ranges and I keep reinterpreting his work in terms familiar to functional programmers. Eric’s range comprehensions are a result of some of this back and forth.

Lazy ranges are such an excellent example of functional programming that it would be foolish for me to pass this opportunity to dissect them. To any functional programmer worth their salt they just scream “monad!” A monad is a higher order pattern that can be built step by step, so that’s what I’m going to do. I’ll start with the functor pattern, then add some functionality that will make it a pointed functor, then add some more to make it an applicative functor, and finally add some more to make it a monad. This gradual buildup of functionality is reminiscent of building a class hierarchy, and indeed it can be modelled as such in Haskell (although Haskell type classes are slightly different than C++ classes). This hierarchy would look something like this:

  • A monad is-an applicative functor
  • An applicative functor is-a pointed functor
  • A pointed functor is-a functor

So let’s start with a functor.


I have a pet peeve about the use of the word “functor” in C++. People keep calling function objects functors. It’s like calling Luciano Pavarotti an “operator,” because he sings operas. The word functor has a very precise meaning in mathematics — moreover, it’s the branch of mathematics that’s extremely relevant to programming. So hijacking this term to mean a function-like object causes unnecessary confusion.

A functor in functional programming is a generic template, which allows the “lifting” of functions. Let me explain what it means. A generic template takes an arbitrary type as a template argument. So a range (whether lazy or eager) is a generic template because it can be instantiated for any type. You can have a range of integers, a range of vectors, a range of ranges, and so on. (We’ll come back to ranges of ranges later when we talk about monads.)

The “lifting” of functions means this: Give me any function from some type T to some other type U and I can apply this function to a range of T and produce a range of U. You may recognize this kind of lifting in the STL algorithm std::transform, which can be used to apply a function to a container. STL containers are indeed functors. Unfortunately, their functorial nature is buried under the noise of iterators. In Eric’s range library, the lifting is done much more cleanly using view::transform. Have a look at this example:

 int total = accumulate(view::iota(1) |
                        view::transform([](int x){return x*x;}) |
                        view::take(10), 0);

Here, view::transform takes an anonymous function that squares its argument, and lifts this function to the level of ranges. The range created by view::iota(1) is piped into it from the left, and the resulting rage of squares emerges from it on the right. The (infinite) range is then truncated by take‘ing the first 10 elements.

The function view::iota(1) is a factory that produces an infinite range of consecutive integers starting from 1. (We’ll come back to range factories later.)

In this form, view::transform plays the role of a higher-order function: one that takes a function and returns a function. It almost reaches the level of terseness and elegance of Haskell, where this example would look something like this:

total = sum $ take 10 $ fmap (\x->x*x) [1..]

(Traditionally, the flow of data in Haskell is from right to left.) The (higher-order) function fmap can be thought of as a “method” of the class Functor that does the lifting in Haskell. In C++ there is no overall functor abstraction, so each functor names its lifting function differently — for ranges, it’s view::transform.

The intuition behind a functor is that it generates a family of objects that somehow encapsulate values of arbitrary types. This encapsulation can be very concrete or very abstract. For instance, a container simply contains values of a given type. A range provides access to values that may be stored in some other container. A lazy range generates values on demand. A future, which is also a functor (or will be, in C++17), describes a value that might not be currently available because it’s being evaluated in a separate thread.

All these objects have one thing in common: they provide means to manipulate the encapsulated values with functions. That’s the only requirement for a functor. It’s not required that a functor provide access to encapsulated values (which may not even exist), although most do. In fact there is a functor (really, a monad), in Haskell, that provides no way of accessing its values other than outputting them to external devices.

Pointed Functor

A pointed functor is a functor with one additional ability: it lets you lift individual values. Give me a value of any type and I will encapsulate it. In Haskell, the encapsulating function is called pure although, as we will see later, in the context of a monad it’s called return.

All containers are pointed, because you can always create a singleton container — one that contains only one value. Ranges are more interesting. You can obviously create a range from a singleton container. But you can also create a lazy range from a value using a (generic) function called view::single, which doesn’t have a backing container behind it.

There is, however, an alternative way of lifting a value to a range, and that is by repeating it indefinitely. The function that creates such infinite (lazy) ranges is called view::repeat. For instance, view::repeat(1) creates an infinite series of ones. You might be wondering what use could there be of such a repetitive range. Not much, unless you combine it with other ranges. In general, pointed functors are not very interesting other than as stepping stones towards applicative functors and monads. So let’s move on.

Applicative Functor

An applicative functor is a pointed functor with one additional ability: it lets you lift multi-argument functions. We already know how to lift a single-argument function using fmap (or transform, or whatever it’s called for a particular functor).

With multi-argument functions acting on ranges we have two different options corresponding to the two choices for pure I mentioned before: view::single and view::repeat.

The idea, taken from functional languages, is to consider what happens when you provide the fist argument to a function of multiple arguments (it’s called partial application). You don’t get back a value. Instead you get something that expects one or more remaining arguments. A thing that expects arguments is called a function (or a function object), so you get back a function of one fewer arguments. In C++ you can’t just call a function with fewer arguments than expected, or you get a compilation error, but there is a (higher-order) function in the Standard Library called std::bind that implements partial application.

This kind of transformation from a function of multiple arguments to a function of one argument that returns a function is called currying.

Let’s consider a simple example. We want to apply std::make_pair to two ranges: view::ints(10, 11) and view::ints(1, 3). To this end, let’s replace std::make_pair with the corresponding curried function of one argument returning a function of one argument:

[](int i) { return [i](int j) { return std::make_pair(i, j); };}

First, we want to apply this function to the first range. We know how to apply a function to a range: we use view::transform.

auto partial_app = view::ints(10, 11) 
                 | view::transform([](int i) { 
                      return [i](int j) { return std::make_pair(i, j); }

What’s the result of this application? Can you guess? Our curried function will be applied to each integer in the range, returning a function that pairs that integer with its argument. So we end up with a range of functions of the form:

[i](int j) { return std::make_pair(i, j); }

So far so good — we have just used the functorial property of the range. But now we have to decide how to apply a range of functions to the second range of values. And that’s the essence of the definition of an applicative functor. In Haskell the operation of applying encapsulated functions to encapsulated arguments is denoted by an infix operator <*>.

With ranges, there are two basic strategies:

  1. We can enumerate all possible combinations — in other words create the cartesian product of the range of functions with the range of values — or
  2. Match corresponding functions to corresponding values — in other words, “zip” the two ranges.

The former, when applied to view::ints(1, 3), will yield:


and the latter will yield:

{(10, 1),(11, 2)}

(when the ranges are not equal length, you stop zipping when the shorter one is exhausted).

To see that those two choices correspond to the two choices for pure, we have to look at some consistency conditions. One of them is that if you encapsulate a single-argument function in a range using pure and then apply it to a range of arguments, you should get the same result as if you simply fmapped this function directly over the range of arguments. For the record, I’ll write it here in Haskell:

pure f <*> xs == fmap f xs

This is sort of an obvious requirement: You have two different ways of applying a single-argument function to a range, they better give the same result.

Let’s try it with the view::single version of pure. When acting on a function, it will create a one-element range containing this function. The “all possible combinations” application will just apply this function to all elements of the argument range, which is exactly what view::transform would do.

Conversely, if we apply view::repeat to the function, we’ll get an infinite range that repeats this function at every position. We have to zip this range with the range of arguments in order to get the same result as view::transform. So this implementation of pure works with the zippy applicative. Notice that if the argument range is finite the result will also be finite. But this kind of application will also work with infinite ranges thanks to laziness.

So there are two legitimate implementations of the applicative functor for ranges. One uses view::single to lift values and uses the all possible combinations strategy to apply a range of functions to a range of arguments. The other uses view::repeat to lift values and the zipping application for ranges of functions and arguments. They are both acceptable and have their uses.

Now let’s go back to our original problem of applying a function of multiple arguments to a bunch of ranges. Since we are not doing it in Haskell, currying is not really a practical option.

As it turns out, the second version of applicative has been implemented by Eric as a (higher-order) function view::zip_with. This function takes a multi-argument callable object as its first argument, followed by a variadic list of ranges.

There is no corresponding implementation for the combinatorial applicative. I think the most natural interface would be an overload of view::transform (or maybe view::fmap) with the same signature as zip_with. Our example would then look like this:

view::transform(std::make_pair, view::ints(10, 11), view::ints(1, 3));

The need for this kind of interface is not as acute because, as we’ll learn next, the combinatorial applicative is supplanted by a more general monadic interface.


Monads are applicative functors with one additional functionality. There are two equivalent ways of describing this functionality. But let me first explain why this functionality is needed.

The range library comes with a bunch of range factories, such as view::iota, view::ints, or view::repeat. It’s also very likely that users of the library will want to create their own range factories. The problem is: How do you compose existing range factories to obtain new range factories?

Let me give you an example that generated a blog post exchange between me and Eric. The challenge was to generate a list of Pythagorean triples. The idea is to take a cross product of three infinite ranges of integers and select those triples that satisfy the equation x2 + y2 = z2. The cross product of ranges is reminiscent of the “all possible combinations” applicative, and indeed that’s the applicative that can be extended to a monad (the zippy one can’t).

To make this algorithm feasible, we have to organize these ranges so we can (lazily) traverse them. Let’s start with a factory that produces all integers from 1 to infinity. That’s the view::ints(1) factory. Then, for each z produced by that factory, let’s create another factory view::ints(1, z). This range will provide our xs — and it makes no sense to try xs that are bigger than zs. These values, in turn, will be used in the creation of the third factory, view::ints(x, z) that will generate our ys. At the end we’ll filter out the triples that don’t satisfy the Pythagorean equation.

Notice how we are feeding the output of one range factory into another range factory. Do you see the problem? We can’t just iterate over an infinite range. We need a way to glue the output side of one range factory to the input side of another range factory without unpacking the range. And that’s what monads are for.

Remember also that there are functors that provide no way of extracting values, or for which extraction is expensive or blocking (as is the case with futures). Being able to compose those kinds of functor factories is often essential, and again, the monad is the answer.

Now let’s pinpoint the type of functionality that would allow us to glue range factories end-to-end. Since ranges are functorial, we can use view::transform to apply a factory to a range. After all a factory is just a function. The only problem is that the result of such application is a range of ranges. So, really, all that’s needed is a lazy way of flattening nested ranges. And that’s exactly what Eric’s view::flatten does.

With this new flattening power at our disposal, here’s a possible beginning of the solution to the Pythagorean triple problem:

view::ints(1) | view::transform([](int z) { 
                view::ints(1, z) | ... } | view::flatten

However, this combination of view::transform and view::flatten is so useful that it deserves its own function. In Haskell, this function is called “bind” and is written as an infix operator >>=. (And, while we’re at it, flatten is called join.)

And guess what the combination of view::transform and view::flatten is called in the range library. This discovery struck me as I was looking at one of Eric’s examples. It’s called view::for_each. Here’s the solution to the Pythagorean triple problem using view::for_each to bind range factories:

auto triples =
  for_each(ints(1), [](int z) {
    return for_each(ints(1, z), [=](int x) {
      return for_each(ints(x, z), [=](int y) {
        return yield_if(x*x + y*y == z*z, std::make_tuple(x, y, z));

And here’s the same code in Haskell:

triples = 
  (>>=) [1..] $ \z -> 
     (>>=) [1..z] $ \x -> 
        (>>=) [x..z] $ \y -> 
           guard (x^2 + y^2 == z^2) >> return (x, y, z)

I have purposefully re-formatted Haskell code to match C++ (A more realistic rendition of it is in my post Getting Lazy with C++). Bind operators >>= are normally used in infix position but here I wanted to match them against for_each. Haskell’s return is the same as view::single, which Eric renamed to yield inside for_each. In this particular case, yield is conditional, which in Haskell is expressed using guard. The syntax for lambdas is also different, but otherwise the code matches almost perfectly.

This is an amazing and somewhat unexpected convergence. In our tweeter exchange, Eric sheepishly called his for_each code imperative. We are used to thinking of for_each as synonymous with looping, which is such an iconic imperative construct. But here, for_each is the monadic bind — the epitome of functional programming. This puppy is purely functional. It’s an expression that returns a value (a range) and has no side effects.

But what about those loops that do have side effects and don’t return a value? In Haskell, side effects are always encapsulated using monads. The equivalent of a for_each loop with side effects would return a monadic object. What we consider side effects would be encapsulated in that object. It’s not the loop that performs side effects, its that object. It’s an executable object. In the simplest case, this object contains a function that may be called with the state that is to be modified. For side effects that involve the external world, there is a special monad called the IO monad. You can produce IO objects, you can compose them using monadic bind, but you can’t execute them. Instead you return one such object that combines all the IO of your program from main and let the runtime execute it. (At least that’s the theory.)

Is this in any way relevant to an imperative programmer? After all, in C++ you can perform side effects anywhere in your code. The problem is that there are some parts of your code where side effects can kill you. In concurrent programs uncontrolled side effects lead to data races. In Software Transactional Memory (STM, which at some point may become part of C++) side effects may be re-run multiple times when a transaction is retried. There is an urgent need to control side effects and to separate pure functions from their impure brethren. Encapsulating side effects inside monads could be the ticket to extend the usefulness of pure functions inside an imperative language like C++.

To summarize: A monad is an applicative functor with an additional ability, which can be expressed either as a way of flattening a doubly encapsulated object, or as a way of applying a functor factory to an encapsulated object.

In the range library, the first method is implemented through view::flatten, and the second through view::for_each. Being an applicative functor means that a range can be manipulated using view::transform and that any value may be encapsulated using view::single or, inside for_each, using yield.

The ability to apply a range of functions to a range of arguments that is characteristic of an applicative functor falls out of the monadic functionality. For instance, the example from the previous section can be rewritten as:

for_each(ints(10, 11), [](int i) {
  return for_each(ints(1, 3), [i](int j) {
    return yield(std::make_pair(i, j));

The Mess We’re In

I don’t think the ideas I presented here are particularly difficult. What might be confusing though is the many names that are used to describe the same thing. There is a tendency in imperative (and some functional) languages to come up with cute names for identical patterns specialized to different applications. It is also believed that programmers would be scared by terms taken from mathematics. Personally, I think that’s silly. A monad by any other name would smell as sweet, but we wouldn’t be able to communicate about them as easily. Here’s a sampling of various names used in relation to concepts I talked about:

  1. Functor: fmap, transform, Select (LINQ)
  2. Pointed functor: pure, return, single, repeat, make_ready_future, yield, await
  3. Applicative functor: <*>, zip_with
  4. Monad: >>=, bind, mbind, for_each, next, then, SelectMany (LINQ)

Part of the problem is the lack of expressive power in C++ to unite such diverse phenomena as ranges and futures. Unfortunately, the absence of unifying ideas adds to the already overwhelming complexity of the language and its libraries. The functional paradigm could be a force capable of building connections between seemingly distant application areas.


I’m grateful to Eric Niebler for reviewing the draft of this blog and correcting several mistakes. The remaining mistakes are all mine.