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We’ve seen several formulations of a monoid: as a set, as a single-object category, as an object in a monoidal category. How much more juice can we squeeze out of this simple concept?

Let’s try. Take this definition of a monoid as a set m with a pair of functions:

μ :: m × m -> m
η :: 1 -> m

Here, 1 is the terminal object in Set — the singleton set. The first function defines multiplication (it takes a pair of elements and returns their product), the second selects the unit element from m. Not every choice of two functions with these signatures results in a monoid. For that we need to impose additional conditions: associativity and unit laws. But let’s forget about that for a moment and just consider “potential monoids.” A pair of functions is an element of a cartesian product of two sets of functions. We know that these sets may be represented as exponential objects:

μ ∈ m m×m
η ∈ m1

The cartesian product of these two sets is:

m m×m × m1

Using some high-school algebra (which works in every cartesian closed category), we can rewrite it as:

m m×m + 1

The plus sign stands for the coproduct in Set. We have just replaced a pair of functions with a single function — an element of the set:

m × m + 1 -> m

Any element of this set of functions is a potential monoid.

The beauty of this formulation is that it leads to interesting generalizations. For instance, how would we describe a group using this language? A group is a monoid with one additional function that assigns the inverse to every element. The latter is a function of the type m->m. As an example, integers form a group with addition as a binary operation, zero as the unit, and negation as the inverse. To define a group we would start with a triple of functions:

m × m -> m
m -> m
1 -> m

As before, we can combine all these triples into one set of functions:

m × m + m + 1 -> m

We started with one binary operator (addition), one unary operator (negation), and one nullary operator (identity — here zero). We combined them into one function. All functions with this signature define potential groups.

We can go on like this. For instance, to define a ring, we would add one more binary operator and one nullary operator, and so on. Each time we end up with a function type whose left-hand side is a sum of powers (possibly including the zeroth power — the terminal object), and the right-hand side being the set itself.

Now we can go crazy with generalizations. First of all, we can replace sets with objects and functions with morphisms. We can define n-ary operators as morphisms from n-ary products. It means that we need a category that supports finite products. For nullary operators we require the existence of the terminal object. So we need a cartesian category. In order to combine these operators we need exponentials, so that’s a cartesian closed category. Finally, we need coproducts to complete our algebraic shenanigans.

Alternatively, we can just forget about the way we derived our formulas and concentrate on the final product. The sum of products on the left hand side of our morphism defines an endofunctor. What if we pick an arbitrary endofunctor F instead? In that case we don’t have to impose any constraints on our category. What we obtain is called an F-algebra.

An F-algebra is a triple consisting of an endofunctor F, an object a, and a morphism

F a -> a

The object is often called the carrier, an underlying object or, in the context of programming, the carrier type. The morphism is often called the evaluation function or the structure map. Think of the functor F as forming expressions and the morphism as evaluating them.

Here’s the Haskell definition of an F-algebra:

type Algebra f a = f a -> a

It identifies the algebra with its evaluation function.

In the monoid example, the functor in question is:

data MonF a = MEmpty | MAppend a a

This is Haskell for 1 + a × a (remember algebraic data structures).

A ring would be defined using the following functor:

data RingF a = RZero
             | ROne
             | RAdd a a 
             | RMul a a
             | RNeg a

which is Haskell for 1 + 1 + a × a + a × a + a.

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