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Comment by jkhdigital | Hacker News Reader

There’s a few gaps in this explanation that my deductive completionist brain is looking for.

Namely, I’m wondering what kind of thing is even capable of being a monoid and an endofunctor at the same time. Mathematically, a monoid is a set (or collection to be more general) closed under a binary operator (the monoidal product) and with a distinguished identity element. An endofunctor is a mapping from a category to itself, which entails a mapping of each object as well as a mapping of each morphism. So I’m not even sure how to parse the phrase “monoid in the category of endofunctors” because the data required to describe these two things seems completely different.

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The categorical definition of a monoid is different than the usual algebraic one (although one can be recovered from the other) and usually treats a monoid as a single object (hence "mono-").

In particular, usually a monoid is defined as a single object category (sometimes a "monoid object" is defined inside a category, but that's a different presentation). The usual multiple objects of an algebraic monoid correspond to different arrows/morphisms within a monoid.

So for example, while the usual algebraic presentation of a monoid would say the integers form a monoid with an identity element `zero` and normal integer addition forming the monoidal operator, the categorical definition says instead that the monoid is the category that consists of only one element Unit and the morphisms (x => x + 0), (x => x + 1), (x => x + 2), etc. We have an identity morphism (x => x + 0), which when composed with any other morphism does not change it (x => x + 2 composed with x => x + 0 still is just x => x + 2). This is a common pattern in category theory: everything is reduced to composition.

So that's a categorical monoid, just a single object with a lot of morphisms and an identity morphism. We can back out the usual algebraic definition by associating with each morphism the "element within" (i.e. associating (x => x + c) with c), but it represents a different way of thinking.

We can also talk about a monoid object in a larger category in much the same way.

Hence now that we can talk of a monoid as a single object, we can then say that an endofunctor is a monoid (given the proper mappings on an endofunctor, i.e. natural transformations).

A monoid is a certain kind of object (potentially one of many) in a certain kind of category (called “monoidal” category).

> We can also talk about a monoid object in a larger category in much the same way.

Indeed that is one way of viewing a monoid. However, the most basic categorical definition of a monoid (and the first one usually introduced in a category theory textbook) views it as a category with only one object, where all the "interesting stuff" happens with the morphisms. A monoidal category and monoid objects are a restatement of that idea in a larger category.

You're right, there's a missing step. By "a monoid M in the category C" we mean that M is an object in C along with two morphisms "unit" : 1 -> M and "multiplication" : M x M -> M satisfying the "monoid laws" translated to C. (And really C should be a so-called "monoidal category".) When C = Set, this gives the usual notion of a monoid, which classically is a set by definition, as you say. However there are lots of interesting monoid objects in other categories!

The trick is that it is not a monoid AND and endofunctor but of monoid OF endofunctors. The whole "monads are monoids in the category of endofunctors" was a joke because it is a needlessly complex way of describing what a monad is.

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