Describing the Bool-And monoid in terms of categories

Normally I put the question before the context, but in this case I want to admit
the possibility that the context and my understanding nullify the question.
Plus it helps me think through my question.

I recently started reading Category Theory for Programmers (Bartosz
Milewski), and this is my
understanding of categories: they are "algebraic structures" which consist of
objects and arrows/morphisms between those objects. The morphisms have to obey
the laws of associativity, so

$$ a \rightarrow ( b \rightarrow c ) = ( a \rightarrow b ) \rightarrow c $$

And there must be an identity morphism for each object.

Now, Milewski goes on to explain that monoids (which I'm pretty comfortable with
in the set-theoretic sense) are also viewable as categories. This is the part I
am having trouble with. One of the exercises in the book is to consider the
Boolean-and monoid (booleans with the and operator) as a category:


Represent the Bool monoid with the AND operator as a category: List the
morphisms and their rules of composition.

I want to give some examples, which I'll do in a SML (though I'll borrow Haskell names).

The monoid could be described set-theoretically with the following signature:

signature MONOID = sig
  type m
  val mempty : m
  val mappend : m -> m -> m
end

Further, the monoid for booleans with the and operator would be given as

structure BoolAnd : MONOID = struct
  type m = bool
  val mempty = true
  fun mappend x y = x andalso y
end

So, here is my understanding of this monoid as a category and its morphisms: is
it correct?

• The objects in the category are booleans (true and false) and functions from

booleans to booleans

• BoolAnd.mappend is a morphism from the former to the latter

• mappend true is an identity morphism for the function objects in the

category (I say "an" because isn't the actual identity function fun id x = x
also an identity morphism for the functions, thanks to polymorphic types? Or
does that not count

Monoids are one-object categories. The elements are morphisms, monoid multiplication is composition and the monoid identity is the identity morphism.

In the case of Booleans-with-AND the monoid is $M = (\{\top, \bot\}, \land, \top)$. The category therefore has a single object (we could call it $M$) with two morphisms $\top : M \to M$ and $\bot : M \to M$. Composition is given by $\land$ and the identity morphism is $\top$.

To give a little more context, there is a way to view a monoid in such a way that the objects are the elements of the monoid: as a discrete (no non-identity morphisms) monoidal category. Category Theory for Programmers covers the topic of monoidal categories in Chapter 22, "Monads Categorically".

StackExchange Computer Science Q#126806, answer score: 2