# Can a thin category with relations be a monoid?

I'm following along with https://bartoszmilewski.com/2014/12/05/categories-great-and-small/ and trying to make sure I understand the content -- I'm still working out the differences between category theory, and (I think it's called) the category of sets.

Consider arrows as relations (not functions) for the following:

1) a category with one object, and one arrow
2) a category with one object, and multiple arrows
3) a category with multiple objects, and no more than one arrow between objects

Then

1) is a monoid (by definition) and a thin category (by definition). The single object in the category would be a set with no more than one element, and it would have to form a valid relationship back to itself. (Could it have zero elements? I don't know)

2) is a monoid (by definition) but not a thin category (by definition, because the homset has more than one element). The single object in the category would have elements that all have relations with every other element. An example might be a set with a,b,c where a,b,c are equal, and composition with arrows is the relation "less than or equal to."

3) not a monoid because there is more than one object, but it is a thin category because every homset has at most one arrow. You could have composition define a relation like "less than or equal to" and this will form a directed acyclic graph, etc. (a partial order I think)

It seems that outside the case of an object with one element, there isn't a monoid that is also a thin category (in the category of sets, where composition of arrows is a relation).

Is the above statement and my understanding of 1,2, and 3 correct?

You're correct in identifying that (1) and (2) are monoids, while (3) is not.

Maybe this will be made more clear by making precise what is meant by saying that a category with one object 'is' a monoid (and a thin category 'is' a preorder).

A (small) category with one object determines a monoid: if $\mathcal{C}$ is a category with a single object $A$, then the hom set $\mathcal{C}(A,A)$ is a monoid with unit $\mathrm{id}_A$ and multiplication given by composition. Conversely, any monoid $(M, {\,\cdot\,}, e)$ determines a category $\mathcal{C}$ with one object (let's call it $A$), with the morphisms from $A$ to $A$ given by elements of $M$, with identity morphism $\mathrm{id}_A=e$ and with composition given by monoid multiplication. The identity and associativity axioms of the category correspond exactly with the unit and associativity axioms of the monoid, respectively.

With this correspondence established, it is then easy to verify that functors between one-object categories correspond with homomorphisms of monoids in a canonical way.

What your article refers to as a 'thin' category, i.e. a category such that between any two objects there is at most one morphism, can be identified with the notion of a preoder, i.e. a set equipped with a reflexive, transitive relation. Indeed, if $\mathcal{C}$ is a thin category, then you can define a preorder $(P, \le)$ by letting $P = \mathrm{ob}(\mathcal{C})$ and defining $A \le B$ if and only if there is a morphism in $\mathcal{C}$ from $A$ to $B$. Conversely, given a preorder $(P, \le)$, you can define a thin category $\mathcal{C}$ whose objects are the elements of $P$, and such that a (unique) morphism $A \to B$ exists if and only if $A \le B$. The identity and associativity axioms of the category correspond with the reflexivity and transitivity axioms of the preorder, respectively.

Again, functors between thin categories correspond with order-preserving maps between preorders in a canonical way.