From Category Theory for Programmers by Bartosz Milewski in section 3.5 titled Monoid as Category we read the following
Every monoid can be described as a single object category with a set of morphisms that follow appropriate rules of composition.
With all the examples provided, such that string concatenation with concatenating empty string as identity operation and natural number addition with adding 0 as identity operation et cetera.
I am new to Category Theory, correct me if I'm wrong, but I understand the point is to not think about what is in the objects, but rather focus on how objects relate it each other.
Distinguishing objects seems fundamental. I don't see how does this Monoid as Category represents a Monoid.
My argument based on given examples. A string after concatenating anything else than empty string has to be a different string, and thus different object. A natural number after adding anything else than zero has to be a different number, and thus different object.
This brings me to my title question, in category theory, is it possible for an object to have an arrow pointing into itself, such that this arrow is not an identity arrow? This feels strange and unnatural to me. I would like someone more experienced to comment on that.
Best Answer
You're essentially asking whether $\operatorname{Hom}_C(A, A)$ is a singleton or not. It doesn't have to be a singleton, and most of the time it isn't. In fact, an arrow from an object to itself is called an endomorphism, and $\operatorname{Hom}_C(A, A)$ is denoted as $\operatorname{End_C}(A)$.
For a concrete example, consider the set of functions from a set $A$ to itself. As long as $A$ contains more than one element, then $\operatorname{End}_{\text{Set}}(A)$ is not a singleton.