Are the arrows of category theory (in general) "intensional"?

Question

Thinking about monoids as single-object categories, I got to wondering about what distinguishes the arrows of a single-object category, especially given that the object might have no internal structure (e.g., it might not be a set with elements).

For instance, the trivial monoid consisting of just the identity element under some binary operation would be a category $\mathbf{M_0}$ with object $M$ and the identity arrow $\mathit{id}_M : M \rightarrow M$ as its sole arrow. But since the natural numbers under addition form a commutative monoid $(\mathbb{N}, +)$ it should be possible to construe that monoid as a single-object category $\mathbf{M}_\mathbb{N}$ with object $M$. Suppose that the single-object is the same in both categories, so that we get from $\mathbf{M_0}$ to $\mathbf{M}_\mathbb{N}$ simply by adding a countable infinity of arrows $1, 2, \dotsc : M \rightarrow M$, identifying $0$ with the identity arrow, requiring composition to be commutative, and ensuring additional things like that $1 \circ 1 = 2$ so that composition behaves like addition.

But assuming that $M$ has no "elements" or any other such internal structure, what differentiates the countable infinity of arrows that make up $\mathbf{M}_\mathbb{N}$? I've read, like in the comments on the linked question, that arrows can be thought of as ordered triples so that $1$ would be $(1, M, M)$ and $2$ would be $(2, M, M)$. Does the "name" (e.g., "1") of an arrow play a distinctive role in distinguishing it from other arrows with the same source and target (a single object, in this case)?

Answer 1

It's composition that plays a role, not the 'names'.

Without thinking your monoid as a one-object category, we can pose exactly the same question:

We have a set $\Bbb N$ of countably infinitely many elements, and an addition (and a distinguished additive identity element). What does differentiate its elements?

We have the $0$ that acts as the additive identity.
Besides that we have $1$ which is not a result of any $a+b$ for nonzero $a,b$.
Then we have $2=1+1$, and so on.

These do describe these elements, and you are free to replace 'addition' to 'composition' everywhere in the above.

Answer 2

When defining a category, you can take any set to be the set of morphisms between two objects (as long as you then define composition appropriately). So if you have your favorite monoid $(A,\cdot_A)$, you can simply define a category which has one object $M$ and whose set of morphisms from $M$ to $M$ is the set $A$, with composition defined by the operation $\cdot_A$. So, if you like, the arrows are distinguished by their "names", but that's not really the right way to think about it. Rather, the arrows are nothing but their names: the arrows literally are just elements of the set $A$ (which could be any set at all, as long as you have an appropriate binary operation on it).

Answer 3

For variety, it's worth noting that any category can be given an extensional-flavored notion of equality:

If $f$ and $g$ are two arrows $X \to Y$, then the following are equivalent:

• $f=g$
• $fx = gx$ for every arrow $x$ with codomain $X$
• $yf = yg$ for every arrow $y$ with domain $Y$

This is one of many ways in which the notion "arrow with codomain $X$" acts as a good substitute for the notion of "element of $X$". When thinking this way, we call the arrow a "generalized element".

So, despite the theorem being a rather trivial one, it's still a rather useful point of view.

As an example, the generalized elements of the object $M$ in $\mathbf{M}_{\mathbb{N}}$ are precisely the natural numbers. (because $\hom(M,M)$ is, by definition, the natural numbers)

Also, many categories allow a useful (and less trivial) variation:

A set of objects $\mathcal{G}$ is a generating set for the category if and only if, for every pair $f,g$ of arrows $X \to Y$, the following are equivalent:

• $f = g$
• $fx = gx$ for every arrow $x:G \to X$, with $G \in \mathcal{G}$

There are a lot of categories where the usual notion of elements coincides with the notion of a generalized element with domain in a generating set. For example, in the category of groups, take $\mathcal{G} = \{ \mathbb{Z} \}$. The notions of "an element of the group $G$" and "an arrow $\mathbb{Z} \to G$" are basically the same.

Another example is $\mathbf{Cat}$. Take $\mathcal{G}$ to consist of the two categories $\bullet$ and $\bullet \to \bullet$. The two kinds of generalized elements are precisely the objects and the arrows.