Definition explanation: Definition of point of a set in category theory I just started studying category theory using the book: "Conceptual Mathematics" by F. WILLIAM LAWVERE and STEPHEN H. SCHANUEL and there's one thing that I can't understand. In the first chapter, before the author gives the definition of a Category he gives the example of the category of finite sets and maps between those sets.
At some point during the example he gives the following definition:

Fix a set singleton set denoted as $1$. Then a point of a set $X$ is a map $1\to X$.

I'll set up a little example so I can explain what got me confused about this definition.

Let $X$ be the set $\{$a, b, c$\}$. Here a,b,c are just letters and not variables representing something.

*

*If we fix a singleton set $1$, then there are $3$ maps from $1$ to $X$. They only say that a point is a map from $1 \to X$ but they never specify what map, so I suppose that the point "a" is the map that associates the only element of $1$ to "a". Is this correct?


*Are the elements of the set $X$ and the points of the set $X$ the same thing? Is a point an element of the set?
If this is the case and the set $X$ is a collection of those points (this is, the collection of maps $1\to X$), then what we are doing is defining a point using the definition of point: We are defining a point as a function from $1$ to a set of other points. So it makes sense that the set $X$ has its elements: "a,b,c" and then there are also the points "a", "b" and "c".


*There are infinite singleton sets that we can choose as our set $1$. Does that mean that each point has an infinite amount of ways it can be represented as functions? Or if $1$ and $1'$ are different singleton sets, and if $1\to X$ and $1'\to X$ are two functions that map the element of the singleton set to "a", does that count as just one point (the point "a"), or as two separate points?
Thank you.
 A: *

*Yes.


*If you introduce the notation $a\in X$ for "a is an element of the set $X$", then $a\in X$ is synonymous with "for all singleton sets $\{*\}$, there is a morphism $f\colon \{*\}\to X$ with $f(*)=a$. So, no, the elements of a set and the points of a set are not the same thing. But, they are in a bijective correspondence. And no, a point is (typically) not an element. The thing to notice here is that in order to speak of the element $a\in X$ we must introduce the notation $\in$. But in order to speak of a point $X$ (in the sense given in the book), we do not need to introduce anything new. All we have is the category of sets, and a point is defined in categorical terms. If the definition of the category of sets can be given without using $\in$ in any way, then we obtain a way of reasoning about sets without ever using $\in$ (this can be done, in a clever way).


*Correct, each point has many different presentations, one for each choice of the actual set $1$. All of these presentations are canonically 'the same'. Instead of identifying them, we simply let all of them coexist in peace. We simply remember that a point is a morphism $1\to X$, where $1$ is some fixed singleton. It really doesn't matter which one (or which $1$) it is.
