Question on the definition of global points in category.

Question

Let $C$ be a category and let $1$ be a terminal object, a global point of of an object $X$ is a morphism $1 \to X$.

I don't have problems with the definition, but I am a little confused when it comes to the category $\bf{Sets}$. In $\bf{Sets}$ the terminal objects are all the singleton elements $\{ \ast \}$ and since for any set $A$ there are many morphisms $\{ \ast \} \to A$, then we have a lot of global points right? But in my book it says that any function $f: A \to B$ is determined by what it does with the global points of $A$. I understand that every element of $A$ is a global points of $A$, and then this makes sense, but isn't there many many more global points? Or are the global points of a set $A$ just the elements of $A$?

Please help to sort out my confusion.

Answer 1

Global points in $A$ are essentially just elements of $A$, in that a map $*\to A$ is determined by a choice of singleton $*$ and the image $a$ of the map. There are many singletons, but they're all isomorphic, so we rarely pay attention to he distinction between them.

Answer 2

I'm going to assume your concern is that you are worried that every element of $A$ gives rise to an arrow $a : 1 = \{*\} \to A$ it also gives rise to an arrow $a' : 1' = \{\star\} \to A$. In other words, we have an arrow into $A$ for every pair of an element of $A$ and a terminal object. This actually forms a proper class (unless $A = \emptyset$).

Basically, what's happening is we choose one particular object to be the terminal object which we notate as $1$ typically. Then the global elements of $A$ are the elements of $\Gamma(A) \equiv \mathbf{Set}(1,A)$. That terminal objects are isomorphic means they are interchangeable with respect to all categorical properties. In other words, there's no way to tell which choice we chose. That they are further isomorphic by a unique isomorphism means if we do want to make a different choice, everything transforms in a canonical way. In other words, we have no choice in how things change when we make a different choice for the terminal object.

So "global elements" doesn't mean arrows from all terminal objects, just arrows from one, and which it is doesn't matter.