I'm starting to study category theory and I don't understand these remarks in this book:
I have the following questions:
- Why does we study the global elements $x:1\to M$, since as we see in example 2, there are categories which doesn't work so well with global elements? so, why are they special to make us care about it?
- I need help to prove 3, I can't avoid to think the morphisms as functions, I don't know how to prove it in a general category.
Thanks in advance
The most important (and in fact, motivating) example is the category of sheaves on a space $X$. Here a global element is the same as a global section (this also explains the name). Of course, not every sheaf is defined via its global sections, but they are definitely interesting. In fact, one of the most important theorems in a first course on algebraic geometry is Riemann-Roch, which just counts (the dimension of) global sections of a given divisor on a curve.
The whole point of generalized elements is that they are an adequate replacement of global elements. They are more general and also more powerful: Every object is determined via its generalized elements - this is the Yoneda Lemma. In 3., simply take $x=\mathrm{id}$ (this is the "universal" generalized element).
When you cannot think about categories without thinking about sets and functions, you should consult a list of abstract categories (path category of a graph, homotopy category, relation category, groups as categories, partial orders as categories, etc.).