Elementary question about subpresheaves

Question

I have encountered the following phrase which I do not understand. The letter $L$ is used for a locale but I guess (?) this holds for other categories as well.

The terminal sheaf $1$ is such that $\mathbf{1}(u)$ is a singleton for each $u\in L$. A subpresheaf $R >\to \mathbf{1}$ is thus entirely determined by those $u\in L$ for which $R(u)$ is a singleton. Each of those subpresheaves corresponds to a downward directed family of elements.

Can someone please explain why this is true? What does it really mean to say that something is "entirely determined" by something else and why do the subpresheaves form such a family? Thanks!

Answer 1

If you have an injective function $f : A \to B$, then suppose I do the following:

I write down an element of $B$. I say this is the image of $f$ applied to some element of $A$.

If true, then there is a unique element of $A$ that has that property: thus, this information "entirely determines" an element of $A$.

(I could imagine the phrase "entirely determined" also being used in the situation where we have a function $g : B \to A$ and specify an element of $B$. However, all of the examples I can think of off the top of my head could naturally be expressed in the way above as well)

In this case, we have

• $A$ is the collection of all subobjects of $\mathbf{1}$
• $B$ is the collection of all subsets of the set of objects of $L$
• $f$ is given by taking a subobject $S$ and producing the set of all $u$ such that $S(u)$ is a singleton

All of your "why" questions are pretty much nothing more than the definitions.