Clarification on definition of a Sheaf

Question

On Wikipedia, the gluing and locality properties of a Sheaf are defined in terms of elements $s$ of the object $S$ associated with $\mathscr{F}(U)$.

I have two points of confusion.

1. I thought objects in a category don't necessarily have elements so does this definition even makes sense for categories outside of sets with structure?

2. My second question is, assuming $S$ is a set. What is even met by the gluing compatibility conditions,

$$res_{V \cap W }(s_i) = res_{V \cap W }(s_j) $$

For instance in the case of the skyscraper sheaf at a point $p$, give an open covering of $U$, $s_i$ may only even exist for the $U_i$ containing $p$.

From the definitions, it feels like you need to elements of a set to make the definition to make things work and implicitly a function associated with each element defined for every open subset of $U$ that maps to the empty set over subsets where an element disappears.

My thinking must be horribly wrong here but I'm hoping someone can clarify these misconceptions.

Answer 1

1. A (pre)sheaf on a category $\mathcal{C}$ is a functor from $\mathcal{C}^{\rm op} \to \mathsf{Set}$ or sets with extra structure: Abelian groups, rings, modules, etc. $\mathcal{C}$ is often the category of open sets on a topological space. In particular, $\mathscr{F}(U)$ is always a set, by definition.

2. In the equation $\operatorname{res}_{V \cap W }(s_i) = \operatorname{res}_{V \cap W }(s_j)$ we assume that $s_i \in \mathscr{F}(V)$ and $s_j \in \mathscr{F}(W)$. Or to use the notation on Wikipedia: $s_i \in \mathscr{F}(U_i)$ and $s_j \in \mathscr{F}(U_j)$ with $$ \operatorname{res}_{U_i \cap U_j}(s_i) = \operatorname{res}_{U_i \cap U_j}(s_j) $$ For a skyscraper sheaf (of sets, let's say) $\mathscr{F}(U) = \{0\}$ (the terminal object of $\mathsf{Set}$ up to isomorphism) for all $U$ not containing $p$ and hence $s_i = 0$ for all $U_i$ not containing $p$. These $s_i$ still exist.

Maybe it would be best for you to read some examples of sheaves and think through the glueing axiom. For example, the sheaf of continuous functions on a topological space or sheaves of smooth/continuously differentiable functions on a manifold.

Answer 2

I assume you are considering $U$ to be an open set in some topological space, so your category is $\mathrm{Top}(X)$, and your presheaf is a functor $\mathscr{F}\colon\mathrm{Top}(X)^{\text{op}}\to \mathrm{Set}$. To be able to discuss "glueing", your source category must have a notion of "cover" of an object, that is a Grothendieck topology (such a category is called a site). In such context, the sheaf condition is expressed by an equalizer diagram for any cover. While this approach requires some more machinery, it allows to express the glueing axiom without having to evaluate the restriction maps at sections.