Intuitive way to understand identity/gluing axioms of sheaf

Question

Is there an intuitive way to understand the identity and gluing axiom of a sheaf, specifically in the setting where the source category is affine schemes? What is the motivation for such a definition? Why is it important?

Answer 1

At least for me, the case of functions on a space was a good starting point for understanding sheaves.

Consider a space $X$ (say, a topological space), and the presheaf $\mathcal{O}_X$ of (continuous) maps: for $U \subset X$ open, $\mathcal{O}_X(U)$ is the set of continuous maps $U \to \mathbb{R}$. Then the sheaf condition says in particular that if you have two open subsets $U,V \subset X$ and two continuous maps $f : U \to \mathbb{R}$ and $f : V \to \mathbb{R}$ that agree on the intersection $U \cap V$ (i.e. $f(x) = g(x)$ for all $x \in U \cap V$), then there is a continuous map $h : U \cup V \to \mathbb{R}$ such that $h(x) = f(x)$ for $x \in U$ and $h(y) = g(y)$ for $y \in V$.

But this is rather obvious (a direct consequence of the definition of subspace topology), and it's probably a fact that you've known for a long time and use without thinking! Well, the gluing axiom for a sheaf is exactly the reformulation of this (and for a bunch of open subsets at once instead of just two). Pretty much all examples of sheaves I can think of are examples of "functions" satisfying some properties (sections of a bundle $E \to X$ are functions $X \supset U \to E$, etc.), and the gluing axiom says that given functions on open subsets that agree on all the intersections, you can define a function on the union of everything. In terms of intuition, it's a good way to think about it.