Sheaf

A sheaf is a presheaf $$\mathscr{F} : \operatorname{Op}(X)^\text{op} \to \mathsf{C}$$ satisfying two additional axioms, locality and gluing.

In the special case that $$\mathsf{C}$$ is a concrete category, we can write these conditions as follows:

Suppose we have an open subset $$U \subseteq X$$ and an open cover $$U = \bigcup_\alpha U_\alpha$$ of $$X$$. Then:


 * (Locality) If $$f, g \in \mathscr{F}(U)$$ are two sections such that $$f|_{U_\alpha} = G|_{U_\alpha}$$ for each $$\alpha$$ then $$f = g$$.
 * (Gluing) If we have sections $$f_\alpha \in \mathscr{F}(U_\alpha)$$ such that $$f_\alpha|_{U_\alpha \cap U_\beta} = f_\beta|_{U_\alpha \cap U_\beta}$$ then there exists some section $$f \in \mathscr{F}(U)$$ with $$f|_{U_\alpha} = f_\alpha$$ for each $$\alpha$$.

That is, any section is determined by its behavior locally, and we can glue together compatible sections to form a larger one.

Etymology
A "sheaf," in non-mathematical contexts, is a bundle of wheat; the various blades of wheat lie parallel to one another. A sheaf of wheat looks something like a line bundle, and the sheaf of sections of a line bundle was a major motivating example of a sheaf.

Motivation
Prior to the current notion of sheaves, vector bundles were known and understood, but lacked certain nice algebraic properties that vector spaces had, e.g. the ability to form quotients. Sheaves were initially defined to provide a "completion" for the category of vector bundles on a space.