Sheaves

Entry conditions

Use sheaves only when you can:

• Start from a site $(C,J)$.
• Define a presheaf on $C$.
• State a concrete gluing requirement for your data.

Definitions

• A sheaf is a presheaf that satisfies the sheaf condition: compatible local data can be uniquely glued into global data.

Vocabulary (plain language)

• Compatible: local pieces agree on overlaps.
• Glue: combine compatible local pieces into one global piece.
• Sheaf condition: the formal rule that guarantees unique gluing.

Symbols used

• $F$: a presheaf or sheaf.
• $\{U_i\to U\}$: a cover of $U$.

Intuition

Sheaves formalize “local-to-global.” If you know data on all pieces of a cover and those pieces agree where they overlap, then there is one global piece of data that explains all the local pieces.

Worked example

Let $C$ be the open sets of a space and $F(U)$ be continuous functions on $U$. If functions agree on overlaps, there is a unique function on the union that restricts to each piece. This presheaf is a sheaf.

How to recognize the structure

• You can define what it means for local data to agree.
• You can define what it means to glue.
• Gluing is unique when it exists.

Common mistakes

• Using sheaves without a clear notion of overlap.
• Assuming gluing is unique when it is not.

Minimal data

• A site $(C,J)$.
• A presheaf $F$.
• A stated sheaf condition.