Presheaves

Entry conditions

Use presheaves only when you can:

• Start from a site $(C,J)$ or at least a category $C$ of contexts.
• Assign data to each object in $C$.
• Define restriction maps along morphisms.

Definitions

• A presheaf on a category $C$ assigns to each object $U$ a set $F(U)$, and to each morphism $f:V\to U$ a restriction map $F(f):F(U)\to F(V)$, satisfying identity and composition laws.

Vocabulary (plain language)

• Section: an element of $F(U)$, i.e., data on $U$.
• Restriction: a way to move data from a larger context to a smaller one.

Symbols used

• $F$: a presheaf.
• $F(U)$: the data assigned to object $U$.
• $F(f)$: the restriction map along $f$.

Intuition

A presheaf is a bookkeeping system for data on each context, together with rules for how to restrict data when you move to a smaller context.

Worked example

Let $C$ be the category of open sets of a space. A presheaf can assign to each open set $U$ the set of real-valued functions on $U$, and restriction maps are literal restriction of functions to smaller open sets.

How to recognize the structure

• You can assign data to every object.
• You can restrict data along every morphism.
• Restrictions compose correctly.

Common mistakes

• Defining restrictions that do not compose.
• Forgetting to define restrictions for some morphisms.

Minimal data

• A category $C$.
• A set assignment $U\mapsto F(U)$.
• Restriction maps for every morphism.