Sites

Entry conditions

Use sites only when you can:

• Define a category of contexts or observations.
• Define what it means for a family of morphisms to “cover” an object.
• State what it means for local pieces to cover a whole.

If there is no notion of coverage, a site is not appropriate.

Definitions

• A site is a category $C$ equipped with a Grothendieck topology $J$.

Vocabulary (plain language)

• Site: a collection of objects and arrows plus a rule for which arrows count as covers.
• Cover: a family of arrows that are declared to give local pieces of an object.

Symbols used

• $C$: a category.
• $J$: a Grothendieck topology on $C$.
• $U_i \to U$: a cover family with arrows into $U$.

Intuition

A site formalizes “local pieces” and how they cover a whole.

If you only have a list of objects but no meaningful notion of coverage, then a site is the wrong tool.

What the axioms mean

• Covering: a family of morphisms $\{U_i \to U\}$ is declared to cover $U$ if the $U_i$ jointly provide the local pieces of $U$.

Worked examples

Example 1: Open sets

Let $C$ be the category of open sets of a topological space, with inclusions as morphisms. A cover is the usual open cover.

Example 2: Contexts as traces

If objects are observational contexts and morphisms are refinements, then a cover says which refinements are enough to reconstruct the original context.

How to recognize the structure

• Can you say when a family of morphisms covers an object?

Common mistakes

• Declaring a topology without checking the Grothendieck axioms.
• Treating any family of maps as a cover without justification.

Minimal data

• A category $C$.
• A topology $J$ that specifies covering families for each object.

Misuse warnings

• Do not use sites when coverage is only metaphorical.