Trace Site

Definition

A trace site is a small category $T$ together with a Grothendieck topology $J$. The objects of $T$ are traces (finite observational contexts), and the topology $J$ declares which families of morphisms count as covers of a trace.

Plain-language meaning

A trace site names the local contexts you can observe and how they relate. The topology tells you which families of local contexts are enough to reconstruct a bigger context.

Minimal structure

• A list of traces (objects).
• A list of refinement or transport steps between traces (morphisms).
• A rule saying which families of morphisms cover a trace.

Entry conditions

Use a trace site only when:

• You can identify discrete traces or contexts.
• You can define how one trace refines or maps to another.
• You can state when a family of refinements is enough to cover a trace.

Example

Let $T$ have objects $t_0,t_1$ and a morphism $n:t_0\to t_1$ (plus identities). Let $J$ be the trivial topology where only identity covers exist. This is a valid trace site.