Lesson 1: Trace Site and Fibers

Goal

Explain what a trace site is, what it encodes, and how recognition fibers attach to it.

What is a trace?

A trace is a finite observational context: a snapshot, perspective, or partial record. GFRTU does not assume a single global viewpoint; it starts with many traces and relationships between them.

What is the trace site $(T,J)$?

• Objects of $T$ are traces.
• Morphisms in $T$ are refinement or transport steps between traces (for example, “forget” or “normalize” operations).
• The topology $J$ tells you when a family of refinements covers a trace, i.e., when local pieces are sufficient to reconstruct information at that trace.

The site is what lets us say “this local data is enough to glue into a global recognition.”

How to specify a trace site

When modeling a concrete situation:

1. List the traces you can actually observe or represent.
2. Decide how one trace refines another (what counts as a valid comparison).
3. Decide which refinements cover a trace (the minimal families that reconstruct it).

You are not required to pick a complex topology. A trivial topology is acceptable if you only need sheaf semantics for coherence, not for reconstruction.

What are recognition fibers?

Recognition fibers are the local logics attached to traces. For each trace $t$ you supply a finite Heyting algebra $H_t$ and reindexing maps along the morphisms of $T$.

Think of $H_t$ as the set of recognitions that can be made at trace $t$, ordered by implication, with meet/join representing logical conjunction/disjunction.

What this gives you

Once the trace site and fibers are specified, you have:

• a notion of locality,
• a local logic per trace,
• and a way to transport recognitions across traces.

This is the minimum input needed for the GFRTU to build global semantics.