Concrete example first
At each trace, imagine a small logic of recognition states: (\bot < m < \top).
- (\bot): not recognized,
- (m): partially recognized,
- (\top): recognized.
If a morphism (f:t\to t’) forgets detail, a reindexing map (H(f):H_{t’}\to H_t) transports recognitions backward along that forgetful move. If these maps preserve logical operations and compose functorially, you have a fibered recognition system.
Formal definition
A recognition fiber at trace (t) is a finite Heyting algebra (H_t). A fibered recognition system is a contravariant assignment [ H:T^{op}\to \mathbf{FinHeyt} ] that sends each trace to a finite Heyting algebra and each morphism (f:t\to t’) to a structure-preserving reindexing map (H(f):H_{t’}\to H_t).
In GFRTU, these fibers carry local recognition logic before sheaf-level gluing.
Why it matters in GFRTU
- It gives each trace an explicit internal logic.
- It makes recognition transport precise across refinements.
- It provides the domain for stabilizer and drift dynamics.