Recognition Fibers

Definition

A recognition fiber at trace $t$ is a finite Heyting algebra $H_t$ representing the recognitions available at that trace. A fibered recognition system assigns a fiber to each trace and provides reindexing maps along morphisms in $T$.

Plain-language meaning

Each trace has its own local logic: a set of recognitions with logical operations like “and”, “or”, and “implies”. When you move between traces, recognitions must transport in a consistent way.

Minimal structure

• For each trace $t$, a finite Heyting algebra $H_t$.
• For each morphism $f:t\to t^{\prime }$, a reindexing map $H(f):H_{t^{\prime }}\to H_t$.
• Reindexing respects the Heyting operations.

Entry conditions

Use recognition fibers only when:

• Each trace has a well-defined local logic.
• You can list recognitions and their order structure.
• You can define how recognitions move along trace morphisms.

Example

Let each $H_t$ be the three-element chain $\{\perp <m<\top \}$. For a morphism $f$, define $H(f)$ as the identity map. This yields a consistent fibered system.