Concrete example first
For a toy case with traces raw and clean:
- define ((T,J)),
- attach two finite fibers and a reindexing map,
- choose (\sigma) and (\Delta),
- compute fixed fibers,
- iterate (UG) until closure,
- optionally restrict to one sub-site as a cell.
That sequence is the practical GFRTU workflow.
Formal workflow
Step 1: Specify the trace site
Define a small category (T) and Grothendieck topology (J).
Step 2: Attach recognition fibers
Provide finite Heyting algebras (H_t), reindexing maps, and dynamic maps (\sigma_t,\Delta_t) compatible with transport.
Step 3: Build the fixed layer
Compute (H_t^*={a\mid \sigma_t(a)=a=\Delta_t(a)}) and verify these fixed fibers form a subsheaf.
Step 4: Run generative closure
Define (UG) on (P(\mathrm{Ob}(\mathbf{Sh}(T,J)))), iterate from (\emptyset), and take the least fixed point.
Step 5: Localize into cells (optional)
For a full subcategory (U\subseteq T), form (\mathbf{Sh}(U,J|_U)) to study local behavior.
Validation checklist
- Site axioms hold.
- Fiber functoriality holds.
- Dynamic operators satisfy required properties.
- Fixed fibers are reindexing-stable.
- Closure iteration reaches a fixed point.