Concrete example first
If the full trace site has traces ({t_0,t_1,t_2}), you might focus on ({t_0,t_1}) because those are the traces relevant to one subsystem. Keep all morphisms between them and inherit the same cover notion. Sheaves on that restricted site form a local universe for that subsystem.
Formal definition
A cell is the local universe obtained from a full subcategory (U\subseteq T) with induced topology (J|_U). Its ambient category is (\mathbf{Sh}(U,J|_U)), and it relates to the global universe (\mathbf{Sh}(T,J)) through restriction/inclusion geometric behavior.
Cells are generated localizations, not separate primitive universes.
Why it matters in GFRTU
- It supports local analysis without abandoning global semantics.
- It provides a way to compare neighborhoods of traces.
- It explains how global behavior can be studied by local sub-sites.