Cells

Definition

A cell is a local GFRTU obtained by restricting the trace site $(T,J)$ to a full subcategory $U\subseteq T$ with the induced topology. The corresponding sheaf topos $\mathbf{Sh}(U,J{|}_U)$ embeds into the global universe.

Plain-language meaning

A cell is a local sub-universe: it describes how recognitions behave inside a chosen region of traces.

Minimal structure

• A full subcategory $U$ of the trace site.
• The induced topology $J{|}_U$.
• The restricted recognition fibers on $U$.

Entry conditions

Use cells only when:

• You can identify a meaningful subcollection of traces.
• The subcollection is closed under the needed morphisms.

Example

If $T$ has traces $\{t_0,t_1,t_2\}$, take $U=\{t_0,t_1\}$ with all morphisms between them. The resulting cell describes recognitions on that subsite.