Meaning
An admissible section module is the family of module fibers belonging to a coherent sector, together with their transport and composition data.
For a sector section s, its fibers are Eα,s=Pα,sHα. The full object retains these fibers, their comparison maps, and the compositors relating successive passages. It is therefore a module-valued pseudosection over the indexing category.
Example (illustrative)
One stage may use coefficients in Aα and another in Aβ, with a specified correspondence linking the two. The section module keeps that relationship. Simply taking the union of the vectors would forget which algebra acts where.
Scope and limits
It is not automatically one module over one coefficient algebra, and it is not yet an ordinary physical Hilbert space.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 3.5.