A refinement is a morphism from a coarser to a finer stratum — a map that extracts more detailed or more local information. In a stratified structure, refinement maps go in a preferred direction (coarser → finer), and this directedness is part of the directed structure of the site.
Refinement maps compose: if t → t’ refines t to t’, and t’ → t” refines t’ to t”, then t → t” refines t to t” directly. The composition of refinements is associative, making the refinements part of the categorical structure. The covering families of the site are built from refinement maps and their compositions.
In the GFRTU, morphisms in the trace site are refinement maps: a morphism t → t’ means t’ is a more detailed record than t. Restriction of recognition fibers runs in the opposite direction (from finer to coarser), extracting the coarser data that is compatible with the refinement.