Functoriality and naturality of arithmetic barcodes under morphisms

Develop a systematic functorial and naturality framework for arithmetic barcodes of network sheaves under graph morphisms and sheaf pullbacks, including the construction of induced maps on cohomology and barcodes and the identification of invariants that are preserved or behave monotonically under such morphisms.

Background

Understanding how arithmetic barcodes behave under morphisms is essential for compositional and comparative analyses across networks and sheaves. Classical persistent homology enjoys functorial properties that facilitate stability proofs and hierarchical analyses.

A formal functoriality theory in the precision-graded, torsion-sensitive setting would clarify when and how barcode information transfers along structural maps of graphs and sheaves.