Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs

Abstract: We initiate a functorial study of ample C$^*$-diagonal pairs and their Weyl groupoids, focusing on how certain well-behaved $$-homomorphisms induce geometric maps between the associated groupoids. Given a morphism between diagonal pairs satisfying compatibility conditions with the diagonal and the canonical conditional expectations, we construct an induced partial morphism between the associated Weyl groupoids and analyze its properties. This provides a way to transfer certain structural information between Cartan-type inclusions. As applications, we study the behaviour of expectation-compatible ideals, faithful conditional expectations, and dynamical comparison under diagonal-preserving morphisms. We further investigate tensor products of ample C$^$-diagonal pairs and prove that the Weyl groupoid of a tensor product is naturally identified with the product of the corresponding Weyl groupoids. Under suitable hypotheses, we obtain a subadditivity result for diagonal dimension via dynamic asymptotic dimension. We also prove that the Weyl functor is faithful on a natural subcategory of \emph{untwisted} pairs, providing a concrete invariant that distinguishes non-isomorphic diagonal pairs. The theory is illustrated through examples arising from AF algebras, graph C$^*$-algebras, crossed products, and recent constructions of exotic diagonals in UHF and Cuntz algebras.