Emergence of Structure via Relations in Relational Quantum Dynamics (RQD)

We develop a rigorous formal framework for how structure emerges through relations in Relational Quantum Dynamics (RQD). In RQD, quantum systems and observers have no intrinsic, absolute identities—only identities defined by their web of interactions. We formalize this idea using category theory and the Yoneda lemma, proving that an object (system or observer) is uniquely determined (up to isomorphism) by its relationships to all others, with no additional intrinsic properties. We then show that networks of relational interactions naturally form structured graphs, which we formalize as enriched categories and bicategories capturing weighted relations and multi-observer interactions. Within these relational graphs, we demonstrate how correlations, mutual awareness, and information flow are encoded and can be quantified using measures from quantum information theory. We prove that cohomological obstructions arising from quantum contextuality endow these relational graphs with nontrivial topology. Using sheaf cohomology, we show that the absence of a global section (a consistent global assignment of outcomes) corresponds to a nonzero cohomology class (an obstruction) on the relational structure. In particular, we prove that in contextual scenarios the first Čech cohomology H1 of the presheaf of measurement outcomes is nonvanishing, formally certifying contextuality and implying a “twisted” topological structure on the network of contexts. Examples and case studies are provided to illustrate these results. Our results integrate and extend existing research in category-theoretic quantum foundations and contextuality, providing a unified formal treatment of structure emergence in RQD for a specialized audience in physics and mathematics.