Categorical Diffusion of Weighted Lattices

Abstract

We introduce a categorical framework for diffusion on network-structured data valued in weighted lattices, extending the Laplacian paradigm beyond the category of Hilbert spaces. Central to our approach is the Lawvere Laplacian, an endofunctor on the category of cochains of a cellular sheaf enriched in a commutative unital quantale. We establish the Tarski-Lawvere Fixed Point Theorem, generalizing Tarski's classical result to show that the suffix and prefix points of a quantale-enriched endofunctor form complete weighted lattices. Leveraging this, we prove the Hodge-Lawvere Theorem, which identifies the suffix points of the Laplacian with weighted global sections, providing a geometric characterization of equilibria. Finally, we derive a discrete-time harmonic flow that evolves data toward these sections, offering a constructive method for information aggregation in systems ranging from discrete event processes to preference dynamics.