Categorical Diffusion of Weighted Lattices
Robert Ghrist, Miguel Lopez, Paige Randall North, Hans Riess
TL;DR
This work develops a categorical diffusion framework for network-structured data valued in weighted lattices by introducing the Lawvere Laplacian, a $ ext{Q}$-endofunctor on cochains of a $ ext{Q}$-enriched cellular sheaf. A central theoretical advance is the Tarski-Lawvere Fixed Point Theorem, which ensures the completeness of prefix/suffix/fixpoint subcategories, and the Hodge-Lawvere Theorem that identifies suffix points with weighted global sections via adjoint bisheaf structure. The paper then derives a constructive harmonic flow that iteratively aggregates information toward these equilibria, providing a versatile method for diffusion in discrete event systems, preference dynamics, and related domains. Collectively, the framework generalizes diffusion beyond Hilbert spaces, integrates weighted limits and adjunctions, and broadens cellular sheaf theory to quantale-enriched settings, with potential extensions to other enrichment bases and the Chu construction.
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.