Categorical Diffusion of Weighted Lattices

TL;DR

This paper develops a categorical framework for diffusion processes on network data valued in weighted lattices, generalizing classical Laplacian methods to a broader mathematical setting with applications in information aggregation.

Contribution

It introduces the Lawvere Laplacian and proves the Tarski-Lawvere Fixed Point and Hodge-Lawvere Theorems, extending diffusion analysis to weighted lattice-valued network data.

Findings

Establishes a categorical diffusion framework using Lawvere Laplacian.

Proves fixed point theorems for quantale-enriched endofunctors.

Derives a discrete harmonic flow for data aggregation.

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.