From Line Knowledge Digraphs to Sheaf Semantics: A Categorical Framework for Knowledge Graphs

TL;DR

This paper introduces a categorical framework that models knowledge graphs using sheaf semantics, linking graph structures with topos theory for enhanced contextual reasoning.

Contribution

It develops a novel mathematical model connecting graph theory, category theory, and sheaf semantics for knowledge graphs.

Findings

Defines a Grothendieck topology on free categories from graphs

Constructs a topos of sheaves for local-to-global reasoning

Unifies graph structure with categorical semantics

Abstract

This paper proposes a categorical framework for knowledge graphs linking combinatorial graph structure with topos-theoretic semantics. Knowledge graphs are represented as labelled directed multigraphs and analysed through incidence matrices and line knowledge digraph constructions. The graph induces a free category whose morphisms correspond to relational paths. To model context-dependent meaning, a Grothendieck topology is defined on the free category generated by the graph leading to a topos of sheaves that supports local-to-global semantic reasoning. The framework connects graph-theoretic structure, categorical composition, and sheaf semantics in a unified mathematical model for contextual relational reasoning.