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

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.