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

Moses Boudourides

TL;DR

A categorical framework for knowledge graphs linking combinatorial graph structure with topos-theoretic semantics and context-dependent meaning in a unified mathematical model for contextual relational reasoning is proposed.

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.

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

TL;DR

A categorical framework for knowledge graphs linking combinatorial graph structure with topos-theoretic semantics and context-dependent meaning in a unified mathematical model for contextual relational reasoning is proposed.

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.
Paper Structure (28 sections, 23 theorems, 89 equations)

This paper contains 28 sections, 23 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Categorical viewpoint
  3. Topos-theoretic perspective
  4. Overview of contributions
  5. Knowledge Graphs as Edge-Labelled Digraphs
  6. Incidence matrices
  7. Line Knowledge Digraphs
  8. Incidence interpretation
  9. Structural decomposition
  10. Knowledge Graphs as Free Categories
  11. Functorial constructions
  12. Sites on Knowledge Graph Categories
  13. Alternative interpretive topologies
  14. Sheaves and Topos Construction
  15. Subobject classifier and internal logic
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let $\mathcal{K} = (E, P, T)$ be a knowledge graph, and let $H^{(h)}$ and $H^{(t)}$ denote the head and tail incidence matrices of $\mathcal{K}$, respectively, as introduced in Section sec:kg. Then the matrices are symmetric and encode shared-head and shared-tail relations between triples.

Theorems & Definitions (57)

  • Definition 2.1: Knowledge graph
  • Definition 2.2: Head and tail incidence matrices
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.1: Out-line and in-line knowledge digraphs
  • Proposition 3.1
  • ...and 47 more