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The Topological Structures of the Orders of Hypergraphs

Robert E. Green, Cliff A. Joslyn, Audun Myers, Michael G. Rawson, Michael Robinson

TL;DR

The paper develops a cohesive, category-theoretic framework that unifies binary relations, hypergraphs, formal contexts, Dowker complexes, and their associated (co)sheaf and homology theories. By establishing functorial correspondences among incidence matrices, concept lattices, and Dowker cosheaves, it proves three key results: (i) the edge-intersection structure of hypergraphs closes into a lattice isomorphic to the concept lattice, (ii) the concept lattice is an isomorphism-invariant of the Dowker cosheaf, and (iii) a novel Dowker cosheaf of chain complexes provides an isomorphism-invariant refinement that generalizes Dowker’s homology; furthermore, it connects these invariants via a running example and general categorical constructions. The approach yields new tools for relational data analysis, linking FCA and topological data analysis through a fully functorial lens, and results in invariant algebro-topological objects (cosheaves and cosheaf-chain complexes) that capture essential lattice structure. Overall, the work advances a unified, structural understanding of how binary relations sit inside hypergraphs, topologies, and homological invariants, with implications for robust, multi-perspective analysis of relational data. The framework provides concrete invariants and computable constructions that can be leveraged for comparative analyses across contexts, lattices, and Dowker-based topological representations.

Abstract

We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept lattices, and as specifying topological cosheaves of simplicial (Dowker) complexes on simplicial (Dowker) complexes. We provide an integrative, functorial framework combining previously known with three new results: 1) given a binary relation, there are order isomorphisms among the bounded edge order of the intersection complexes of its dual hypergraphs and its concept lattice; 2) the concept lattice of a context is an isomorphism invariant of the Dowker cosheaf (of abstract simplicial complexes) of that context; and 3) a novel Dowker cosheaf (of chain complexes) of a relation is an isomorphism invariant of the concept lattice of the context that generalizes Dowker's original homological result. We illustrate these concepts throughout with a running example, and demonstrate relationships to past results.

The Topological Structures of the Orders of Hypergraphs

TL;DR

The paper develops a cohesive, category-theoretic framework that unifies binary relations, hypergraphs, formal contexts, Dowker complexes, and their associated (co)sheaf and homology theories. By establishing functorial correspondences among incidence matrices, concept lattices, and Dowker cosheaves, it proves three key results: (i) the edge-intersection structure of hypergraphs closes into a lattice isomorphic to the concept lattice, (ii) the concept lattice is an isomorphism-invariant of the Dowker cosheaf, and (iii) a novel Dowker cosheaf of chain complexes provides an isomorphism-invariant refinement that generalizes Dowker’s homology; furthermore, it connects these invariants via a running example and general categorical constructions. The approach yields new tools for relational data analysis, linking FCA and topological data analysis through a fully functorial lens, and results in invariant algebro-topological objects (cosheaves and cosheaf-chain complexes) that capture essential lattice structure. Overall, the work advances a unified, structural understanding of how binary relations sit inside hypergraphs, topologies, and homological invariants, with implications for robust, multi-perspective analysis of relational data. The framework provides concrete invariants and computable constructions that can be leveraged for comparative analyses across contexts, lattices, and Dowker-based topological representations.

Abstract

We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept lattices, and as specifying topological cosheaves of simplicial (Dowker) complexes on simplicial (Dowker) complexes. We provide an integrative, functorial framework combining previously known with three new results: 1) given a binary relation, there are order isomorphisms among the bounded edge order of the intersection complexes of its dual hypergraphs and its concept lattice; 2) the concept lattice of a context is an isomorphism invariant of the Dowker cosheaf (of abstract simplicial complexes) of that context; and 3) a novel Dowker cosheaf (of chain complexes) of a relation is an isomorphism invariant of the concept lattice of the context that generalizes Dowker's original homological result. We illustrate these concepts throughout with a running example, and demonstrate relationships to past results.

Paper Structure

This paper contains 19 sections, 46 theorems, 54 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary Relations, Hypergraphs, and Formal Contexts
  4. Running Example
  5. Categories
  6. Order Theory
  7. Hypergraphs and Dowker Complexes
  8. Galois Connections: Polar Maps and Closure Operators
  9. Concept Lattices and Hypergraphs
  10. The Category of Formal Contexts
  11. Concept Lattices
  12. The Intersection Complexes of Hypergraphs
  13. Intersection Complexes and Concept Lattices
  14. Contexts and Topology
  15. Cosheaf structures on set systems
  16. ...and 4 more sections

Key Result

Lemma 1

Composition of hypergraph morphisms is well-defined. If $m_1 : H_1 \to H_2$ and $m_2 : H_2 \to H_3$ are hypergraph morphisms, then $(m_2 \circ m_1) : H_1 \to H_3$ is itself a hypergraph morphism.

Figures (11)

  • Figure 1: Functorial diagram of the categories used in this paper: nodes are categories, and arrows indicate functors. Note that this may not be commutative.
  • Figure 2: Diagram relating the mathematical structures developed in this paper, that is, the objects of the categories from Figure \ref{['figures/categories']}. Nodes are types of objects, and edges indicate functional relationships, that is, the ability to construct one object uniquely from another. Labeled edges indicate the specific nature of a relationship and/or propositional support in the text.
  • Figure 3: (Left) Euler diagram of a multihypergraph $H$ in Example \ref{['eg:running']}. (Right) Its collapsed hypergraph.
  • Figure 4: (Left) The Hasse diagram of the full concept lattice for our example. Concepts are nodes, Galois pairs of extents (black, left) and intents (red, right). (Right) Its reduced form (see Sec. \ref{['sec:cl']}).
  • Figure 5: The Hasse diagram of the poset $(E,\subseteq)$ for our example, using compact set notation.
  • ...and 6 more figures

Theorems & Definitions (104)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Edge Poset
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 94 more