Incidence hypergraphs: the categorical inconsistency of set-systems and a characterization of quiver exponentials

This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e., qu...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2023-08, Vol.58 (1), p.1-36
Main Authors: Grilliette, Will, Rusnak, Lucas J.
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Graph theory
Group Theory and Generalizations
Homomorphisms
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary:This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e., quivers). The category of incidence hypergraphs is introduced as a “graph-like” remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is shown that the category of quivers embeds into the category of incidence hypergraphs via a logical functor that is the inverse image of an essential geometric morphism between the topoi. Consequently, the quiver exponential is shown to be simply represented using incidence hypergraph homomorphisms.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-023-01232-8