A Functorial Link between Quivers and Hypergraphs

TL;DR

This paper explores the categorical relationships between hypergraphs, multigraphs, and quivers, establishing adjoint functors and analyzing their structural properties within category theory.

Contribution

It introduces functorial links between hypergraphs, multigraphs, and quivers, revealing adjoint relationships and differences in categorical properties such as cartesian closure and projectivity.

Findings

Existence of right adjoint functor from multigraphs to hypergraphs.

Adjoint pair between multigraphs and quivers via underlying graph operations.

Hypergraphs lack enough projective objects, unlike multigraphs.

Abstract

This paper discusses some issues arising from the category $\mathfrak{H}$ of hypergraphs, the category $\mathfrak{M}$ of (undirected) multigraphs, and the topos $\mathfrak{Q}$ of quivers. First, the natural inclusion of $\mathfrak{M}$ into $\mathfrak{H}$ admits a right adjoint functor by deleting all nontraditional edges. Dually, the operations of taking the underlying multigraph of a quiver and taking the associated digraph of a multigraph form an adjoint pair between $\mathfrak{M}$ and $\mathfrak{Q}$. On the other hand, neither $\mathfrak{H}$ nor $\mathfrak{M}$ is cartesian closed, meaning that neither is a topos like $\mathfrak{Q}$. Moreover, despite $\mathfrak{M}$ being a subcategory of $\mathfrak{H}$, $\mathfrak{H}$ does not have enough projective objects while $\mathfrak{M}$ admits a projective cover for every object.