Universal Properties of Some Quivers

TL;DR

This paper explores the universal properties of specific classes of quivers, revealing how certain graph structures relate to adjoint functors in category theory.

Contribution

It characterizes four classes of quivers as images under adjoint functors to vertex and edge functors, unifying their properties.

Findings

Independent vertex sets correspond to left adjoints of the vertex functor.

Independent edge sets correspond to left adjoints of the edge functor.

Complete digraphs and bouquets relate to right adjoints of vertex and edge functors.

Abstract

In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In particular, the following notions coincide: (1) independent sets of vertices with a left adjoint functor to the vertex functor, (2) independent sets of edges with a left adjoint functor to the edge functor, (3) complete digraphs with a right adjoint functor to the vertex functor, (4) bouquets with a right adjoint functor to the edge functor.