Sheaves on Quivers via a Grothendieck Topology on the Path Category

TL;DR

This paper introduces Grothendieck topologies on the path category of a finite graph, linking sheaf theory to quiver representations and exploring how different topologies reflect various representation-theoretic structures.

Contribution

It constructs and analyzes coarse and discrete Grothendieck topologies on quiver path categories, connecting sheaves to dual and groupoid representations, and proposes intermediate topologies for further study.

Findings

Coarse topology sheaves correspond to dual quiver representations.

Discrete topology sheaves are locally constant and relate to groupoid representations.

Both topologies satisfy Grothendieck axioms and have well-characterized sheaf categories.

Abstract

We construct Grothendieck topologies on the path category of a finite graph, examining both coarse and discrete cases that offer different perspectives on quiver representations. The coarse topology declares each vertex covered by all incoming morphisms, giving the minimal non-trivial Grothendieck topology where sheaves correspond to dual representations via dualization. The discrete topology is the finest possible, forcing sheaves to be locally constant with isomorphic restriction maps. We verify these satisfy Grothendieck's axioms, characterize their sheaf categories, and establish functorial relationships between them. Sheaves on the coarse site arise naturally from quiver representations through dualization, while discrete sheaves correspond to representations of the groupoid completion. This work suggests intermediate topologies could capture subtler representation-theoretic phenomena.