Sheaves on Quivers via a Grothendieck Topology on the Path Category

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.