Sheaves on Quivers via a Grothendieck Topology on the Path Category
Eric M. Schmid, Fernando Tohmé, William Chin
TL;DR
This work studies Grothendieck topologies on the path category $C_Q$ of a finite quiver to connect sheaf theory with quiver representations. It introduces two extremal topologies—the coarse site with covering sieves $R_v = Hom(-,v)$ and the discrete site $J_{disc}$—and analyzes their sheaf categories, showing that coarse-sheaves arise from dualization of representations while discrete-sheaves correspond to representations of the groupoid completion. The paper verifies the Grothendieck axioms for both topologies, constructs an adjunction between the corresponding sheaf categories, and proposes intermediate topologies (e.g., edge-generated, graded) to capture subtler representation-theoretic phenomena. By embedding quiver representations into a topos-theoretic framework, the approach offers new structural insights and tools for representation theory, with potential applications to graded or stability-sensitive questions.
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.