Sheaf Subcategories of Quiver Representations

Abstract

Let $Q$ be a finite quiver without oriented cycles, $C$ its path category, and $k$ a field. This paper is expository. We assemble in one place, and in elementary combinatorial form, the dictionary between Grothendieck topologies on $C$ and subcategories of $\mathrm{mod}\text{-}kQ$. Every Grothendieck topology on $C$ is rigid, so the sheaf categories are presheaf categories on full subcategories and are indexed by subsets $\Sigma$ of the vertices; this is due to Murfet in the quiver case and, in far greater generality, to Di-Li-Liang. Passing to $k$-linear coefficients, each topology cuts out a full subcategory $\mathcal{S}_\Sigma \subseteq \mathrm{mod}\text{-}kQ$. We identify $\mathcal{S}_\Sigma$ as the perpendicular category, in the sense of Geigle-Lenzing, of the set of simples off $\Sigma$; it is therefore wide, and equivalent to $\mathrm{mod}\text{-}kQ_\Sigma$ for an explicit reduced quiver $Q_\Sigma$. We record when $\mathcal{S}_\Sigma$ is a Serre subcategory (exactly when $\Sigma$ is closed under successors), observe that $\Sigma \mapsto \mathcal{S}_\Sigma$ embeds the Boolean lattice $2^{Q_0}$ into the lattice of wide subcategories, and note that in Dynkin type $A_n$ this captures $2^n$ of the $C_{n+1}$ wide subcategories. Intrinsically, a wide subcategory $\mathcal{W}$ is a sheaf subcategory exactly when ${}^{\perp}\mathcal{W}$ is a Serre subcategory and $\mathcal{W} = ({}^{\perp}\mathcal{W})^{\perp}$. All of these results are known, or follow readily from known results; the aim is a concrete self-contained account of the quiver case, with attributions collected in Section 8.