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 $\mathrm{mod}\text{-}k{Q}_{\Sigma }$ 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.

19 pages