A generalization of Gabriel's Galois covering functors and derived equivalences

Abstract

Let $G$ be a group acting on a category $\mathcal{C}$. We give a definition for a functor $F:\mathcal{C}\to {\mathcal{C}}^{\prime }$ to be a $G$-covering and three constructions of the orbit category $\mathcal{C}/G$, which generalizes the notion of a Galois covering of locally finite-dimensional categories with group $G$ whose action on $\mathcal{C}$ is free and locally bonded defined by Gabriel. Here $\mathcal{C}/G$ is defined for any category $\mathcal{C}$ and we do not require that the action of $G$ is free or locally bounded. We show that a $G$-covering is a universal "$G$-invariant" functor and is essentially given by the canonical functor $\mathcal{C}\to \mathcal{C}/G$. By using this we improve a covering technique for derived equivalence. Also we prove theorems describing the relationships between smash product construction and the orbit category construction by Cibils and Marcos (2006) without the assumption that the $G$-action is free. The orbit category construction by a cyclic group generated by an auto-equivalence modulo natural isomorphisms (e.g., the construction of cluster categories) is justified by a notion of the "colimit orbit category". In addition, we give a presentation of the orbit category of a category with a monoid action by a quiver with relations, which enables us to calculate many examples.

arXiv commentTitle changed. Definitions of ${\mathrm{Mod}}^G\mathcal{C}$ and ${\mathrm{Mod}}_G\mathcal{B}$ in section 6 were corrected. Proof of Theorem 8.1 is slightly changed to make it more readable. Other minor changes

References in corpus (2)

• On triangulated orbit categories
• A generalization of Gabriel's Galois covering functors II: 2-categorical Cohen-Montgomery duality

Cited by in corpus (2)

• The Grothendieck Construction and Gradings for Enriched Categories
• Mirror symmetry between orbifold curves and cusp singularities with group action