Abstract representation theory via coherent Auslander-Reiten diagrams

Abstract

We provide a general method to study representations of quivers over abstract stable homotopy theories (e.g. arbitrary rings, schemes, dg algebras, or ring spectra) in terms of Auslander-Reiten diagrams. For a finite acyclic quiver $Q$ and a stable $\infty$-category $\mathcal{C}$, we prove an abstract equivalence of the representations $\mathcal{C}^Q$ with a certain mesh $\infty$-category $\mathcal{C}^{\mathbb{Z}Q,\, \mathrm{mesh}}$ of representations of the repetitive quiver $\mathbb{Z}Q$, that we build inductively using abstract reflection functors. This allows to produce, from the symmetries of the Auslander-Reiten quiver, universal autoequivalences of representations $\mathcal{C}^Q$ in any stable $\infty$-category $\mathcal{C}$, which are the elements of the spectral Picard group of $Q$. In particular, we get abstract versions of key functors in classical representation theory -- e.g. reflection functors, the Auslander-Reiten translation, the Serre functor, etc. Moreover, for representations of trees this enables us to realize the whole derived Picard group over a field as a factor of the spectral Picard group.