2025/11/04 by Álvaro Sánchez, Sánchez, Álvaro
Mathematics · #16G20 (Primary) #18G80 #55P43 (Secondary) #55U35 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2511.02731
openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28
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 ∞-category C, we prove an abstract equivalence of the representations CQ with a certain mesh ∞-category CℤQ, mesh of representations of the repetitive quiver ℤQ, that we build inductively using abstract reflection functors. This allows to produce, from the symmetries of the Auslander-Reiten quiver, universal autoequivalences of representations CQ in any stable ∞-category 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.