Reflection

This talk will introduce a notion of a stratification of a (stable presentable) category. A stratification of a scheme determines a stratification of its category of quasi-coherent sheaves. A stratification of a topological space determines a stratification of its category of linear sheaves. Such a stratification determines two reconstruction theorems, each reconstructing the category in terms of its strata and gluing data. An abelian group can be reconstructed in terms of its p-completions and its rationalization, as well as from its p-torsion and its corationalization. Concatenating these two reconstruction theorems results in a sort of duality — reflection — which is a categorification of Möbius inversion for posets. Examples of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors. In topology, reflection is closely related to Verdier duality, which generalizes Poincaré duality. This talk will discuss all this, through many examples. This is a report on joint work with Aaron Mazel-Gee and Nick Rozenblyum.