Schopieray's Galois-modular extension conjecture
Theo Johnson-Freyd
TL;DR
The paper addresses Schopieray's conjecture that every premodular (braided) fusion category can be embedded as a Galois-closed subcategory of a modular category, proving this for pseudounitary categories. It develops a constructive framework: in the Tannakian case it computes cohomological obstructions and builds explicit $G$-equivariant and $G$-graded constructions (notably $\mathbf{Vec}^\psi[\tilde{G}]$ and its equivariantization) to produce a Galois nondegenerate extension with integral centralizer, then upgrades to modularity via pseudounitarity. In the non-Tannakian case, it leverages Deligne's super fibre functor and the MNE theory of minimal nondegenerate extensions, together with deequivariantization arguments, to obtain a Galois extension whose centralizer remains integral, ensuring modular extendability. These results provide a concrete path to control and classify premodular subcategories through Galois symmetries and illustrate the role of the centralizer and obstruction theory in the higher Morita landscape.
Abstract
Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.