Capacity, Coherence, and the Thin Radical in Equivariant Categories

Let G be a transitive permutation group on a finite set X. How many essentially different G-equivariant category structures exist on X with finite group endomorphism monoid M_0? We identify a combinatorial invariant — the thin radical D of the Schurian coherent configuration CC(G,X) — and conjecture that the isomorphism classes biject with H^2(D, Z(M_0)). We prove this classification at two structural extremes and verify it computationally in nine additional cases. At one extreme, G = Sym(X): ev