math.LO — Logic

Definability and Lifting in Model Subgroupoids

Contributed by Zhenyu Sun

Restriction to a wide subgroupoid of a model groupoid can introduce invariant relations, maps, and objects. For open topological groupoids, a map between restricted objects lifts exactly when its graph is invariant under the full groupoid, and a specified quotient presentation lifts exactly when its kernel relation is invariant. Under coherent restriction, these criteria recover definable maps and coherent quotient presentations. For finite continuous actions of a group with an open subgroup of finite index, coinduction identifies extensions with invariant subsets on which evaluation is bijective. For a finite group $G$ and a normal subgroup $N$, we classify extensions of finite $N$-sets by normalizer extensions of their orbit types. Multiplicities must be constant on conjugacy orbits of types; within each such block, the liftable multiplicities form the additive monoid generated by indices of subgroups over which the normalizer extension splits. The same subgroup data determine the groupoid of lifts and its isomorphism-class and mass generating functions. This classification also applies to finite continuous actions of profinite groups with an open normal subgroup. For a fixed action, a centralizer extension gives the splitting obstruction, with an $H^2$ obstruction and an $H^1$ classification when its kernel is abelian. The center of $\mathrm{SL}_2(\mathbb F_5)$ gives the monoid $\langle12,20\rangle$; Heisenberg groups over fields of odd prime order give unbounded lifting indices at fixed cohomological period. Finite torsor theories and infinite pure sets realize the definability criteria. For profinite extensions with finite kernel, one finite quotient detects all liftable multiplicities, though it need not carry every lift.

Author comments: 37 pages. Exact finite-group computations and their output are included as ancillary files.