Relative ω-stability, relative categoricity and internal covers

2026/07/25 by Mostafa Mirabi, Anand Pillay
#math.LO

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Abstract

We study a special case of a relatively categorical theory (T,P), namely when T is an internal cover of T^P. We give a structure theory for relatively categorical internal covers. After passing to T^eq and naming a parameter from the P-part, they are precisely the "pure torsor covers" of T^P, obtained simply by adjoining a new sort S for a principal homogeneous space or torsor of an ∅-definable group G in T^P, and a symbol for the action. We prove that T is relatively ω-stable (or ω-stable over P) as defined in [4] iff the dual binding group H has the DCC on definable subgroups.

Citations

• Definable Galois theory for bimeromorphic geometry
• Stable amalgamation over a predicate and the Gaifman property
• On the existence property over a predicate

Related

• Groupoids and Relative Internality
• Group covers, o-minimality, and categoricity
• Groups definable in two orthogonal sorts
• Covers of groups definable in o-minimal structures
• An automorphism group of an omega-stable structure that is not locally (OB)