Canonical forking in AECs

Abstract

Boney and Grossberg [BG] proved that every nice AEC has an independence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.

33 pages

References in corpus (4)

• Building independence relations in abstract elementary classes
• Infinitary stability theory
• Forking and superstability in tame AECs
• Chains of saturated models in AECs

Cited by in corpus (21)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Forking independence from the categorical point of view
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Saturation and solvability in abstract elementary classes with amalgamation
• Tameness and frames revisited
• Chains of saturated models in AECs
• Downward categoricity from a successor inside a good frame
• Toward a stability theory of tame abstract elementary classes
• Abstract elementary classes stable in ${\aleph }_0$
• Building prime models in fully good abstract elementary classes
• The Γ-Ultraproduct and Averageable Classes
• Good Frames in the Hart-Shelah Example
• NSOP${}_1$-like independence in AECats
• Accessible categories, set theory, and model theory: an invitation
• Unstable independence from the categorical point of view
• Non-forking w-good frames
• Lifting independence along functors
• Tameness, Uniqueness and amalgamation
• An NIP-like Notion in Abstract Elementary Classes
• Semi-Good Frames with Amalgamation and Tameness in lambda^+