Forking independence from the categorical point of view

Abstract

Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in $\mu$-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible - and useful - to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a category and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.

50 pages

References in corpus (9)

• Sheafifiable homotopy model categories
• Canonical forking in AECs
• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Infinitary stability theory
• $\mu$-Abstract Elementary Classes and other generalizations
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Tameness and frames revisited
• Universal abstract elementary classes and locally multipresentable categories

Cited by in corpus (7)

• The Kim-Pillay theorem for Abstract Elementary Categories
• NSOP${}_1$-like independence in AECats
• Independence relations for exponential fields
• Unstable independence from the categorical point of view
• Accessible categories, set theory, and model theory: an invitation
• Lifting independence along functors
• Notions of amalgamation for AECs and categoricity