Forking independence from the categorical point of view

TL;DR

This paper develops a category-theoretic framework for forking independence, aiming to generalize and make accessible the concept beyond traditional model theory, especially for accessible categories with monomorphisms.

Contribution

It introduces an axiomatic, purely category-theoretic definition of stable independence, extending forking concepts to broader categorical contexts.

Findings

Axiomatic definition of stable independence in categories

Category-theoretic axiomatization of model-theoretic forking

Framework applicable to accessible categories with monomorphisms

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.