Forking independence from the categorical point of view
Michael Lieberman, Ji\v{r}\'i Rosick\'y, and Sebastien Vasey

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 -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.
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