This is the title of a talk given at the Harvard Logic Seminar on September 24, 2018.
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 accessible categories, a category-theoretic framework encompassing many examples of non-elementary classes) 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. We connect the definition of stable independence to an exactness property, having effective unions, and use this to give new examples where forking occurs: any coregular locally presentable category with effective unions admits a forking-like independence notion. This includes the cases of Grothendieck toposes and Grothendieck abelian categories.
This is joint work with Michael Lieberman and Jiří Rosický.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Preprint: pdf arXiv.
Michael Makkai and Robert Paré, Accessible Categories: The Foundations of Categorical Model Theory, Contemporary Mathematics, vol. 104, American Mathematical Society, 1989.
Tibor Beke and Jiří Rosický, Abstract elementary classes and accessible categories, Annals of Pure and Applied Logic 163 (2012), 2008–2017.
Will Boney, Rami Grossberg, Michael Lieberman, Jiří Rosický, and Sebastien Vasey, μ-Abstract elementary classes and other generalizations, Journal of Pure and Applied Algebra 220 (2016), no. 9, 3048–3066.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Internal sizes in μ-abstract elementary classes, Preprint: pdf arXiv.
Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams, Preprint: pdf arXiv.