This is the title of a talk given at the Seminář z algebry at Masaryk university on December 13, 2018.
Forking independence is a joint generalization of linear independence in vector spaces and algebraic independence in fields. It was introduced by Saharon Shelah in the seventies and is now a central pillar of modern model theory.
Shelah initially developed the theory of forking for classes of structures axiomatized in first-order logic. However, there are other setups where a theory of forking is possible. This includes classes axiomatized by certain infinitary logics, classes with well-behaved closure operators (quasiminimal classes, used by Zilber to study exponential fields and Schanuel's conjecture), and even accessible categories. In this talk, I will attempt to give a flavor for what forking is, starting with linear independence and ending with the category-theoretic axiomatization developed by Lieberman, Rosický, and myself.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Preprint: pdf arXiv.
Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams, Preprint: pdf arXiv.
Sebastien Vasey, Quasiminimal abstract elementary classes, Archive for Mathematical Logic 57 (2018), nos. 3-4, 299–315. Publisher version pdf arXiv.
John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.