This is the title of a talk given at the Harvard Logic seminar on April 22, 2019.
We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated weak factorization systems (those that are, in a precise sense, generated by a set) are exactly those that give rise to stable independence notions. This two way connection yields a powerful new tool to build tame and stable abstract elementary classes. In particular, we generalize a construction of Baldwin-Eklof-Trlifaj to prove that the category of flat modules with flat monomorphisms has a stable independence notion, and explain how this connects to the fact that every module has a flat cover. Joint work with Jiří Rosický and Michael Lieberman.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Weak factorization systems and stable independence, Preprint: pdf arXiv.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Advances in Mathematics 346 (2019), 719–772. Publisher version pdf arXiv.