Forking independence from the categorical point of view

This is the title of a talk given at the Harvard Logic Seminar on September 24, 2018.

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 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.

References