The Foundation of a Generalized Parallel Connection

The foundation of a matroid is an algebraic invariant that controls representations over any partial field, hyperfield, or more generally, any pasture. We show that, under certain conditions, the foundation of a generalized parallel connection of two matroids is the relative tensor product of their foundations. We use this to show that the foundation of a 2-sum of two matroids is the absolute tensor product of their foundations. This is joint work with Matt Baker, Oliver Lorscheid, and Tianyi Zhang.