On cubes of Frobenius extensions

TL;DR

This paper introduces a diagrammatic approach to understanding natural transformations in Frobenius extension hypercubes, connecting algebraic structures with topological diagrams, inspired by Soergel bimodule theory.

Contribution

It provides a novel diagrammatic description of natural transformations in Frobenius extension hypercubes, linking algebraic and topological methods.

Findings

Diagrammatic description of natural transformations

Relations derived from Soergel bimodule work

Enhanced understanding of induction and restriction functors

Abstract

Given a hypercube of Frobenius extensions between commutative algebras, we provide a diagrammatic description of some natural transformations between compositions of induction and restriction functors, in terms of colored transversely-intersecting planar 1-manifolds. The relations arise in the first and third authors' work on (singular) Soergel bimodules.