Diagramatics of $A_1 \times A_1$ Folded Soergel Bimodules

Abstract

There is an action of $\mathbb{Z}/2$ on the category of Soergel Bimodules of type $A_1 \times A_1$ induced by the nontrivial automorphism of its Dynkin diagram. We give an isotopy presentation by local generators and relations of the equivariantization of the category of Soergel Bimodules of type $A_1 \times A_1$ under this action. This is the first step in a program to describe and study the categories of equivariant Soergel Bimodules.