Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category

In this paper, we introduce a rewriting theory for linear monoidal categories. Those categories are a particular case of linear (n,p)-categories that we define in this paper. We also define linear (n,p)-polygraphs, a linear adaptation of n-polygraphs, to present linear (n−1,p)-categories. We focus then on linear (3,2)-polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory to prove a basis theorem on the category AOB. Our method uses decreasingness, a property introduced by van Ostroom.