The nil-Brauer category

We introduce the nil-Brauer category and prove a basis theorem for its morphism spaces. This basis theorem is an essential ingredient required to prove that nil-Brauer categorifies the split iquantum group of rank one. As this iquantum group is a basic building block for $\mathrm{ı}$-quantum groups of higher rank, we expect that the nil-Brauer category will play a role in future developments related to the categorification of quantum symmetric pairs.