Alistair Savage and I have uploaded our paper The spin Brauer category to the arXiv.
The Brauer category only sees the representations of 
, which come from representation of the group 
. In particular, it misses the spin representation 
(which is a tensor generator of the category of finite dimensional -modules). The starting point of this paper is to define a new category, the spin Brauer category, which sees the spin, and hence all, representations of . Here is the abstract:
We introduce a diagrammatic monoidal category, the spin Brauer category, that plays the same role for the spin and pin groups as the Brauer category does for the orthogonal groups. In particular, there is a full functor from the spin Brauer category to the category of finite-dimensional modules for the spin and pin groups. This functor becomes essentially surjective after passing to the Karoubi envelope, and its kernel is the tensor ideal of negligible morphisms. In this way, the spin Brauer category can be thought of as an interpolating category for the spin and pin groups. We also define an affine version of the spin Brauer category, which acts on categories of modules for the pin and spin groups via translation functors.
If you go to the arXiv and download the source, you will see the following line:
%\toggletrue{details} % To include details (default is false)
If you uncomment this line (and fix the bibliography, which for reasons unknown to me appears at the start when you download source), and then compile the file yourself, you will see a version with more details.