In this [paper](https://academic.oup.com/ptep/article/2018/5/053A01/5025801) also the [journal front page](https://academic.oup.com/ptep/article/2018/5/053A01/5025801), eq. 2.14, it introduces 
  the Grothendieck group $K^0$ of the category of boundary conditions of topological field theory.  

[![enter image description here][1]][1]

> My question is that 
> - what exactly is the Grothendieck group $K^0$ of the category of boundary conditions of topological field theory really mean? 
> - Does this say that the boundary conditions of topological field theory can be related or classified by a Grothendieck group? Is this an abelian group or nonabelian group? ($K^0(BCondμ(M^{d−1})  )=?$) 
> - Should the boundary conditions of topological field theory classified by some bimodule of certain modular tensor category? how is this related to a Grothendieck group?


  [1]: https://i.sstatic.net/TKwr1.png