Deligne categories and the limit of categories Rep(GL(m|n))

Deligne categories Rep(GL_t) (for a complex parameter t) have been constructed by Deligne and Milne in 1982 as a polynomial extrapolation of the categories of algebraic representations of the general linear groups GL_n(C). 
In this talk, we will show how to construct a "free abelian tensor category generated by one object of dimension t", which will be, in a sense, the smallest abelian tensor category which contains the respective Deligne's category Rep(GL_t). 
The construction is based on an interesting stabilization phenomenon occurring in categories of representations of supergroups GL(m|n) when t is an integer and m-n=t. 
This is based on a joint work with V. Seganova and V. Hinich.