Deligne categories and the limit of categories rep(gl(m|n))

For each integer t a tensor category V t is constructed, such that exact tensor functors V t - C classify dualizable t-dimensional objects in C not annihilated by any Schur functor. This means that V t is the “abelian envelope” of the Deligne category D t = Rep(GLt). Any tensor functor Rep(GLt) -C is proved to factor either through V t or through one of the classical categories Rep(GL(m|n)) with m - n = t. The universal property of V t implies that it is equivalent to the categories RepDt1 Dt2 (GL(X),), (t = t1 + t2, t1 not an integer) suggested by Deligne as candidates for the role of abelian envelope.