  For each integer $t$ a tensor category $V_t$ is constructed, such that exact
tensor functors $V_t \longrightarrow C$ 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 $Rep(GL_t)$. Any tensor functor
$Rep(GL_t)\longrightarrow C$ 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
  $Rep_{Rep(GL_{t_1})\otimes Rep(GL_{t_2})}(GL(X),\epsilon)$, ($t=t_1+t_2$,
$t_1$ not integer) suggested by Deligne as candidates for the role of abelian
envelope.
