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

TL;DR

This paper constructs tensor categories $V_t$ that serve as abelian envelopes for Deligne categories $Rep(GL_t)$, classifying dualizable objects and factoring tensor functors through classical or Deligne categories.

Contribution

It introduces the categories $V_t$ as universal abelian envelopes for $Rep(GL_t)$, providing a new framework for understanding tensor functors and their factorizations.

Findings

$V_t$ classifies dualizable $t$-dimensional objects.

Tensor functors from $Rep(GL_t)$ factor through $V_t$ or classical categories.

$V_t$ is equivalent to categories $Rep_{Rep(GL_{t_1}) imes Rep(GL_{t_2})}(GL(X), ext{"}epsilon")},

Abstract

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.