  Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We
study a monoidal category $\mathbb{T}_\alpha$ which is universal among all
symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$
and $B$ such that $A$ has a, possibly transfinite, filtration. We construct
$\mathbb{T}_\alpha$ as a category of representations of the Lie algebra
$\mathfrak{gl}^M(V_*,V)$ consisting of endomorphisms of a fixed diagonalizable
pairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ of
dimension $\alpha$. Here $\alpha$ is an arbitrary cardinal number. We describe
explicitly the simple and the injective objects of $\mathbb{T}_\alpha$ and
prove that the category $\mathbb{T}_\alpha$ is Koszul. We pay special attention
to the case where the filtration on $A$ is finite. In this case
$\alpha=\aleph_t$ for $t\in\mathbb{Z}_{\geq 0}$.
