Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Westudy a monoidal category $\mathbb{T}_\alpha$ which is universal among allsymmetric $\mathbb{K}$-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 diagonalizablepairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ ofdimension $\alpha$. Here $\alpha$ is an arbitrary cardinal number. We describeexplicitly the simple and the injective objects of $\mathbb{T}_\alpha$ andprove that the category $\mathbb{T}_\alpha$ is Koszul. We pay special attentionto the case where the filtration on $A$ is finite. In this case$\alpha=\aleph_t$ for $t\in\mathbb{Z}_{\geq 0}$.
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