  We find for each simple finitary Lie algebra $\mathfrak{g}$ a category
$\mathbb{T}_\mathfrak{g}$ of integrable modules in which the tensor product of
copies of the natural and conatural modules are injective. The objects in
$\mathbb{T}_\mathfrak{g}$ can be defined as the finite length absolute weight
modules, where by absolute weight module we mean a module which is a weight
module for every splitting Cartan subalgebra of $\mathfrak{g}$. The category
$\mathbb{T}_\mathfrak{g}$ is Koszul in the sense that it is antiequivalent to
the category of locally unitary finite-dimensional modules over a certain
direct limit of finite-dimensional Koszul algebras. We describe these
finite-dimensional algebras explicitly. We also prove an equivalence of the
categories $\mathbb{T}_{o(\infty)}$ and $\mathbb{T}_{sp(\infty)}$ corresponding
respectively to the orthogonal and symplectic finitary Lie algebras
$o(\infty)$, $sp(\infty)$.
