  The category of weight modules $L_k(\mathfrak{sl}_2)\text{-wtmod}$ of the
simple affine vertex algebra of $\mathfrak{sl}_2$ at an admissible level $k$ is
neither finite nor semisimple and modules are usually not lower-bounded and
have infinite dimensional conformal weight subspaces. However this vertex
algebra enjoys a duality with $W_\ell(\mathfrak{sl}_{2|1})$, the simple
prinicipal $W$-algebra of $\mathfrak{sl}_{2|1}$ at level $\ell$ (the $N=2$
super conformal algebra) where the levels are related via $(k+2)(\ell+1)=1$.
Every weight module of $W_\ell(\mathfrak{sl}_{2|1})$ is lower-bounded and has
finite-dimensional conformal weight spaces. The main technical result is that
every weight module of $W_\ell(\mathfrak{sl}_{2|1})$ is $C_1$-cofinite. The
existence of a vertex tensor category follows and the theory of vertex
superalgebra extensions implies the existence of vertex tensor category
structure on $L_k(\mathfrak{sl}_2)\text{-wtmod}$ for any admissible level $k$.
As applications, the fusion rules of ordinary modules with any weight module
are computed and it is shown that $L_k(\mathfrak{sl}_2)\text{-wtmod}$ is a
ribbon category if and only if $L_{k+1}(\mathfrak{sl}_2)\text{-wtmod}$ is, in
particular it follows that for admissible levels $k = - 2 + \frac{u}{v}$ and $v
\in \{2, 3\}$ and $u = -1 \mod v$ the category
$L_k(\mathfrak{sl}_2)\text{-wtmod}$ is a ribbon category.
