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The category of weight modules $L_k(\mathfrak{sl}_2)\text{-wtmod}$ of thesimple affine vertex algebra of $\mathfrak{sl}_2$ at an admissible level $k$ isneither finite nor semisimple and modules are usually not lower-bounded andhave infinite dimensional conformal weight subspaces. However this vertexalgebra enjoys a duality with $W_\ell(\mathfrak{sl}_{2|1})$, the simpleprinicipal $W$-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 hasfinite-dimensional conformal weight spaces. The main technical result is thatevery weight module of $W_\ell(\mathfrak{sl}_{2|1})$ is $C_1$-cofinite. Theexistence of a vertex tensor category follows and the theory of vertexsuperalgebra extensions implies the existence of vertex tensor categorystructure on $L_k(\mathfrak{sl}_2)\text{-wtmod}$ for any admissible level $k$.As applications, the fusion rules of ordinary modules with any weight moduleare computed and it is shown that $L_k(\mathfrak{sl}_2)\text{-wtmod}$ is aribbon category if and only if $L_{k+1}(\mathfrak{sl}_2)\text{-wtmod}$ is, inparticular 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.

Tensor categories of weight modules of $\widehat{\mathfrak{sl}}_2$ at admissible level