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Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$modules for the Virasoro Lie algebra whose composition factors are irreduciblequotients of reducible Verma modules. Recently, it has been shown that$\mathcal{O}_c$ admits vertex algebraic tensor category structure for any$c\in\mathbb{C}$. Here, we determine the structure of this tensor category when$c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that$\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreduciblemodules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then computeall tensor products involving irreducible modules and their projective covers.Using these tensor product formulas, we show that $\mathcal{O}_c$ has asemisimplification which, as an abelian category, is the Deligne product of twotensor subcategories that are tensor equivalent to the Kazhdan-Lusztigcategories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as astraightforward consequence of the braided tensor category structure on$\mathcal{O}_c$ together with the theory of vertex operator algebra extensions,we rederive known results for triplet vertex operator algebras$\mathcal{W}(p)$, including rigidity, fusion rules, and construction ofprojective covers. Finally, we prove a recent conjecture of Negron that$\mathcal{O}_c^0$ is braided tensor equivalent to the$PSL(2,\mathbb{C})$-equivariantization of the category of$\mathcal{W}(p)$-modules.

Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$