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

Let $mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $mathcal{O}_c$ admits vertex algebraic tensor category structure for any $cinmathbb{C}$. 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 irreducible modules in a natural tensor subcategory $mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $mathfrak{sl}_2$ at levels $-2+p^{pm 1}$. Next, as a straightforward 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 of projective 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.