Projective covers of the simple modules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$

We study the structure of the abelian category of modules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$. Using the logarithmic deformation by Fjelstad et al., we construct logarithmic $\mathcal{W}_{p_+,p_-}$-modules that have $L_0$ nilpotent rank three or two. By using the structure of these logarithmic modules and the results on logarithmic Virasoro modules by Kyt\"{o}l\"{a} and Ridout, we compute ${\rm Ext}^1$ groups between certain indecomposable modules and simple modules. Based on these ${\rm Ext}^1$ groups we determine the structure of the projective covers of all $\mathcal{W}_{p_+,p_-}$-simple modules.