  Given a vertex operator algebra $V$, one can construct two associative
algebras, the Zhu algebra $A(V)$ and the $C_2$-algebra $R(V)$. This gives rise
to two abelian categories $A(V)-\text{Mod}$ and $R(V)-\text{Mod}$, in addition
to the category of admissible modules of $V$. In case $V$ is rational and
$C_2$-cofinite, the category of admissible $V$-modules and the category of all
$A(V)$-modules are equivalent. However, when $V$ is not rational, the
connection between these two categories is unclear. The goal of this paper is
to study the triplet vertex operator algebra $\mathcal{W}(p)$, as an example to
compare these three categories, in terms of abelian categories. For each of
these three abelian categories, we will determine the associated Ext quiver,
the Morita equivalent basic algebra, i.e., the algebra $ \text{End}
(\oplus_{L\in \text{Irr}} P_L)^{op}$, and the Yoneda algebra
$\text{Ext}^{*}(\oplus_{L\in \text{Irr}}L, \oplus_{L\in \text{Irr}}L)$. As a
consequence, the category of admissible log-modules for the triplet VOA $
\mathcal W(p)$ has infinite global dimension, as do the Zhu algebra $A(\mathcal
W(p))$, and the associated graded algebra $\text{gr} \ A(\mathcal W(p))$ which
is isomorphic to $R(\mathcal W(p))$. We also describe the Koszul properties of
the module categories of $ \mathcal W(p)$, $A(\mathcal W(p))$ and $\text{gr} \
A(\mathcal W(p))$.
