  We study the braided monoidal structure that the fusion product induces on
the abelian category $\mathcal{W}_p$-mod, the category of representations of
the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a
family of vertex operator algebras that form the simplest known examples of
symmetry algebras of logarithmic conformal field theories. We formalise the
methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch,
that are widely used in the physics literature and illustrate a systematic
approach to calculating fusion products in non-semi-simple representation
categories. We apply these methods to the braided monoidal structure of
$\mathcal{W}_p$-mod, previously constructed by Huang, Lepowsky and Zhang, to
prove that this braided monoidal structure is rigid. The rigidity of
$\mathcal{W}_p$-mod allows us to prove explicit formulae for the fusion product
on the set of all simple and all projective $\mathcal{W}_p$-modules, which were
first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and
Runkel.
