  Graham and Lehrer (1998) introduced a Temperley-Lieb category
$\mathsf{\widetilde{TL}}$ whose objects are the non-negative integers and the
morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes.
The Temperley-Lieb algebra $\mathsf{TL}_{n}$ is identified with
$\mathsf{Hom}(n,n)$. The category $\mathsf{\widetilde{TL}}$ is shown to be
monoidal. We show that it is also a braided category by constructing explicitly
a commutor. A twist is also defined on $\mathsf{\widetilde{TL}}$. We introduce
a module category ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$ whose objects are
functors from $\mathsf{\widetilde{TL}}$ to $\mathsf{Vect}_{\mathbb C}$ and
define on it a fusion bifunctor extending the one introduced by Read and Saleur
(2007). We use the natural morphisms constructed for $\mathsf{\widetilde{TL}}$
to induce the structure of a ribbon category on ${\text{
Mod}_{\mathsf{\widetilde{TL}}}}(\beta=-q-q^{-1})$, when $q$ is not a root of
unity. We discuss how the braiding on $\mathsf{\widetilde{TL}}$ and
integrability of statistical models are related. The extension of these
structures to the family of dilute Temperley-Lieb algebras is also discussed.
