  We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder
relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror
equivalence with the category of perfect complexes on a crepant resolution
(over $k[t_0,\ldots, t_n]$) of the singularity $uv=t_0t_1\ldots t_n$. Upon
making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any
crepant resolution of the $cA_{n}$ singularity given by the equation $uv=
f_0\ldots f_n$. These categories inherit braid group actions via the action on
$\mathcal{W}(T^*S^1,D)$ of the mapping class group of $T^*S^1$ fixing $D$. We
also give a geometric model of the derived contraction algebra of a $cA_n$
singularity in terms of the relative Fukaya category of the disc.
