In this paper, we construct a Fukaya category of any infinite type surface
whose objects are gradient sectorial Lagrangians. This class of Lagrangian
submanifolds is introduced by one of the authors in [Oh21b] which can serve as
an object of a Fukaya category of any Liouville manifold that admits an
exhausting proper Morse function, in particular on the Riemann surface of
infinite type. We describe a generating set of the Fukaya category in terms of
the end structure of the surface when the surface has countably many limit
points in its ideal boundary, the latter of which can be described in terms of
a subset of the Cantor set. We also show that our Fukaya category is not
quasi-equivalent to the limit of the Fukaya category of surfaces of finite type
appearing in the literature.Comment: 48 pages, 13 figure