  Ginzburg algebras associated to triangulated surfaces provide a means to
categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the
finite derived category of such a Ginzburg algebra can be embedded into the
Fukaya category of the total space of a Lefschetz fibration over the surface.
Inspired by this perspective we provide a description of the full derived
category in terms of a perverse schober. The main novelty is a gluing formalism
describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras
associated to discs. As a first application we give a new proof of the derived
invariance of these Ginzburg algebras under flips of an edge of the
triangulation. Finally, we note that the perverse schober as well as the
resulting gluing construction can also be defined over the sphere spectrum.
