A Homotopical Invariant of Weinstein Surfaces

One generally expects that the techniques of arboreal singularities and gluing of local differential graded categories will result in a useful global invariant for all Weinstein manifolds. In this paper we construct explicit models for the homotopy limits of diagrams of microlocal sheaf categories which arise from Weinstein surfaces with arboreal skeleta. This is done by characterizing all relevant Reedy model structures on the categories of diagrams that we care about. We prove invariance using a complete set of moves for Weinstein homotopies in this setting. Finally we give combinatorial presentations of the invariant for all topological surfaces.