Bar-Natan Modules and Tunneling Graphs

We describe a general method for presentations of colimit modules of functors into module categories. This is applied to the Bar-Natan functor, which is defined on a category of surfaces embedded in a 3-manifold M with morphisms defined by certain 3-manifolds embedded in M x [0,1] and takes values in a category of modules defined from a commutative Frobenius algebra. The colimit of the Bar-Natan functor is the Bar-Natan module of M. Our approach naturally leads to the definition of the tunneling graph of M, which contains the geometric data necessary to deduce the structure of the Bar-Natan module.