The signature of a path, first introduced by Chen, is the holonomy of a universal translation invariant connection on Euclidean space and it provides a transform whereby paths are represented by non-commutative power series. Chen proved that the signature characterizes a path up to thin homotopy—an equivalence relation that essentially consists of reparametrization and cancellation of retracings. Kapranov generalized the signature by introducing a universal higher connection whose holonomy represents surfaces (or higher dimensional membranes) through formal series of tensors. I will report on joint work with Darrick Lee in which we study the signature of piecewise linear surfaces and characterize the kernel in terms of thin homotopy.