A multiplicative surface signature through its Magnus expansion

In the last decade, the concept of path signature has found great success in data science applications, where it provides features describing the path. This is partly explained by the fact that it is possible to compute the signature of a path in linear time, owing to a dynamic programming principle, based on Chen's identity. The path signature can be regarded as a specific example of product or time-/path-ordered integral. In other words, it can be seen as a 1-parameter object build on iterated integrals over a path. Increasing the number of parameters by one, which amounts to considering iterated integrals over surfaces, is more complicated. An observation that is familiar in the context of higher gauge theory where multiparameter iterated integrals play an important role. The 2-parameter case is naturally related to a non-commutative version of Stokes' theorem, which is understood to be fundamentally linked to the concept of crossed modules of groups. Indeed, crossed modules with non-trivial kernel of the feedback map permit to compute features of a surface that go beyond what can be expressed by computing line integrals along the boundary of a surface. A good candidate for the crossed analog of free Lie algebra then seems to be a certain free crossed module over it. Building on work by Kapranov, we study the analog to the classical path signature taking values in such a free crossed module of Lie algebra. In particular, we provide a Magnus-type expression for the logarithm of surface signature as well as a sewing lemma for the crossed module setting.