A representation on the space of paths is a map which is compatible with the concatenation operation of paths, such as the path signature and Cartan development (or equivalently, parallel transport), and has been used to define characteristic functions for the law of stochastic processes. In this talk, we consider representations of surfaces which are compatible with the two distinct algebraic operations on surfaces: horizontal and vertical concatenation. To build these representations, we use the notion of higher parallel transport, which was first introduced to develop higher gauge theories. We will not assume any background in geometry or category theory. This is a continuation of the previous talk based on a recent preprint (https://arxiv.org/abs/2311.08366) with Harald Oberhauser.

The curvature-dimension condition, CD(K,N) for short, and the (weaker) measure contraction property, or MCP(K,N), are two synthetic notions for a metric measure space to have Ricci curvature bounded from below by K and dimension bounded from above by N. In this talk, we investigate the validity of these conditions in sub-Finsler geometry, which is a wide generalization of Finsler and sub-Riemannian geometry. Firstly, we show that sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure can not satisfy the CD(K,N) condition for any K and N. Secondly, we focus on the sub-Finsler Heisenberg group, where we show that, on the one hand, the CD(K,N) condition can not hold for any reference norm and, on the other hand, the MCP(K,N) may hold or fail depending on the regularity of the reference norm.