A combinatorial approach to geometric rough paths and their controlled paths

We develop the structure theory for transformations of weakly geometric rough paths bounded 1 < p $1 p$ -variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. derive an explicit combinatorial expression path lift a path, use to obtain fundamental identities such associativity integral, adjunction between pushforwards pullbacks change variables formula differential equations (RDEs). As applications we define paths, integration RDEs manifolds, extending results Cass, Driver, Litterer (Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 1471–1518) case arbitrary $p$ .