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We develop the structure theory for transformations of weakly geometric roughpaths of bounded $1 < p$-variation and their controlled paths. Our approachdiffers from existing approaches as it does not rely on smooth approximations.We derive an explicit combinatorial expression for the rough path lift of acontrolled path, and use it to obtain fundamental identities such as theassociativity of the rough integral, the adjunction between pushforwards andpullbacks, and a change of variables formula for rough differential equations(RDEs). As applications we define rough paths, rough integration and RDEs onmanifolds, extending the results of [CDL15] to the case of arbitrary $p$.

A combinatorial approach to geometric rough paths and their controlled paths