  We introduce the class of "smooth rough paths" and study their main
properties. Working in a smooth setting allows us to discard sewing arguments
and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan
perspective is the key to a purely algebraic form of Lyons extension theorem,
the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz,
Prei{\ss}, A rough path perspective on renormalization, J. Funct. Anal.
277(11), 2019] as well as a related notion of "sum of rough paths". We first
develop our ideas in a geometric rough path setting, as this best resonates
with recent works on signature varieties, as well the renormalization of
geometric rough paths. We then explore extensions to the quasi-geometric and
the more general Hopf algebraic setting.
