  The $p$-modulus of curves, test plans, upper gradients, charts,
differentials, approximations in energy and density of directions are all
concepts associated to the theory of Sobolev functions in metric measure
spaces. The purpose of this paper is to give an analogous geometric and
``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where
$\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti
representations play a central role. We give a new definition of fragment-wise
charts and prove that they exists for spaces with finite Hausdorff dimension.
We give a replacement for $p$-duality in terms of Alberti representations and
$\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we
give new and sharper results for approximations of Lipschitz functions, which
yields the density of directions. Our results are applicable to all complete
and separable metric measure spaces. In the process, we show that there are
strong parallels between the Sobolev and Lipschitz worlds.
