  The aim of the present paper is to define a notion of weakly differentiable
cochain in the generality of metric measure spaces and to study basic
properties of such cochains. Our cochains are (sub-)linear functionals on a
subspace of chains, and a suitable notion of chains in metric spaces is given
by Ambrosio-Kirchheim's theory of metric currents. The notion of weak
differentiability we introduce is in analogy with Heinonen-Koskela's concept of
upper gradients of functions. In one of the main results of our paper, we prove
continuity estimates for cochains with $p$-integrable upper gradient in
$n$-dimensional Lie groups endowed with a left-invariant Finsler metric. Our
result generalizes the well-known Morrey-Sobolev inequality for Sobolev
functions. Finally, we prove several results relating capacity and modulus to
Hausdorff dimension.
