  A common approach for describing classes of functions and probability
measures on a topological space $\mathcal{X}$ is to construct a suitable map
$\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be
applied to address both problems. The case where $\mathcal{X}$ is a space of
paths $[0,1] \to \mathbb{R}^n$ and $\Phi$ is the path signature map has
received much attention in stochastic analysis and related fields. In this
article we develop a generalized $\Phi$ for the case where $\mathcal{X}$ is a
space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show
that the map $\Phi$ generalizes many of the desirable algebraic and analytic
properties of the path signature to $d \ge 2$. The key ingredient to our
approach is topological; in particular, our starting point is a generalisation
of K-T Chen's path space cochain construction to the setting of cubical mapping
spaces.
