  The goal of this paper is to show that fundamental concepts in higher-order
Fourier analysis can be nauturally extended to the non-commutative setting. We
generalize Gowers norms to arbitrary compact non-commutative groups. On the
structural side, we show that nilspace theory (the algebraic part of
higher-order Fourier analysis) can be naturally extended to include all
non-commutative groups. To this end, we introduce generalized nilspaces called
"groupspaces" and demonstrate that they possess properties very similar to
nilspaces. We study $k$-th order generalizations of groups that are special
groupspaces called {\it k-step} groupspaces. One step groupspaces are groups.
We show that $k$-step groupspaces admit the structure of an iterated principal
bundle with structure groups $G_1,G_2,\dots,G_k$. A similar, but somewhat more
technical statement holds for general groupspaces, with possibly infinitely
many structure groups. Structure groups of groupspaces are in some sense
analogous to higher homotopy groups. In particular we use a version of the
Eckmann-Hilton argument from homotopy theory to show that $G_i$ is abelian for
$i\geq 2$. Groupspaces also show some similarities with $n$-groups from higher
category theory (also used in physics) but the exact relationship between these
concepts is a subject of future research.
