  We study the stability of covers of simplicial complexes. Given a map $f:Y\to
X$ that satisfies almost all of the local conditions of being a cover, is it
close to being a genuine cover of $X$? Complexes $X$ for which this holds are
called cover-stable. We show that this is equivalent to $X$ being a cosystolic
expander with respect to non-abelian coefficients. This gives a new
combinatorial-topological interpretation to cosystolic expansion which is a
well studied notion of high dimensional expansion. As an example, we show that
the $2$-dimensional spherical building $A_{3}(\mathbb{F}_q)$ is cover-stable.
We view this work as a possibly first example of "topological property
testing", where one is interested in studying stability of a topological notion
that is naturally defined by local conditions.
