  We introduce a functional covariant differential as a tool for studying field
space geometry in a manifestly covariant way. We then touch upon its role in
gauge theories and general relativity over bounded regions, and in BRST
symmetry. Due to the Gribov problem, we argue that our formalism ---allowing
for a non-vanishing functional curvature---is necessary for a global treatment
of gauge-invariance in field space. We conclude by suggesting that the
structures we introduce satisfactorily implement the notion of a
(non-asymptotic) observer in gauge theories and general relativity.
