  We study a new notion of trace operators and trace spaces for abstract
Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators.
We characterize the kernels and images of the related trace operators and
discuss duality relationships between trace spaces. We elaborate that many
properties of the classical boundary traces associated with the Euclidean de
Rham complex on bounded Lipschitz domains are rooted in the general structure
of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be
formulated using quotient spaces/annihilators. We show that, if a Hilbert
complex admits stable "regular decompositions" with compact lifting operators,
then the associated trace Hilbert complex is Fredholm. Incarnations of abstract
concepts and results in the concrete case of the de Rham complex in
three-dimensional Euclidean space will be discussed throughout.
