We  relativize the notion of degree spectra by considering
multi-component spectra, i.e. a degree spectrum with
respect to a given sequence of sets of natural numbers. We study this
under the omega-enumeration reducibility.  It is a uniform
reducibility between sequences of sets of natural numbers, introduced
and studied by Soskov, H. Ganchev, M. Soskova, etc.
The notion of omega-degree spectrum generalizes the notion of relative
spectrum. The omega-co-spectrum is the set of omega-enumeration
degrees which are lower bounds of the elements of the omega-spectrum.

We prove that most  of the  properties of the degree spectrum such as
the minimal pair  theorem and the existence of quasi-minimal degree
are true for the omega-degree spectrum.
We give an explicit form  of the elements of the omega-co-spectrum of
a structure by means of  computable $\Sigma^+_k$ formulae.