Denote by \begin{align} D(n) = \dots \to \underline{\text{U}(1)} \to \Omega^1 \to \dots \to \Omega^n \to0 \to \dots \end{align} the Deligne complex over a manifold $M$, where $\Omega^n$ sits in degree $n$. The hypercohomology $H^n(M;D(n))$ is the so-called Deligne cohomology of $M$ and classifies $n$-gerbes with connection.
Is there a very explicit way to define RELATIVE Deligne cohomology, so cohomology for a pair? I would like to write down a short exact sequence for complexes of sheaves that will give me long exact sequences.
And what is the interpretation of relative classes with respect to the above higher-gerbe-interpretation? Is there such a thing as a relative gerbe?
Thank you for any hints.