The formalism of Segal sections

Given a family of model categories $\cal E \to \cal C$, we associate to it a homotopical category of derived, or Segal, sections $DSect(\cal C,\cal E)$ that models the higher-categorical sections of the localisation $L\cal E \to \cal C$. The derived sections provide an alternative, strict model for various higher algebra objects appearing in the work of Lurie. We prove a few results concerning the properties of the homotopical category $DSect(\cal C,\cal E)$, and as an example, study its behaviour with respect to the base-change along a select class of functors.