  The description of algebraic structure of n-fold loop spaces can be done
either using the formalism of topological operads, or using variations of
Segal's $\Gamma$-spaces. The formalism of topological operads generalises well
to different categories yielding such notions as $\mathbb E_n$-algebras in
chain complexes, while the $\Gamma$-space approach faces difficulties.
  In this paper we discuss how, by attempting to extend the Segal approach to
arbitrary categoires, one arrives to the problem of understanding "weak"
sections of a homotopical Grothendieck fibration. We propose a model for such
sections, called derived sections, and study the behaviour of homotopical
categories of derived sections under the base change functors. The technology
developed for the base-change situation is then applied to a specific class of
"resolution" base functors, which are inspired by cellular decompositions of
classifying spaces. For resolutions, we prove that the inverse image functor on
derived sections is homotopically full and faithful.
