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

Derived sections of Grothendieck fibrations and the problems of homotopical algebra