Derived sections of Grothendieck fibrations and the problems of homotopical algebra

TL;DR

This paper explores the extension of Segal's approach to homotopical algebra via Grothendieck fibrations, introducing derived sections and analyzing their behavior under base change, with applications to resolution functors.

Contribution

It proposes a model for weak sections of homotopical Grothendieck fibrations, called derived sections, and studies their properties under base change, extending the Segal approach.

Findings

Derived sections form a homotopical category.

Base change functors preserve derived sections.

Inverse image functors are homotopically full and faithful for resolutions.

Abstract

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.