A general framework for homotopic descent and codescent

TL;DR

This paper develops a comprehensive homotopy-theoretic framework for descent, codescent, and their duals within the context of $mbda$-categories, unifying various spectral sequences and relating them to algebraic K-theory and étale cohomology.

Contribution

It introduces a general $mbda$-category-based framework for homotopic descent and codescent, providing criteria and spectral sequences that unify many classical and modern results.

Findings

General criteria for homotopic descent and codescent.

Construction of descent and codescent spectral sequences.

Identification of classical spectral sequences as instances of the framework.

Abstract

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as $\infty$-category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for $(\infty, 1)$-categories. We provide general criteria, reminiscent of Mandell's theorem on $E_{\infty}$-algebra models of $p$-complete spaces, under which homotopic (co)descent is satisfied. Furthermore, we construct general descent and codescent spectral sequences, which we interpret in terms of derived (co)completion and homotopic (co)descent. We show that a number of very well-known spectral sequences, such as the unstable and stable Adams spectral sequences, the Adams-Novikov spectral sequence and the descent spectral sequence of a map, are examples of general (co)descent spectral sequences. There is also a close relationship between the Lichtenbaum-Quillen conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic K-theory to \'etale K-theory. Moreover, there are intriguing analogies between derived cocompletion (respectively, completion) and homotopy left (respectively, right) Kan extensions and their associated assembly (respectively, coassembly) maps.