Homotopy limits in the category of dg-categories in terms of $\mathrm{A}_{\infty}$-comodules

TL;DR

This paper develops an explicit model for homotopy limits of dg-categories using simplicial constructions and applies it to homotopy descent, confirming a conjecture related to $ ext{A}_ ext{infty}$-comodules.

Contribution

It introduces a new explicit construction for homotopy limits of dg-categories and applies it to homotopy descent, proving a conjecture in the field.

Findings

Explicit model for homotopy limits of dg-categories.

Application to homotopy descent in terms of $ ext{A}_ ext{infty}$-comodules.

Proof of a conjecture by Block, Holstein, and Wei.

Abstract

In this paper, we apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (2016) and Arkhipov and Poliakova (2018), as well as our own results on homotopy ends (Arkhipov and {\O}rsted 2018), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $\mathrm{A}_{\infty}$-comodules, proving a conjecture by Block, Holstein, and Wei (2017) in the process.