# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.03583