# Galois descent criteria

## TL;DR

This paper explores homotopy descent in algebraic $K$-theory, focusing on Galois cohomological methods and the construction of fibrant models via homotopy fixed points over fields.

## Contribution

It introduces a pro-categorical approach to achieve full homotopy descent from finite Galois descent conditions in algebraic $K$-theory computations.

## Key findings

- Homotopy fixed point spaces define finite Galois descent.
- A pro-categorical construction is necessary for full homotopy descent.
- Fibrant models can be constructed from naive Galois cohomological objects.

## Abstract

This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects given by homotopy fixed point constructions, but only up to pro-equivalence. The homotopy fixed point spaces define finite Galois descent for simplicial presheaves (and their relatives) over a field, but a pro-categorical construction is a necessary second step for passage from finite descent conditions to full homotopy descent in a Galois cohomological setting.