# Cosimplicial spaces and cocycles

## TL;DR

This paper explores the relationship between cosimplicial spaces, non-abelian cohomology, and stacks, establishing new connections between global sections, total complexes, and Postnikov invariants using cocycle methods.

## Contribution

It develops a cocycle-theoretic framework linking cosimplicial groupoids, stacks, and cohomology, providing new insights into the structure of cosimplicial spaces and their invariants.

## Key findings

- Global sections of stack completions are weakly equivalent to Bousfield-Kan total complexes.
- Postnikov k-invariants are elements of stack cohomology associated to the fundamental groupoid.
- Cocycle techniques unify various aspects of cosimplicial space theory.

## Abstract

Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the Bousfield-Kan total complex of $BG$ for all cosimplicial groupoids $G$. The $k$-invariants for the Postnikov tower of a cosimplicial space $X$ are naturally elements of stack cohomology for the stack associated to the fundamental groupoid $\pi(X)$ of $X$. Cocycle-theoretic ideas and techniques are used throughout the paper.