Categorical and combinatorial aspects of descent theory

TL;DR

This paper explores the combinatorial and categorical foundations of descent theory across all dimensions, focusing on strict and weak n-categories, and integrates concepts like non-abelian cohomology and stacks.

Contribution

It provides a detailed combinatorial description of descent constructions for strict n-categories and outlines the extension to weak n-categories, advancing higher-dimensional descent theory.

Findings

Precise combinatorial description for strict n-categories

Outline for weak n-categories

Development of descent theory in low dimensions

Abstract

There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict n-categories and outlined for weak n-categories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves non-abelian cohomology, stacks, torsors, homotopy, and higher-dimensional categories. Many of the ideas are scattered through the literature or are folklore; a few are new.