Higher and derived stacks: a global overview

TL;DR

This paper provides a comprehensive overview of higher and derived stacks in algebraic geometry, covering foundational concepts, examples, recent developments, and open questions to serve as a broad introduction to the subject.

Contribution

It offers a global, accessible overview of the theory of higher and derived stacks, integrating motivations, foundational material, and recent advances in the field.

Findings

Summarizes key concepts and examples of higher and derived stacks.

Highlights recent developments and open questions in the field.

Provides foundational material for further research and study.

Abstract

These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.