Characterizing algebraic stacks

TL;DR

This paper generalizes the concept of algebraic stacks to arbitrary subcanonical sites, providing new characterizations and extending the framework to algebraic n-stacks, with comparisons across different sites.

Contribution

It introduces a generalized definition of algebraic stacks on subcanonical sites and characterizes them via weak equivalences to representable presheaves, also defining algebraic n-stacks.

Findings

Algebraic stacks are characterized as weakly equivalent to certain representable presheaves.

The paper extends algebraic stack theory to arbitrary subcanonical sites.

It compares various naturally associated sites to a stack.

Abstract

We extend the notion of algebraic stack to an arbitrary subcanonical site C. If the topology on C is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to representable presheaves of groupoids whose domain map is a cover. This leads naturally to a definition of algebraic n-stacks. We also compare different sites naturally associated to a stack.