Functorial reconstruction theorems for stacks

TL;DR

This paper investigates conditions under which stacks can be reconstructed from their functors of isomorphism classes, revealing that many moduli stacks are uniquely determined by these functors and uncovering new anabelian-like phenomena.

Contribution

It introduces new criteria for reconstructing stacks from functors and demonstrates that standard moduli stacks are often uniquely determined by their associated functors.

Findings

Many standard moduli stacks are determined by their functors

New anabelian-type phenomena in the category of schemes

Structures encoding automorphism data in groupoids

Abstract

We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their functors. Our methods seem to exhibit new anabelian-type phenomena, in the form of structures in the category of schemes that encode automorphism data in groupoids.