A Representability Theorem for Stacks in Derived Geometry Contexts

TL;DR

This paper proves a general representability theorem for derived stacks across various geometric contexts, including algebraic and analytic, providing practical criteria for their geometricity.

Contribution

It introduces a unified representability theorem applicable to both derived algebraic and analytic geometries, broadening the scope of existing theorems.

Findings

The theorem applies to a wide range of derived geometric contexts.

Provides verifiable conditions for a derived stack to be n-geometric.

Lays groundwork for future moduli stack studies in derived analytic geometry.

Abstract

The representability theorem for stacks, due to Artin in the underived setting and Lurie in the derived setting, gives conditions under which a stack is representable by an $n$-geometric stack. In recent work of Ben-Bassat, Kelly, and Kremnizer, a new theory of derived analytic geometry has been proposed as geometry relative to the $(\infty,1)$-category of simplicial commutative Ind-Banach $R$-modules, for $R$ a Banach ring. In this paper, we prove a representability theorem which holds in a very general context, which we call a representability context, encompassing both the derived algebraic geometry context of To\"en and Vezzosi and these new derived analytic geometry contexts. The representability theorem gives natural and easily verifiable conditions for checking that derived stacks in these contexts are $n$-geometric, such as having an $n$-geometric truncation, being nilcomplete, and having an obstruction theory. Future work will explore representability of certain moduli stacks arising in derived analytic geometry, for example moduli stacks of Galois representations.