Representability theorem in derived analytic geometry

TL;DR

This paper proves a representability theorem in derived analytic geometry, establishing conditions under which a moduli functor corresponds to a derived analytic stack, thereby confirming the naturalness and practicality of derived analytic spaces.

Contribution

The paper introduces a representability theorem for derived analytic geometry, providing verifiable conditions that ensure a moduli functor is a derived analytic stack, applicable to complex and non-archimedean settings.

Findings

The theorem confirms the naturalness of derived analytic spaces.

Conditions for representability are practical and easy to verify.

Enables enhancement of classical moduli spaces with derived structures.

Abstract

We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theorem enables us to enhance various classical moduli spaces with derived structures, thus provides plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem.