  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.
