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.

We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-étale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case.for the relative case in complex geometry would be more involved because Stein algebras are not noetherian in general. Related works.In the classical sense, complex analytic stacks were considered in [4], and non-archimedean analytic stacks were considered in [49,47] to the best of our knowledge.The general theory of higher stacks was studied extensively by Simpson [42], Lurie [33] and Toën-Vezzosi [45,46]. Our Section 2 follows mainly [46]. However, we do not borrow directly the HAG context of [46], because the latter is based on symmetric monoidal model categories which is not suitable for analytic geometry.Our definition of properness for analytic stacks follows an idea of Kiehl in rigid analytic geometry [26]. The coherence of derived direct images under proper morphisms (i.e. Grauert's theorem) was proved in [17,27,15,23,30] for complex analytic spaces and in [26] for rigid analytic spaces.Analytification of algebraic spaces and classical algebraic stacks was studied in [1,31,44,10]. Analogs and generalizations of Serre's GAGA theorems are found in [19,28,37,5,9,8,21]. Our proofs use induction on the geometric level of higher stacks. We are very much inspired by the strategies of Brian Conrad in [8].In [39], Mauro Porta deduced from this paper a GAGA theorem for derived complex analytic stacks. Several results in this paper are also used in the work [40].In [50], Tony Yue Yu applied the GAGA theorem for non-archimedean analytic stacks to the enumerative geometry of log Calabi-Yau surfaces.Acknowledgements. We are grateful to for very useful discussions. The authors would also like to thank each other for the joint effort.2. Higher geometric stacks 2.1. Notations. We refer to Lurie [32,36] for the theory of ∞-category. The symbol S denotes the ∞-category of spaces. Definition 2.1 (cf. [43, 00VH]). An ∞-site (C, τ ) consists of a small ∞-category C and a set τ of families of morphisms with fixed target {U i → U } i∈I , called τ -coverings of C, satisfying the following axioms:Remark 2.3. In all the examples that we will consider in this paper, the site (C, τ ) is a classical Grothendieck site, i.e. the category C is a 1-category. We state Definition 2.2 for general ∞-sites because it will serve as a geometric context for derived stacks in our subsequent works (cf. [39,40]). Proposition 2.4. Let (C, τ ) be an ∞-site and let D be an (n + 1, 1)-category for n ≥ 0 (cf. [32, §2.3.4]). Then a functor F : C op → D satisfies d...

Abstract. We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition turns out to be insufficient. In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie. We prove the following three fundamental properties of derived non-archimedean analytic spaces:(1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the ∞-category of derived non-archimedean analytic spaces.(2) The ∞-category of derived non-archimedean analytic spaces admits fiber products.(3) The ∞-category of higher non-archimedean analytic Deligne-Mumford stacks embeds fully faithfully into the ∞-category of derived non-archimedean analytic spaces. The essential image of this embedding is spanned by n-localic discrete derived non-archimedean analytic spaces.We will further develop the theory of derived non-archimedean analytic geometry in our subsequent works. Our motivations mainly come from intersection theory, enumerative geometry and mirror symmetry.