A Representability Theorem for Stacks in Derived Geometry Contexts

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.