Representability of derived stacks

TL;DR

This paper extends Lurie's representability theorem for derived stacks, simplifying its conditions and broadening its applicability, including to dg-manifolds and related objects.

Contribution

The authors establish variants of Lurie's theorem that make the criteria easier to verify, including a pre-representability result for dg-manifolds.

Findings

Simplified criteria for derived stack representability.

Reduced verification to underived parts and cohomology groups.

Applicable to dg-manifolds and related geometric objects.

Abstract

Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify for many applications. Provided a derived analogue of Schlessinger's condition holds, the theorem reduces to verifying conditions on the underived part and on cohomology groups. Another simplification is that functors need only be defined on nilpotent extensions of discrete rings. Finally, there is a pre-representability theorem, which can be applied to associate explicit geometric stacks to dg-manifolds and related objects.