Comparison results for derived Deligne-Mumford stacks

TL;DR

This paper compares two definitions of derived Deligne-Mumford stacks by To"en-Vezzosi and Lurie, clarifying their equivalence which is often assumed but not explicitly documented.

Contribution

It provides a precise comparison and clarification of the equivalence between the two existing frameworks for derived Deligne-Mumford stacks.

Findings

The two notions are essentially equivalent.

Clarifies the relationship between To"en-Vezzosi and Lurie's definitions.

Facilitates switching between the two frameworks in research.

Abstract

In this short note we write down a comparison between the notion of derived Deligne-Mumford stack in the sense of B. To\"en and G. Vezzosi [HAG-II] and the one introduced by J. Lurie in [DAG-V]. It is folklore that the two theories yield essentially the same objects, but it is difficult to locate in the literature a precise result, despite being sometimes useful to be able to switch between the two frameworks.