Derived algebraic stack

Definition

A derived algebraic stack, or derived Artin stack, is a stack $\mathcal X$ on derived affine schemes such that:

1. the diagonal $\mathcal X\to\mathcal X\times\mathcal X$ is representable by derived algebraic spaces; and
2. there is a family of derived affine schemes $U_i$ and a smooth surjective morphism
   $$\coprod_i U_i\longrightarrow\mathcal X.$$

The precise definition is made recursively by geometric level: the representability required of the diagonal is one level lower than that required of $\mathcal X$. It also depends on a chosen model of derived commutative rings and a Grothendieck topology.