  We introduce a formalism for derived moduli functors on differential graded
associative algebras, which leads to non-commutative enhancements of derived
moduli stacks and naturally gives rise to structures such as Hall algebras.
Descent arguments are not available in the non-commutative context, so we
establish new methods for constructing various kinds of atlases. The formalism
permits the development of the theory of shifted bi-symplectic and shifted
double Poisson structures in the companion paper.
