higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
…
A derived moduli stack is a moduli space in the context of derived algebraic geometry.
The existence of derived moduli stacks was first conjectured by Maxim Kontsevich, see hidden smoothness principle.
classifying space, classifying stack, moduli space, moduli stack
universal principal bundle, universal principal infinity-bundle
The first examples of derived moduli spaces, using the language of dg-schemes, were constructed in
Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Quot schemes, 1999, arXiv:math/9905174.
Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Hilbert schemes, 2000, arXiv:math/0005155.
In the language of derived algebraic geometry they were considered in
Jacob Lurie, Moduli Problems and DG-Lie Algebras
J. P. Pridham, Representability of derived stacks, arxiv/1011.2189
For the construction of the derived moduli stack of perfect complexes on a smooth proper scheme, see