higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
…
A derived moduli space is a moduli space in the context of derived geometry.
classifying space, classifying stack, moduli space, moduli stack
universal principal bundle, universal principal infinity-bundle
Derived moduli spaces and stacks were considered in
Jacob Lurie, Moduli Problems and DG-Lie Algebras
J. P. Pridham, Representability of derived stacks, arxiv/1011.2189
I. Ciocan-Fontanine, M. Kapranov, Derived Hilbert schemes, math.AG/0005155