nLab derived moduli stack of objects in a dg-category

Given any dg-category $\mathcal{C}$, there is an associated derived moduli stack $\mathcal{M}_\mathcal{C}$ which parametrizes the pseudo-perfect? dg-modules over $\mathcal{C}^{op}$. When $\mathcal{C}$ is smooth? and proper?, $\mathcal{M}_\mathcal{C}$ classifies the objects of $\mathcal{C}$.

In the case where $\mathcal{C}$ is the dg-category enhancing the derived category of a smooth proper scheme $X$, the derived Artin stack? $\mathcal{M}_X$ is called the derived moduli stack of perfect complexes on $X$.

Proposition (Toen-Vezzosi, 3.4). The functor $\mathcal{M}_- : Ho(DGCat(k))^{op} \to DSt(k)$ admits a left adjoint

$$Pf : \DSt(k) \to \Ho(\DGCat(k))^{op}$$

which associates to a derived stack its dg-category of perfect complexes.

References

• Bertrand Toen, Michel Vaquie, Moduli of objects in dg-categories, 2007, arXiv:math/0503269.