# MODULI OF OBJECTS IN DG-CATEGORIES BY BERTRAND TOËN

@inproceedings{Ton2007MODULIOO, title={MODULI OF OBJECTS IN DG-CATEGORIES BY BERTRAND TO{\"E}N}, author={Bertrand To{\"e}n}, year={2007} }

– The purpose of this work is to prove the existence of an algebraic moduli classifying objects in a given triangulated category. To any dg-category T (over some base ring k), we define a D−-stack MT in the sense of [TOËN B., VEZZOSI G., Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc., in press], classifying certain T -dg-modules. When T is saturated, MT classifies compact objects in the triangulated category [T ] associated to T . The main result of…

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## References

SHOWING 1-10 OF 24 REFERENCES

The homotopy theory of dg-categories and derived Morita theory

- Mathematics
- 2004

The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in…

A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations

- Mathematics
- 1998

We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a…

Moduli of complexes on a proper morphism

- Mathematics
- 2005

Given a proper morphism X -> S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat…

Derived Hilbert schemes

- Mathematics
- 2000

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X)…

Monoidal model categories

- Mathematics
- 1998

A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a…

Equivalences of monoidal model categories

- Mathematics
- 2002

We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation…

Algebras and Modules in Monoidal Model Categories

- Mathematics
- 1998

In recent years the theory of structured ring spectra (formerly known as A∞‐ and E∞‐ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and…