nLab dg-localization

Idea

dg-localization is the analogue in the world of dg-categories to the notion of simplicial localization. In good cases it is presented by a dg-model category.

Definition

Definition

The dg-localization of a dg-category $T$ at a subset of morphisms $S$ is the data of a morphism

$$\gamma : T \longrightarrow T[S^{-1}]$$

in the (infinity,1)-category of dg-categories $dg-cat$ such that for any dg-category $T' \in dg-cat$ the induced morphism

$$\gamma^* : Map(T[S^{-1}], T') \longrightarrow Map(T, T')$$

of mapping spaces is injective on connected components and has image the subset of morphisms $T \to T'$ in $dgcat$ that send morphisms of $S$ to equivalences in $T'$.

Proposition (Toen)

The dg-localization of any dg-category at a set of morphisms exists.

Related concepts

• dg-quotient?
• simplicial localization

References

See section 8.2 of

• B. Toen, The homotopy theory of dg-categories and derived Morita theory, arXiv:math/0408337.

and section 4.3 of

• B. Toen, Lectures on dg-categories, pdf.