# 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.

## References

See section 8.2 of

and section 4.3 of

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