dg-localization

*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.

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 $T'$.

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

- dg-quotient?
- simplicial localization

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

Last revised on February 4, 2015 at 12:45:38. See the history of this page for a list of all contributions to it.