nLab 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 TT at a subset of morphisms SS is the data of a morphism

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

in the (infinity,1)-category of dg-categories dgcatdg-cat such that for any dg-category TdgcatT' \in dg-cat the induced morphism

γ *:Map(T[S 1],T)Map(T,T) \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 TTT \to T' in dgcatdgcat that send morphisms of SS to equivalences in TT'.

Proposition (Toen)

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


See section 8.2 of

and section 4.3 of

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