nLab local equivalence

In the context of localization of an (∞,1)-category or the corresponding 11-category-theoretic Bousfield localization, local equivalences are those morphisms that are seen as equivalences by local objects.

More concretely, let SS be a subset of morphisms. Recall that an SS-local object XX is one such that for all s:ABs : A \to B in SS the induced morphism

Hom(s,X):Hom(B,X)Hom(A,X) Hom(s,X) : Hom(B,X) \to Hom(A,X)

is an equivalence.

Conversely, a morphism f:VWf : V \to W is is an SS-local equivalence if for every SS-local object XX the induced morphism

Hom(f,X):Hom(W,X)Hom(V,X) Hom(f,X) : Hom(W,X) \to Hom(V,X)

is an equivalence.

In the context of simplicial model categories “equivalence” means: weak equivalence of simplicial sets.


The model category theoretic notion is discussed in section A.3.7 of

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