nLab local equivalence

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

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

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

is an equivalence.

Conversely, a morphism $f : V \to W$ is is an $S$-local equivalence if for every $S$-local object $X$ the induced morphism

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

is an equivalence.

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

References

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

• Jacob Lurie, Higher Topos Theory .