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.

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

Last revised on July 16, 2018 at 13:00:38. See the history of this page for a list of all contributions to it.