An object and a morphism in a category
are called orthogonal if the arrow is bijective.
In the framework of simplicial model categories, the notion
of simplicially enriched orthogonality plays an important role:
and are said to be simplicially orthogonal if the arrow
is a weak equivalence of simplicial sets.
We will explain why the condition that a given class of objects
and a given class of morphisms be the simplicial orthogonal
complement of each other is necessary and sufficient to ensure
the existence of a homotopy localization functor such that
is the class of -local objects and is the class of
-equivalences, under suitable assumptions on the
model category and possibly using large-cardinal axioms.