# Simplicial orthogonality

Carles Casacuberta

Universitat de Barcelona
carles.casacuberta@ub.edu

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.

2005-05-23