Simplicial orthogonality

Carles Casacuberta

Universitat de Barcelona

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