Orthogonality of morphisms

Let \(C\) be a category. A morphism \(l \colon A \to B\) is said to be left orthogonal to a morphism \(r \colon X \to Y\), written \(l \perp r\), if the following square is a pullback square:

Equivalently, for every commutative square

the anima of dashed fillers making both triangles commute is contractible. In this case, we also say that \(r\) is right orthogonal to \(l\).

Given a collection of morphisms \(S\), we denote by \(S^\perp\) the collection of all morphisms that are right orthogonal to every morphism in \(S\), and by \({}^\perp S\) the collection of those that are left orthogonal to every morphism in \(S\).