This result exhibits orthogonal factorization systems as a special case of double infinity-categories: a fully faithful functor sends each factorization system to the double infinity-category whose horizontal and vertical morphisms are its two morphism classes and whose squares are commutative squares, characterizing the image as those double infinity-categories with a unique square filling every bottom-left corner. Students learn how the unique-factorization axiom is re-expressed as a filling condition, and how fibrations over a factorization system correspond, via an (un)straightening equivalence, to cartesian and cocartesian fibrations over the associated double infinity-category.
This higher-category-theory paper relates two classical categorical structures in the setting of infinity-categories. An orthogonal factorization system is a category equipped with two classes of mor…