Abstract

A basic definition of a factorization system on a category could be a pair (E,M) of classes of morphisms such that every morphism in the category factorizes as the composite of something from E followed by something from M. For example, the (epi,mono)-factorization in the category of sets arises from expressing a function as the composite of ‘surjection onto image’ followed by ‘inclusion of image into codomain’. (See also Q4 CT Sheet 1). Note that in these cases, the factorizations of a given morphism are (essentially) unique.

Orthogonality is a simple binary relation on the morphisms of a category, which will allow us to define the notion of an ‘Orthogonal Factorization System’ (OFS). I will justify the definition by showing that it is (almost) equivalent to ‘factorization system with unique factorizations’ and go on to describe the basic properties and examples of OFS ’s. I hope to explain the connection to reflective subcategories and the orthogonal subcategory problem and talk about some existence theorems.