Abstract
I will present a variation of Quillen small object argument, inspired by Kelly, which is better suited to construct unique factorisation systems. The construction uses the pushout product and pullback hom structure on the category of arrows. I will explain how this construction recovers the + construction involved in sheafification and I will apply it to construct modalities, localizations and left exact localizations explicitely from generators. All constructions are valid in 1-categories and as well as infinity-categories. This is a joint work with C. L. Subramaniam.