Small object arguments, plus-construction, and left-exact localizations

Abstract

We present a variant of the small object argument, inspired by Kelly, better suited to construct unique factorisation systems. Our main result is to compare it to the plus-construction involved in sheafification. We apply this to construct localizations, modalities and left-exact localizations explicitly from generators.

v2. Add a couple of examples

References in corpus (2)

• Higher Topos Theory
• The join construction

Cited by in corpus (1)

• Left-exact Localizations of $\infty$-Topoi I: Higher Sheaves