How nice are free completions of categories?

Abstract

Every category K has a free completion PK under colimits and a free completion ΣK under coproducts. A number of properties of K transfer to PK and ΣK (e.g., completeness or cartesian closedness). We prove that PK is often a pretopos, but, for K large, seldom a topos. Moreover, for complete categories K we prove that PK is locally cartesian closed whenever K is additive or cartesian closed or dual to an extensive category. We also prove that PK is (co)wellpowered if K is a ‘set-like’ category, but it is neither wellpowered nor cowellpowered for a number of important categories.