How nice are free completions of categories?

Abstract

Every category $\mathcal K$ has a free completion $\mathcal P \mathcal K$ under colimits and a free completion $\Sigma\mathcal K$ under coproducts. A number of properties of $\mathcal K$ transfer to $\mathcal P \mathcal K$ and $\Sigma\mathcal K$ (e.g., completeness or cartesian closedness). We prove that $\mathcal P\mathcal K$ is always a pretopos, but, for $\mathcal K$ large, seldom a topos. Moreover, for complete categories $\mathcal K$ we prove that $\mathcal P\mathcal K$ is locally cartesian closed whenever $\mathcal K$ is additive or cartesian closed or dual to an extensive category. We also study the question whether $\mathcal P \mathcal K$ is (co)wellpowered. The answer is affirmative for "set-like" categories. But for a number of categories $\mathcal K$ the answer turns out to be negative.