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.