How nice are free completions of categories?

TL;DR

This paper investigates the properties of free completions of categories under colimits and coproducts, showing they are often pretoposes and analyzing conditions under which they are topoi or locally cartesian closed.

Contribution

It establishes that free completions are always pretoposes and characterizes when they are topoi or locally cartesian closed, depending on properties of the original category.

Findings

 Pf6topos for all categories

 Locally cartesian closed f6r complete categories with specific properties

 (Co)wellpoweredness depends on category type

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.