2018/06/30 by Jiří Adámek, Jiří Rosický
Mathematics · #math.CT
published as Top. Appl. 273 (2020), 106972
arxiv created 2019/04/06 · arxiv updated 2020/12/04
Every category \mathcal K has a free completion \mathcal P \mathcal K under colimits and a free completion Σ\mathcal K under coproducts. A number of properties of \mathcal K transfer to \mathcal P \mathcal K and Σ\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.