TLDR: There are two different intuitive definitions of predictive fairness. Given populations with different base crime rates, it is impossible to meet both fairness criteria.
If you can survive the WP payroll, the clearest article I've found on this is here:
It defines 4 separate objectives (profit maximisation and 3 types of fairness), and shows that they each result in different thresholds. Just click any of the 4 red buttons.
It then goes on to advocate a particular kind of fairness.
If you can survive the WP payroll, the clearest article I've found on this is here:
https://www.washingtonpost.com/news/monkey-cage/wp/2016/10/1...
The full paper is here: https://arxiv.org/pdf/1610.07524v1.pdf