See Chapter 6 of the book 
[**Derived Categories**][1]
for various notions of localization, including the Ore conditions. (Also available at the arXiv at https://arxiv.org/abs/1610.09640.) 

Here's a sample of the type of stuff in loc. cit. It captures the 2-categorical nature of localization 
[![enter image description here][2]][2]


  [1]: https://www.cambridge.org/il/academic/subjects/mathematics/algebra/derived-categories
  [2]: https://i.sstatic.net/TwLdW.jpg