I will give a leisurely introduction to double ∞-categories, provide a range of examples, and explain why these structures are useful. In particular, I will sketch how they play a key role in an abstract framework that systematically develops different generalizations of ∞-category theories (like equivariant and enriched versions). This was part of my PhD thesis.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/TopologySeminar