Here is a MO post asking a question that I’ve had in mind for a while: Why higher category theory?
Having studied some elementary topos before, I have been interested in higher topos since I attended a summer school lecture last year. Besides formal generalisations, I expect to see applications which provide new results or meaningful insights. Though the meaning of “new” and “useful” very much diverse between different mathematical cultures.
Here are some important applications of higher categories in K-theory (added on Mar 05, 2020), suggested by my supervisor Schlichting.
Dustin-Mathew-Morrow, Algebraic K-theory and descent for blow-ups
Nikolaus-Scholze, On topological cyclic homology
Clausen-Mathew-Morrow, K-theory and topological cyclic homology of henselian pairs
Antieau-Gepner-Heller, K-theoretic obstructions to bounded t-structures
Blumberg-Gepner-Tabuada, A universal characterization of higher algebraic K-theory
Barwick, On exact infinity-categories and the Theorem of the Heart
