Higher Homotopy Categories, Higher Derivators, and K-Theory

I will discuss the construction of the homotopy n-category and its properties, especially in connection with the existence of higher weak colimits. Inspired by this construction, I will introduce higher categorical generalizations of the notion of a (pre)derivator which take values in n-categories for any fixed n. These may be regarded as a sequence of approximations that bridge the gap between $\infty$-categories and derivators. Then I will define $K$-theory for higher homotopy categories and for higher derivators, and present some results on the comparison with Waldhausen $K$-theory.