There is an established notion of strict $n$-category and of $(\infty,n)$-category for finite n. We will describe how there are two possibly competing ways of making sense of these for $n=\infty$, depending on whether one takes the “inductive” or “coinductive” approach. The inductive and coinductive homotopy theories obtained from the strict homotopy theory of n-categories turn out to be equivalent, and are modeled by $\omega$-categories. Instead, we will discuss how the inductive and coinductive homotopy theories obtained from the homotopy theory of $(\infty,n)$-categories are not equivalent, as the latter is a proper localization of the former. This is joint work with Viktoriya Ozornova and Tashi Walde.