A unified approach to different generalizations of $\infty$-category theory

The language of Joyal and Lurie’s $\infty$-categories has now become indispensable in homotopy theory. However, for some purposes, it is convenient to pass to indexed or enriched versions of $\infty$-categories. For instance, homotopy theories of mathematical objects that admit symmetries governed by some group $G$, are usually better organized in so-called $G$-equivariant $\infty$-categories. We will see some examples of this principle. There are specialized notions of categorical concepts in the equivariant context, such as adjunctions, (co)limits, and Kan extensions. This subsumes some well-known constructions: for instance, taking equivariant coproducts corresponds to induction. Instead of redeveloping such a “category theory” in the equivariant setting, we will explore an abstract framework that produces well-behaved category theories for a wide variety of flavors of $\infty$-categories, including enriched, fibered, and internal variants. We will touch on some examples.