Higher internal category theory

Results which concern the classification of parametrized structures over a given base by means of internal constructions within that base are fairly ubiquitous in homotopy theory. These internalization results are generally formal consequences of reflection properties of an associated externalization functor (that is, usually, some kind of Yoneda embedding). In this talk, we use a suitable externalization functor to define the $(\infty,2)$-category of $\infty$-categories internal to some base $C$. By construction, it can be embedded in the $(\infty,2)$-category of $\infty$-categories indexed over $C$. We show that this embedding is $\infty$-cosmological whenever $C$ is (finitely) complete. That means it creates a plethora of "synthetic" $\infty$-categorical structures which arise in practice as an immediate formal consequence. We furthermore prove a Yoneda lemma for internal $\infty$-categories, discuss applications, and, as time permits, define a lift of this theory to the context of model categorical presentations.