We define and study the $(\infty,2)$-category $\mathbf{ Cat}_{ \infty}(\mathcal{ C})$ of $(\infty,1)$-categories internal to a general $(\infty,1)$-category $\mathcal{ C}$ via an associated externalization construction. In the first part, we show various formal closure properties of $\mathbf{ Cat}_{ \infty}(\mathcal{ C})$ regarding limits, tensors, cotensors and internal mapping objects under the assumption of various suitable closure properties of $\mathcal{ C}$. In particular, we show that $\mathbf{ Cat}_{ \infty}(\mathcal{ C})$ defines a cartesian closed full sub-$\infty$-cosmos of the $\infty$-cosmos $\mathbf{ Fun}(\mathcal{ C}^{ op},\mathbf{ Cat}_{ \infty})$ of $\mathcal{ C}$-indexed $(\infty,1)$-categories under suitable assumptions on $\mathcal{ C}$. We furthermore characterize the objects of $\mathbf{ Cat}_{ \infty}(\mathcal{ C})$ by means of a Yoneda lemma that expresses indexed diagrams of internal shape over $\mathcal{ C}$ in terms of an $(\infty,1)$-categorical totalization. In the second part, we relate the general theory developed to this point to results in the model categorical literature. We show that every model category $\mathbb{ M}$ gives rise to a ''hands-on'' $\infty$-cosmos $\mathbf{ Cat}_{ \infty}(\mathbb{ M})$ (of not-necessarily cofibrant objects) directly by restriction of the Reedy model structure on $\mathbb{ M}^{ \Delta^{ op}}$. We then define an according right derived model categorical externalization functor, and use it to show that the $(\infty,1)$-categorical and the model categorical constructions correspond to one another whenever $\mathcal{ C}$ is presentable and $\mathbb{ M}$ is a suitable presentation thereof.
Categories: math.CT math.AT