The $(\infty ,2)$-category of internal $(\infty ,1)$-categories

Abstract

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 ${\mathbf{Cat}}_{\infty }(\mathbb{M})$ directly by restriction of the Reedy model structure on ${\mathbb{M}}^{\hat{I}{”}^{op}}$. We then define a corresponding 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 $\mathbb{M}$ is a suitable model category.

Final version, to appear in Higher Structures