5.5.4 The $\infty $-Category of $\infty $-Categories
Let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of (small) $\infty $-categories (Construction 4.5.1.1). Recall that the objects of $\mathrm{h} \mathit{\operatorname{QCat}}$ are (small) $\infty $-categories, and a morphism from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in $\mathrm{h} \mathit{\operatorname{QCat}}$ is an isomorphism class of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. In this section, we show that $\mathrm{h} \mathit{\operatorname{QCat}}$ can be realized as the homotopy category of an $\infty $-category $\operatorname{\mathcal{QC}}$, which we will refer to as the $\infty $-category of $\infty $-categories. Proceeding as in ยง5.5.1, we will realize $\operatorname{\mathcal{QC}}$ as the homotopy coherent nerve of a simplicial category.
Construction 5.5.4.1 (The $\infty $-Category of $\infty $-Categories). We define a simplicial category $\operatorname{QCat}$ as follows:
The objects of $\operatorname{QCat}$ are (small) $\infty $-categories.
If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then the simplicial set $\operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the core $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
If $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories, then the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{QCat}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})_{\bullet } \times \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})_{\bullet } \]
is induced by the composition map $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
We let $\operatorname{\mathcal{QC}}$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$. We will refer to $\operatorname{\mathcal{QC}}$ as the $\infty $-category of $\infty $-categories.
Proposition 5.5.4.3. The simplicial set $\operatorname{\mathcal{QC}}$ is an $\infty $-category.
Proof.
For every pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the core $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ is a Kan complex (Corollary 4.4.3.11). It follows that the simplicial category $\operatorname{QCat}$ of Construction 5.5.4.1 is locally Kan, so its homotopy coherent nerve $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ is an $\infty $-category by virtue of Theorem 2.4.5.1.
$\square$
By virtue of Proposition 1.5.3.3, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding of simplicial categories $\operatorname{Pith}(\mathbf{Cat} ) \hookrightarrow \operatorname{QCat}$. Passing to homotopy coherent nerves (and invoking Example 2.4.3.11), we obtain a functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) ) \rightarrow \operatorname{\mathcal{QC}}$. Unwinding the definitions, we see that this functor induces an isomorphism from the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) )$ to the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by those $\infty $-categories of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is an ordinary category.
Variant 5.5.4.10. Let $\kappa $ be an uncountable cardinal. We let $\operatorname{\mathcal{QC}}^{< \kappa }$ denote the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by the $\infty $-categories which are $\kappa $-small. We will refer to $\operatorname{\mathcal{QC}}^{< \kappa }$ as the $\infty $-category of essentially $\kappa $-small $\infty $-categories.
To simplify the exposition, we will often implicitly assume that we are in case $(a)$, as suggested in Variant 5.5.4.10. However, it will be convenient to also allow case $(c)$ when working with $\infty $-categories which are not necessarily small (such as $\operatorname{\mathcal{QC}}$ itself).
Variant 5.5.4.12. Let $\kappa $ be an uncountable cardinal. We let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the full subcategory of $\operatorname{\mathcal{S}}$ spanned by the $\kappa $-small Kan complexes, which we also regard as a full subcategory of $\operatorname{\mathcal{QC}}^{< \kappa }$. Similarly, we let $\operatorname{\mathcal{S}}^{< \kappa }_{\ast }$ denote the full subcategory of $\operatorname{\mathcal{S}}_{\ast }$ spanned by those pointed Kan complexes $(X,x)$ where $X$ is $\kappa $-small.