Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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.

Remark 5.5.4.2. Many authors use the term quasicategory for what we refer to as an $\infty $-category (see Remark 1.3.0.2); the notations of Construction 5.5.4.1 reflect this alternative terminology.

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 4.4.3.9). 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. $\square$

Remark 5.5.4.4. The low-dimensional simplices of $\operatorname{\mathcal{QC}}$ are simple to describe:

  • An object of $\operatorname{\mathcal{QC}}$ is a (small) $\infty $-category $\operatorname{\mathcal{C}}$.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are objects of $\operatorname{\mathcal{QC}}$, then a morphism from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in $\operatorname{\mathcal{QC}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

  • A $2$-simplex of $\operatorname{\mathcal{QC}}$ can be identified with a diagram

    \[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{D}}\ar [dr]^{G} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{\mu }_{\sim } & \\ \operatorname{\mathcal{C}}\ar [ur]^{F} \ar [rr]_{H} & & \operatorname{\mathcal{E}}} \]

    where $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are (small) $\infty $-categories, $F$, $G$, and $H$ are functors, and $\mu : G \circ F \rightarrow H$ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.

Remark 5.5.4.5. Let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1). Then there is a tautological comparison map $\operatorname{\mathcal{QC}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{QCat}} )$, which carries each $\infty $-category $\operatorname{\mathcal{C}}$ to itself and each functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to its isomorphism class $[F] \in \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$. This functor induces an isomorphism of homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \simeq \mathrm{h} \mathit{\operatorname{QCat}}$ (see Proposition 2.4.6.8).

Remark 5.5.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10) if and only if it is an isomorphism in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Remark 5.5.4.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then Remark 4.6.7.6 supplies a homotopy equivalence of Kan complexes $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{QC}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Beware that this homotopy equivalence is generally not an isomorphism.

Remark 5.5.4.8 (Comparison with Kan Complexes). Every Kan complex is an $\infty $-category (Example 1.3.0.3). Moreover, if $X$ and $Y$ are Kan complexes, then the simplicial set $\operatorname{Fun}(X,Y)$ is also a Kan complex (Corollary 3.1.3.4), and therefore coincides with its core $\operatorname{Fun}(X,Y)^{\simeq }$. It follows that we can regard the simplicial category $\operatorname{Kan}$ of Construction 5.5.1.1 as a full simplicial subcategory of $\operatorname{QCat}$. Passing to homotopy coherent nerves, we deduce that the $\infty $-category $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ is the full subcategory of $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ spanned by the Kan complexes.

Remark 5.5.4.9 (Comparison with Categories). Let $\mathbf{Cat}$ denote the strict $2$-category of small categories (Example 2.2.0.4), let $\operatorname{Pith}( \mathbf{Cat} )$ denote its pith (Construction 2.3.2.10), and let us abuse notation by identifying $\operatorname{Pith}( \mathbf{Cat} )$ with the simplicial category described in Example 2.4.2.7. Concretely, this simplicial category can be described as follows:

  • The objects of $\operatorname{Pith}( \mathbf{Cat} )$ are small categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are objects of $\operatorname{Pith}( \mathbf{Cat} )$, then the simplicial set $\operatorname{Hom}_{\operatorname{Pith}( \mathbf{Cat} )}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the nerve of the groupoid $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ whose objects are functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are natural isomorphisms.

By virtue of Proposition 1.4.3.3, 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.