Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

5.4.5 The $(\infty ,2)$-Category of $\infty$-Categories

For some applications, it will be convenient to work with a variant of Construction 5.4.4.1, which retains information about non-invertible natural transformations of functors.

Construction 5.4.5.1 (The $(\infty ,2)$-Category of $\infty$-Categories). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, endowed with the simplicial enrichment of Example 2.4.2.1. We let $\operatorname{\mathbf{QCat}}$ denote the full simplicial subcategory of $\operatorname{Set_{\Delta }}$ spanned by the (small) $\infty$-categories, which we can describe concretely as follows:

• The objects of $\operatorname{\mathbf{QCat}}$ are (small) $\infty$-categories.

• If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty$-categories, then the simplicial set $\operatorname{Hom}_{\operatorname{\mathbf{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }$ is the $\infty$-category of functors $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

We let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}})$. We will refer to $\operatorname{ \pmb {\mathcal{QC}} }$ as the $(\infty ,2)$-category of $\infty$-categories.

Proposition 5.4.5.2. The simplicial set $\operatorname{ \pmb {\mathcal{QC}} }$ is an $(\infty ,2)$-category.

Proof. For every pair of $\infty$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, Theorem 1.4.3.7 guarantees that the simplicial set $\operatorname{Hom}_{\operatorname{\mathbf{QCat}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is an $\infty$-category. The desired result is now a special case of Theorem 5.3.8.1. $\square$

Remark 5.4.5.3. The low-dimensional simplices of $\operatorname{ \pmb {\mathcal{QC}} }$ are simple to describe:

• An object of $\operatorname{ \pmb {\mathcal{QC}} }$ 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 $\operatorname{ \pmb {\mathcal{QC}} }$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

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

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

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 a morphism in the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. Moreover, $\sigma$ is thin if and only if $\mu$ is an isomorphism of functors (Proposition 5.3.8.7).

Remark 5.4.5.4 (Comparison with $\operatorname{\mathcal{QC}}$). Let $\operatorname{QCat}$ and $\operatorname{\mathbf{QCat}}$ be the simplicial categories defined in Constructions 5.4.4.1 and 5.4.5.1, respectively. There is an evident comparison map $\operatorname{QCat}\hookrightarrow \operatorname{\mathbf{QCat}}$ which is the identity at the level of objects, and which is given on morphism spaces by the inclusion maps

$\operatorname{Hom}_{\operatorname{QCat}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } \hookrightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \operatorname{Hom}_{\operatorname{\mathbf{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}).$

Passing to the homotopy coherent nerve, we obtain a functor of $(\infty ,2)$-categories $\operatorname{\mathcal{QC}}\hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }$ which restricts to an isomorphism of $\infty$-categories $\operatorname{\mathcal{QC}}\simeq \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} })$ (Corollary 5.3.8.8).

Remark 5.4.5.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories. Then Theorem 4.6.7.5 supplies an equivalence of $\infty$-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{ \pmb {\mathcal{QC}} }}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Beware that this equivalence is generally not an isomorphism at the level of simplicial sets.

Remark 5.4.5.6 (Comparison with Kan Complexes). Since every Kan complex is an $\infty$-category (Example 1.3.0.3), we can identify the simplicial category $\operatorname{Kan}$ of Construction 5.4.1.1 with a full simplicial subcategory of $\operatorname{\mathbf{QCat}}$. Passing to homotopy coherent nerves, we can identify $\infty$-category of spaces $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$ with the full subcategory of $\operatorname{ \pmb {\mathcal{QC}} }= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}})$ spanned by the Kan complexes.

Remark 5.4.5.7 (Comparison with Categories). Let $\mathbf{Cat}$ denote the strict $2$-category of small categories (Example 2.2.0.4). By virtue of Proposition 1.4.3.3, the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ induces an isomorphism from the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{Cat} )$ to the full subcategory of $\operatorname{ \pmb {\mathcal{QC}} }$ spanned by those $\infty$-categories of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is an ordinary category.

Remark 5.4.5.8 (Passage to the Homotopy Category). Let $\operatorname{Cat}_{\bullet }$ denote the simplicial category associated to the strict $2$-category $\mathbf{Cat}$ (see Example 2.4.2.7). For every pair of $\infty$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, Corollary 1.4.3.5 supplies a comparison map

$\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{D}}} ) ) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{\mathcal{D}}} ) ).$

This construction is compatible with composition, and therefore determines a functor of simplicial categories

$\operatorname{\mathbf{QCat}}\rightarrow \operatorname{Cat}_{\bullet } \quad \quad \operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}.$

Passing to homotopy coherent nerves (and invoking Example 2.4.3.11), we obtain a functor of $(\infty ,2)$-categories

$\operatorname{ \pmb {\mathcal{QC}} }= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathbf{QCat}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Cat}_{\bullet } ) \simeq \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{Cat} ).$

Stated more informally, the construction $\operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ determines a functor from the $(\infty ,2)$-category $\operatorname{ \pmb {\mathcal{QC}} }$ to the ordinary $2$-category $\mathbf{Cat}$.