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

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

Construction (The $(\infty ,2)$-Category of $\infty $-Categories). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets, endowed with the simplicial enrichment of Example 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 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 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 $\square$

Remark 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

Remark (Comparison with $\operatorname{\mathcal{QC}}$). Let $\operatorname{QCat}$ and $\operatorname{\mathbf{QCat}}$ be the simplicial categories defined in Constructions and, 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

Remark Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then Theorem 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 (Comparison with Kan Complexes). Since every Kan complex is an $\infty $-category (Example, we can identify the simplicial category $\operatorname{Kan}$ of Construction 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 (Comparison with Categories). Let $\mathbf{Cat}$ denote the strict $2$-category of small categories (Example By virtue of Proposition, 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 (Passage to the Homotopy Category). Let $\operatorname{Cat}_{\bullet }$ denote the simplicial category associated to the strict $2$-category $\mathbf{Cat}$ (see Example For every pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, Corollary 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, 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 $\mathbf{Cat}$.

Variant Let $\kappa $ be an uncountable cardinal. We let $\operatorname{ \pmb {\mathcal{QC}} }^{< \kappa }$ denote the full simplicial subset of $\operatorname{ \pmb {\mathcal{QC}} }$ spanned by those $\infty $-categories $\operatorname{\mathcal{C}}$ which are $\kappa $-small. Then $\operatorname{ \pmb {\mathcal{QC}} }^{< \kappa }$ is an $(\infty ,2)$-category, which we will refer to as the $(\infty ,2)$-category of essentially $\kappa $-small $\infty $-categories.