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4.5.1 Equivalences of $\infty $-Categories

The collection of $\infty $-categories can be organized into a category, in which the morphisms are given by isomorphism classes of functors.

Construction 4.5.1.1 (The Homotopy Category of $\infty $-Categories). We define a category $\mathrm{h} \mathit{\operatorname{QCat}}$ as follows:

  • The objects of $\mathrm{h} \mathit{\operatorname{QCat}}$ are $\infty $-categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$ is the set of isomorphism classes of objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (or, equivalently, of the homotopy category $\mathrm{h} \mathit{\operatorname{Fun}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). If $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor, we denote its isomorphism class by $[F] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

  • If $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ are $\infty $-categories, then the composition law

    \[ \circ : \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{D}}_{}, \operatorname{\mathcal{E}}_{} ) \times \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}_{}, \operatorname{\mathcal{D}}_{} ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}_{}, \operatorname{\mathcal{E}}_{} ) \]

    is characterized by the formula $[G] \circ [F] = [G \circ F]$.

We will refer to $\mathrm{h} \mathit{\operatorname{QCat}}$ as the homotopy category of $\infty $-categories.

Remark 4.5.1.2. We will later study a refinement of Construction 4.5.1.1. The collection of (small) $\infty $-categories can itself be organized into a (large) $\infty $-category $\operatorname{\mathcal{QC}}$, whose homotopy category can be identified with the ordinary category $\mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 4.5.1.1. See Construction 5.6.4.1.

Remark 4.5.1.3. Let $\mathbf{Cat}$ denote the (strict) $2$-category of categories (Example 2.2.0.4) and let $\mathrm{h} \mathit{\operatorname{Cat}}$ denote its homotopy category (Construction 2.2.8.12). Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful functor from $\mathrm{h} \mathit{\operatorname{Cat}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 4.5.1.1. This functor admits a left adjoint, which carries an $\infty $-category $\operatorname{\mathcal{C}}$ to its homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Remark 4.5.1.4. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.5.10). Then we can regard $\mathrm{h} \mathit{\operatorname{Kan}}$ as a full subcategory of the $\infty $-category $\mathrm{h} \mathit{\operatorname{QCat}}$ (Construction 4.5.1.1), spanned by those $\infty $-categories which are Kan complexes. This follows from the observation that if $Y$ is a Kan complex, then a pair of morphisms $f,g: X \rightarrow Y$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(X,Y)$ if and only if they are homotopic (Proposition 3.1.5.4).

The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ has both left and right adjoints.

Proposition 4.5.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}^{\simeq }$ denote its core (Construction 4.4.3.1). For every Kan complex $X$, composition with the inclusion map $\iota : \operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( X, \operatorname{\mathcal{C}}^{\simeq } ) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}} }( X, \operatorname{\mathcal{C}}). \]

Proof. By virtue of Proposition 4.4.3.20, postcomposition with $\iota $ induces an isomorphism of Kan complexes $\operatorname{Fun}(X, \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq }$. Proposition 4.5.1.5 follows by passing to connected components. $\square$

Corollary 4.5.1.6. The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Remark 4.5.1.4 admits a right adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$.

Remark 4.5.1.7. The right adjoint $\mathrm{h} \mathit{\operatorname{QCat}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ of Corollary 4.5.1.6 can be described more explicitly as follows:

  • To each $\infty $-category $\operatorname{\mathcal{C}}$, it associates the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$ of Construction 4.4.3.1.

  • To each morphism $[F]: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ (given by the isomorphism class of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$), it associates the homotopy class $[F^{\simeq }]$ of the underlying map of cores $F^{\simeq } = F|_{ \operatorname{\mathcal{C}}^{\simeq } }$ (note that the homotopy class of $F^{\simeq }$ depends only on the isomorphism class of $F$, by virtue of Remark 4.4.4.5).

Proposition 4.5.1.8. The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ of Remark 4.5.1.4 admits a left adjoint.

Proof. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We wish to show that there exists a Kan complex $X$ and a morphism $u: \operatorname{\mathcal{C}}\rightarrow X$ with the following property: for every Kan complex $Y$, precomposition with $u$ induces a bijection

\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( X, Y) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{QCat}}}( X, Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, Y ). \]

Unwinding the definitions, we see that this is a reformulation of the requirement that $u$ is a weak homotopy equivalence of simplicial sets. The existence of $u$ now follows from Corollary 3.1.7.2. $\square$

Remark 4.5.1.9. The left adjoint $\mathrm{h} \mathit{\operatorname{QCat}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ of Proposition 4.5.1.8 admits a category-theoretic interpretation: it carries an $\infty $-category $\operatorname{\mathcal{C}}$ to the localization $\operatorname{\mathcal{C}}[W^{-1}]$ obtained by formally inverting the collection $W$ of all morphisms in $\operatorname{\mathcal{C}}$ (see Proposition 6.3.1.20).

Definition 4.5.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is homotopy inverse to $F$ if the isomorphism class $[G]$ is an inverse to $[F]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$: that is, if $G \circ F$ and $F \circ G$ are isomorphic to the identity functors $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as objects of the $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively. We will say that $F$ is an equivalence of $\infty $-categories if $[F]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$: that is, if $F$ admits a homotopy inverse $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. We say that $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent if there exists an equivalence from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

Example 4.5.1.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isomorphism of simplicial sets. Then $F$ is an equivalence of $\infty $-categories. In particular, for every $\infty $-category $\operatorname{\mathcal{C}}$, the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ is an equivalence of $\infty $-categories.

Example 4.5.1.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then the induced map $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories if and only if $F$ is an equivalence of categories.

Example 4.5.1.13. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Then $f$ is a homotopy equivalence if and only if it is an equivalence of $\infty $-categories (see Remark 4.5.1.4). In this case, a morphism $g: Y \rightarrow X$ is a homotopy inverse to $f$ in the sense of Definition 4.5.1.10 if and only if it is a homotopy inverse to $f$, in the sense of Definition 3.1.6.1.

Warning 4.5.1.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. If $F$ is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10), then it is a homotopy equivalence of simplicial sets (in the sense of Definition 3.1.6.1). More precisely, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a homotopy inverse to the functor $F$ (in the sense of Definition 4.5.1.10), then $G$ is also a simplicial homotopy inverse to $F$ (in the sense of Definition 3.1.6.1). Beware that the converse assertion is false in general. For example, the projection map $\Delta ^1 \rightarrow \Delta ^0$ is a homotopy equivalence of simplicial sets (with homotopy inverse given by the inclusion $\Delta ^0 \simeq \{ 0\} \hookrightarrow \Delta ^1$), but not an equivalence of $\infty $-categories.

Remark 4.5.1.15. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $F,G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors which are isomorphic when regarded as objects of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Then $F$ is an equivalence of $\infty $-categories if and only if $G$ is an equivalence of $\infty $-categories.

Remark 4.5.1.16. Let $X$ be an arbitrary simplicial set. Then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ determines a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ to itself. In particular, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories, then the induced map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})$ is also an equivalence of $\infty $-categories.

Remark 4.5.1.17. Let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_{i} \} _{i \in I}$ be a collection of functors between $\infty $-categories indexed by a set $I$. If each $F_{i}$ is an equivalence of $\infty $-categories, then the product functor $\prod _{i \in I} \operatorname{\mathcal{C}}_{i} \rightarrow \prod _{i \in I} \operatorname{\mathcal{D}}_ i$ is also an equivalence of $\infty $-categories.

Remark 4.5.1.18 (Two-out-of-Three). Let $F: \operatorname{\mathcal{C}}_{} \rightarrow \operatorname{\mathcal{D}}_{}$ and $G: \operatorname{\mathcal{D}}_{} \rightarrow \operatorname{\mathcal{E}}_{}$ be functors between $\infty $-categories. If any two of the functors $F$, $G$, and $G \circ F$ is an equivalence of $\infty $-categories, then so is the third. In particular, the collection of equivalences is closed under composition.

Remark 4.5.1.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. If $F$ is an equivalence of $\infty $-categories, then the induced map of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence of Kan complexes. This follows from Corollary 4.5.1.6 (and Remark 4.5.1.7): if the isomorphism class $[F]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$, then the homotopy class $[ F^{\simeq } ]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Remark 4.5.1.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then the induced functor $\mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is an equivalence of ordinary categories. In particular, a morphism $u$ in the $\infty $-category $\operatorname{\mathcal{C}}$ is an isomorphism if and only if $F(u)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.

Remark 4.5.1.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. If $\operatorname{\mathcal{D}}$ is a Kan complex, then $\operatorname{\mathcal{C}}$ is a Kan complex. To prove this, it suffices to show that every morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism (Proposition 4.4.2.1). By virtue of Remark 4.5.1.20, this is equivalent to the assertion that $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in $\operatorname{\mathcal{D}}$, which is automatic when $\operatorname{\mathcal{D}}$ is a Kan complex (Proposition 1.3.6.10). Similarly, if $\operatorname{\mathcal{C}}$ is a Kan complex, then $\operatorname{\mathcal{D}}$ is a Kan complex (this follows by applying the same argument to an inverse equivalence $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$).

Proposition 4.5.1.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The functor $F$ is an equivalence of $\infty $-categories.

$(2)$

For every simplicial set $X$, composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}(X, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})$.

$(3)$

For every simplicial set $X$, composition with $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$.

$(4)$

For every $\infty $-category $\operatorname{\mathcal{B}}$, composition with $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq }$.

$(5)$

For every $\infty $-category $\operatorname{\mathcal{B}}$, composition with $F$ induces a bijection of sets $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq } )$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 4.5.1.16, the implication $(2) \Rightarrow (3)$ from Remark 4.5.1.19, the implication $(3) \Rightarrow (4)$ is immediate, and the implication $(4) \Rightarrow (5)$ follows from Remark 3.1.6.5, and the implication $(5) \Rightarrow (1)$ follows from Yoneda's lemma (applied to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$). $\square$

We close this section by introducing a refinement of Construction 4.5.1.1:

Construction 4.5.1.23 (The Homotopy $2$-Category of $\infty $-Categories). We define a strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are $\infty $-categories.

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then $\underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}} }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})}$ is the homotopy category 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 on $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ is given by

    \begin{eqnarray*} \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) & = & (\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{D}},\operatorname{\mathcal{E}})}) \times (\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}) \\ & \simeq & \mathrm{h} \mathit{(\operatorname{Fun}(\operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \times \operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}}))} \\ & \xrightarrow {\circ } & \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})} \\ & = & \underline{\operatorname{Hom}}_{\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \end{eqnarray*}

We will refer to $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ as the homotopy $2$-category of $\infty $-categories. We let $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ denote the pith $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$, in the sense of Construction 2.2.8.9; we will refer to $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ as the homotopy $(2,1)$-category of $\infty $-categories.

Remark 4.5.1.24. We can describe the strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ more informally as follows:

  • The objects of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are $\infty $-categories.

  • The morphisms of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ are functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

  • If $F_0, F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ are functors between $\infty $-categories, then a $2$-morphism $F_0 \Rightarrow F_1$ in $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ is a homotopy class of natural transformations from $F_0$ to $F_1$.

The strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{QCat}}$ can be described in a similar way, except that its $2$-morphisms are homotopy classes of natural isomorphisms (rather than general natural transformations).

Remark 4.5.1.25. The homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$ of Construction 4.5.1.1 can be identified with the homotopy category of the $2$-category $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ (in the sense of Construction 2.2.8.12); see Remark 2.4.6.18).

Remark 4.5.1.26. Let $\mathbf{Cat}$ denote the (strict) $2$-category of categories (see Example 2.2.0.4). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ defines an isomorphism from $\mathbf{Cat}$ to the full subcategory of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$ spanned by those objects of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a (small) category.

Remark 4.5.1.27. Let $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ denote the homotopy $2$-category of Kan complexes (Construction 3.1.5.13). Then $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ can be identified with the full subcategory of $\mathrm{h}_{2} \mathit{\operatorname{ \pmb {\mathcal{QC}} }}$ spanned by the Kan complexes. Since $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is a $(2,1)$-category, this subcategory is contained in the pith $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{QC}}} = \operatorname{Pith}( \mathrm{h}_{2} \mathit{\operatorname{ \pmb {\mathcal{QC}} }} )$; we can therefore also view $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ as a full subcategory of $\mathrm{h}_{2} \mathit{\operatorname{\mathcal{QC}}}$.