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.
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 4.4.3.22, 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 $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$.
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.
$\square$
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.
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 of $\mathrm{h}_{2} \mathit{\operatorname{\mathbf{QCat}}}$, 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.
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).