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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 $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ as follows:

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

  • If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $\infty $-categories, then $\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) = \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )$ 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 $[F] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Cat}}_{\infty }}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

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

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

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

We will refer to $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ 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 $\operatorname{Cat}_{\infty }$, whose homotopy category can be identified with the ordinary category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Construction 4.5.1.1. We refer the reader to Chapter for more details.

Remark 4.5.1.3. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.4.10). Then we can regard $\mathrm{h} \mathit{\operatorname{Kan}}$ as a full subcategory of the $\infty $-category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ (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 3.1.4.4).

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

Proposition 4.5.1.4. 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

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

Proof. By virtue of Proposition 4.4.3.16, 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 4.5.1.4 follows by passing to connected components. $\square$

Corollary 4.5.1.5. The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Remark 4.5.1.3 admits a right adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$.

Remark 4.5.1.6. The right adjoint $\mathrm{h} \mathit{\operatorname{Cat}_{\infty } } \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ of Corollary 4.5.1.5 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 $\mathrm{h} \mathit{\operatorname{Cat}_{\infty } }$ (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 4.5.1.7. The inclusion functor $\mathrm{h} \mathit{\operatorname{Kan}} \hookrightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Remark 4.5.1.3 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

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

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 3.1.6.2. $\square$

Remark 4.5.1.8. The left adjoint $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ of Proposition 4.5.1.7 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 5.3.1.18).

Definition 4.5.1.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We will say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories if the homotopy class $[F]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ of Construction 4.5.1.1.

Remark 4.5.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if there exists a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ such that $G \circ F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$) and $F \circ G$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}$ (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$). If these conditions are satisfied, then the functor $G$ is also an equivalence of $\infty $-categories, and its isomorphism class (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$) depends only on the isomorphism class of $F$ (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). We will emphasize this dependence by referring to $G$ as an inverse to the functor $F$.

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 4.5.1.3). In this case, a morphism $g: Y \rightarrow X$ is an inverse to $f$ (in the sense of Remark 4.5.1.10) if and only if it is a homotopy inverse to $f$ (in the sense of Definition 3.1.5.1).

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 4.5.1.9), then it is a homotopy equivalence of simplicial sets (in the sense of Definition 3.1.5.1). More precisely, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is an inverse to the functor $F$ (in the sense of Remark 4.5.1.10), then $G$ is also a homotopy inverse to $F$ (in the sense of Definition 3.1.5.1). 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 $X$ 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 $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ to itself. In particular, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be 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 4.5.1.16 (Two-out-of-Six). Let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$, $G: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$, and $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors between $\infty $-categories. If $G \circ F$ and $H \circ G$ are equivalences of $\infty $-categories, then $F$, $G$, and $H$ are equivalences of $\infty $-categories.

Remark 4.5.1.17 (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 4.5.1.18. 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 4.5.1.5 (and Remark 4.5.1.6): if the isomorphism class $[F]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$, then the homotopy class $[ F^{\simeq } ]$ is an invertible morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

Remark 4.5.1.19. 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, the functor $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$.

Remark 4.5.1.20. 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 4.5.1.19, 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 1.3.6.11). 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 $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$).