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4.5.4 Detecting Equivalences of $\infty $-Categories

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 $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is a homotopy equivalence of Kan complexes (Remark 4.5.1.18). The converse assertion is not true in general. For example, the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism on cores, but is never an equivalence of $\infty $-categories unless $\operatorname{\mathcal{C}}$ is a Kan complex. However, we have the following result:

Theorem 4.5.4.1. 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 $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$.

$(4)$

Composition with $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq }$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 4.5.1.15, the implication $(2) \Rightarrow (3)$ from Remark 4.5.1.18, and the implication $(3) \Rightarrow (4)$ is immediate. If $(3)$ is satisfied, then for every $\infty $-category $\operatorname{\mathcal{E}}$, composition with the isomorphism class $[F] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ induces a bijection

\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}}) = \pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})^{\simeq } ) \simeq \pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}})^{\simeq } ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}), \]

so that $[F]$ is an isomorphism in the category $\mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ and $(1)$ is satisfied. We will complete the proof by showing that $(4)$ implies $(3)$.

For every simplicial set $X$, let $\theta _{X}: \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{D}})^{\simeq }$ denote the map given by postcomposition with the functor $F$. Let us say that $X$ is good if the morphism $\theta _{X}$ is a homotopy equivalence. We now make the following observations:

$(a)$

Let $X$ be the colimit of a diagram of monomorphisms

\[ X(0) \hookrightarrow X(1) \hookrightarrow X(2) \hookrightarrow \cdots \]

We then obtain a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=48pt{ \operatorname{Fun}( X(0), \operatorname{\mathcal{C}})^{\simeq } \ar [d]^{ \theta _{X(0)}} & \operatorname{Fun}(X(1),\operatorname{\mathcal{C}})^{\simeq } \ar [l] \ar [d]^{ \theta _{X(1)}} & \operatorname{Fun}( X(2), \operatorname{\mathcal{C}})^{\simeq } \ar [l] \ar [d]^{ \theta _{X(2)}} & \cdots \ar [l] \\ \operatorname{Fun}( X(0), \operatorname{\mathcal{D}})^{\simeq } & \operatorname{Fun}(X(1),\operatorname{\mathcal{D}})^{\simeq } \ar [l] & \operatorname{Fun}( X(2), \operatorname{\mathcal{D}})^{\simeq } \ar [l] & \cdots , \ar [l] } \]

where the horizontal maps are Kan fibrations (Corollary 4.4.5.4). Moreover, the induced map of inverse limits can be identified with the map $\theta _{X}: \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$ (Corollary 4.4.4.6). If each $X(n)$ is good, then the vertical maps appearing in the diagram are homotopy equivalences, so that $\theta _{X}$ is also a homotopy equivalence (Proposition 3.3.9.6). It follows that $X$ is also good.

$(b)$

Let $X$ be a simplicial set which is given as a coproduct $\coprod _{\alpha } X(\alpha )$ of a collection of simplicial sets $X(\alpha )$. Then $\theta _{X}$ can be identified with the product of the maps $\theta _{X(\alpha ) }$ (Corollary 4.4.4.6). Consequently, if each of the summands $X(\alpha )$ is good, then $X$ is also good (Remark 3.1.5.8).

$(c)$

Let $u: X \rightarrow Y$ be an inner anodyne morphism of simplicial sets. Then we have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(Y, \operatorname{\mathcal{C}})^{\simeq } \ar [d] \\ \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq } \ar [r] & \operatorname{Fun}(Y, \operatorname{\mathcal{D}})^{\simeq }, } \]

where the horizontal maps are homotopy equivalences (Proposition 4.5.2.8). It follows that $X$ is good if and only if $Y$ is good.

$(d)$

Suppose are given a categorical pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d] & X' \ar [d] \\ Y \ar [r] & Y'. } \]

If $X$, $X'$, and $Y$ are good, then $Y'$ is also good (see Corollary 3.4.1.10).

$(e)$

Let $X$ be a retract of a simplicial set $Y$. If $Y$ is good, then $X$ is also good.

Now suppose that condition $(4)$ is satisfied, so that the simplex $\Delta ^1$ is good. We wish to show that every simplicial set $X$ is good. Writing $X$ as the direct limit of its skeleta $\{ \operatorname{sk}_{n}(X) \} _{n \geq 0}$ and using $(a)$, we can reduce to the case where $X$ is has dimension $\leq n$ for some integer $n$. We proceed by induction on $n$. The case $n=-1$ is trivial (in this case, the simplicial set $X$ is empty and the morphism $\theta _{X}$ is an isomorphism). We may therefore assume that $n \geq 0$. Let $S$ be the collection of nondegenerate $n$-simplices of $X$, so that Proposition 1.1.3.13 supplies a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in S}{\coprod } \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \underset { \sigma \in S }{\coprod } \Delta ^{n} \ar [d] \\ \operatorname{sk}_{n-1}( X ) \ar [r] & X. } \]

Since the horizontal maps in this diagram are monomorphisms, it is also a categorical pushout diagram (Example 4.5.3.8). Moreover, our inductive hypothesis guarantees that the simplicial sets $\operatorname{sk}_{n-1}(X)$ and $\underset { \sigma \in S }{\coprod } \operatorname{\partial \Delta }^{n}$ are good. Applying $(d)$, we are reduced to showing that the coproduct $\underset { \sigma \in S }{\coprod } \Delta ^{n}$ is good. Using $(b)$, we are reduced to showing that the standard simplex $\Delta ^{n}$ is good. If $n \geq 2$, then the inner horn inclusion $\Lambda ^ n_{1} \hookrightarrow \Delta ^ n$ is a categorical equivalence, so that the desired result follows from our inductive hypothesis together with $(c)$. We are therefore reduced to showing that the standard simplices $\Delta ^0$ and $\Delta ^1$ are good. In the second case this follows from our assumption $(4)$, and in the first case it follows from $(e)$ (since the $0$-simplex $\Delta ^0$ is a retract of $\Delta ^1$). $\square$

Corollary 4.5.4.2. Let $\operatorname{\mathcal{W}}$ denote the full subcategory of $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$ spanned by those morphisms of simplicial sets $f: X \rightarrow Y$ which are categorical equivalences. Then $\operatorname{\mathcal{W}}$ is closed under the formation of filtered colimits in $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$.

Proof. By virtue of Corollary 4.1.3.3, there exists a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ which commutes with filtered colimits and a natural transformation of functors $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the property that, for every simplicial set $X$, the simplicial set $Q(X)$ is an $\infty $-category and the morphism $u_ X: X \rightarrow Q(X)$ is inner anodyne. For every morphism of simplicial sets $f: X \rightarrow Y$, we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{ u_ X } & Y \ar [d]^{ u_{Y} } \\ Q(X) \ar [r]^-{ Q(f) } & Q(Y) } \]

where the vertical maps are categorical equivalences (Corollary 4.5.2.11). It follows from Remark 4.5.2.6 that $f$ is a categorical equivalence if and only if the functor $Q(f)$ is an equivalence of $\infty $-categories. Using the criterion of Theorem 4.5.4.1, we see that $f$ is a categorical equivalence if and only if the induced map $\operatorname{Fun}( \Delta ^1, Q(X) )^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, Q(Y) )^{\simeq }$ is a homotopy equivalence of Kan complexes. The desired result now follows by observing that the construction $X \mapsto \operatorname{Fun}( \Delta ^1, Q(X) )^{\simeq }$ commutes with filtered colimits, since the collection of homotopy equivalences between Kan complexes is closed under filtered colimits (Proposition 3.2.7.3). $\square$

Corollary 4.5.4.3. Suppose we are given a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \operatorname{\mathcal{D}}' \ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]

where $q$ is an isofibration. If $F$ is an equivalence of $\infty $-categories, then $F'$ is also an equivalence of $\infty $-categories.

Proof. Let $X$ be an arbitrary simplicial set. Then we have a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}') \ar [r] \ar [d]^{F'_{X}} & \operatorname{Fun}(X,\operatorname{\mathcal{C}}) \ar [d]^{q_ X} \\ \operatorname{Fun}(X,\operatorname{\mathcal{D}}') \ar [r]^-{F_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{D}}), } \]

where $q_{X}$ is also an isofibration (Corollary 4.4.5.5). Invoking Corollary 4.4.3.14, we conclude that the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}')^{\simeq } \ar [r] \ar [d]^{F'^{\simeq }_{X}} & \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } \ar [d]^{q^{\simeq }_ X} \\ \operatorname{Fun}(X,\operatorname{\mathcal{D}}')^{\simeq } \ar [r]^-{F^{\simeq }_ X} & \operatorname{Fun}(X,\operatorname{\mathcal{D}})^{\simeq }, } \]

is a homotopy pullback square. Since $F$ is an equivalence of $\infty $-categories, the morphism $F^{\simeq }_{X}$ is a homotopy equivalence of Kan complexes (Theorem 4.5.4.1). Applying Corollary 3.4.1.3, we conclude that $F'^{\simeq }_{X}$ is also a homotopy equivalence of Kan complexes. Allowing $X$ to vary, we conclude that $F'$ is an equivalence of $\infty $-categories. $\square$

Corollary 4.5.4.4. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r]^-{q'} \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \operatorname{\mathcal{D}}' \ar [r]^-{F} & \operatorname{\mathcal{D}}, } \]

where $q'$ and $q$ are isofibrations and the functors $F$ and $F'$ are equivalences of $\infty $-categories. Then, for every object $D' \in \operatorname{\mathcal{D}}$ having image $D = F(D') \in \operatorname{\mathcal{D}}$, the induced functor

\[ \operatorname{\mathcal{C}}'_{D'} = \{ D'\} \times _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{C}}' \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_{D} \]

is an equivalence of $\infty $-categories.

Proof. For every simplicial set $X$, we have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}')^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(X,\operatorname{\mathcal{C}})^{\simeq } \ar [d] \\ \operatorname{Fun}(X,\operatorname{\mathcal{D}}')^{\simeq } \ar [r] & \operatorname{Fun}(X,\operatorname{\mathcal{D}})^{\simeq }. } \]

Our assumption that $F$ and $F'$ are equivalences of $\infty $-categories guarantee that the horizontal maps are homotopy equivalences (Theorem 4.5.4.1), and our assumption that $q$ and $q'$ are isofibrations guarantee that the vertical maps are Kan fibrations (Proposition 4.4.3.6). Let $\underline{D} \in \operatorname{Fun}(X,\operatorname{\mathcal{D}})$ and $\underline{D}' \in \operatorname{Fun}(X, \operatorname{\mathcal{D}}')$ be the constant diagrams taking the values $D$ and $D'$, respectively, so that the induced map of fibers

\[ \theta : \{ \underline{D}' \} \times _{ \operatorname{Fun}(X,\operatorname{\mathcal{D}}')^{\simeq } } \operatorname{Fun}(X, \operatorname{\mathcal{C}}')^{\simeq } \rightarrow \{ \underline{D} \} \times _{ \operatorname{Fun}(X,\operatorname{\mathcal{D}})^{\simeq } } \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \]

is also a homotopy equivalence (Proposition 3.2.7.1). Using Corollary 4.4.3.15 (and Corollary 4.4.5.5), we can identify $\theta $ with the natural map $\operatorname{Fun}(X, \operatorname{\mathcal{C}}'_{D'})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}}_{D})^{\simeq }$. Allowing $X$ to vary and applying Theorem 4.5.4.1, we conclude that the functor $\operatorname{\mathcal{C}}'_{D'} \rightarrow \operatorname{\mathcal{C}}_{D}$ is an equivalence of $\infty $-categories. $\square$

Warning 4.5.4.5. Suppose we are given a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r]^-{f'} \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r]^-{f} & S, } \]

where $q$ and $q'$ are Kan fibrations and $f$ is a homotopy equivalence. By virtue of Proposition 3.2.7.1, the following conditions are equivalent:

$(1)$

The morphism $f'$ is a homotopy equivalence of Kan complexes.

$(2)$

For each vertex $s' \in S'$ having image $s = f(s') \in S$, the induced map of fibers $X'_{s'} \rightarrow X_{s}$ is a homotopy equivalence of Kan complexes.

Corollary 4.5.4.4 can be regarded as a generalization of the implication $(1) \Rightarrow (2)$, where we allow $\infty $-categories in place of Kan complexes and isofibrations in place of Kan fibrations. Beware that the implication $(2) \Rightarrow (1)$ does not generalize. For example, we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d]^{ \operatorname{id}} \\ \Delta ^1 \ar [r]^-{\operatorname{id}} & \Delta ^1, } \]

where the vertical maps are isofibrations, the bottom horizontal map is an isomorphism, and the upper horizontal map restricts to an isomorphism on each fiber, but is nevertheless not an equivalence of $\infty $-categories.