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4.5.7 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 4.5.1.19). 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 slightly weaker result:

Theorem 4.5.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is an equivalence of $\infty $-categories if and only if the induced map of Kan complexes $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})^{\simeq }$ is a homotopy equivalence.

Proof. 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. By virtue of Proposition 4.5.1.22, the functor $F$ is an equivalence of $\infty $-categories if and only if every simplicial set $X$ is good. In particular, if $F$ is an equivalence of $\infty $-categories, then $\Delta ^1$ is good. To prove the converse, we 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 (Example 4.5.6.16). 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 3.1.6.8).

$(c)$

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 4.5.3.8). It follows that $X$ is good if and only if $Y$ is good.

$(d)$

Suppose we 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 3.4.1.12).

$(e)$

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

Now suppose that the simplicial set $\Delta ^1$ is good. We will show that every simplicial set $X$ is good, so that $F$ is an equivalence of $\infty $-categories by virtue of Proposition 4.5.1.22. 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 $X$ 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 square (Example 4.5.4.12). 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 4.5.7.2. 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 4.5.3.14). It follows from Remark 4.5.3.5 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 4.5.7.1, 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 3.2.8.3). $\square$

Corollary 4.5.7.3. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr] & & Y \ar [dl] \\ & S & } \]

with the following property: for every simplex $\sigma : \Delta ^ k \rightarrow S$, the induced map $f_{\sigma }: \Delta ^ k \times _{S} X \rightarrow \Delta ^ k \times _{S} Y$ is a categorical equivalence of simplicial sets. Then $f$ is a categorical equivalence of simplicial sets.

Proof. We will prove the following stronger assertion: for every morphism of simplicial sets $S' \rightarrow S$, the induced map

\[ f_{S'}: S' \times _{S} X \rightarrow S' \times _{S} Y \]

is a categorical equivalence of simplicial sets. By virtue of Corollary 4.5.7.2 (and Remark 1.1.3.6), we may assume without loss of generality that $S'$ has dimension $\leq k$ for some integer $k \geq -1$. We proceed by induction on $k$. In the case $k=-1$, the simplicial set $S'$ is empty and there is nothing to prove. Assume therefore that $k \geq 0$. Let $S''$ denote the $(k-1)$-skeleton of $S'$ and let $I$ be the set of nondegenerate $d$-simplices of $S'$, so that Proposition 1.1.3.13 supplies a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { i \in I }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { i \in I }{\coprod } \Delta ^{k} \ar [d] \\ S'' \ar [r] & S', } \]

where the horizontal maps are monomorphisms. It follows that the front and back faces of the diagram

\[ \xymatrix@R =50pt@C=40pt{ (\underset { i \in I }{\coprod } \operatorname{\partial \Delta }^{k}) \times _{S} X \ar [rr] \ar [dd] \ar [dr]^{u} & & \underset { i \in I }{\coprod } (\Delta ^{k} \times _{S} X) \ar [dd] \ar [dr]^{v} & \\ & \underset { i \in I }{\coprod } (\operatorname{\partial \Delta }^{k} \times _{S} Y) \ar [dd] \ar [rr] & & \underset { i \in I }{\coprod } (\Delta ^{k} \times _{S} Y) \ar [dd] \\ S'' \times _{S} X \ar [rr] \ar [dr]^{f_{S''}} & & S' \times _{S} X \ar [dr]^{ f_{S'} } & \\ & S'' \times _{S} Y \ar [rr] & & S' \times _{S} Y } \]

are categorical pushout squares (Proposition 4.5.4.11). Consequently, to show that $f_{S'}$ is a categorical equivalence, it will suffice to show that $f_{S''}$, $u$, and $v$ are categorical equivalences (Proposition 4.5.4.9). In the first two cases, this follows from our inductive hypothesis. We may therefore replace $S'$ by the coproduct $\coprod _{i \in I} \Delta ^{k}$, and thereby reduce to the case of a coproduct of simplices. Using Corollary 4.5.3.10, we can further reduce to the case where $S' \simeq \Delta ^{k}$ is a standard simplex, in which case the desired result follows from our hypothesis on $f$. $\square$

Corollary 4.5.7.4. A commutative diagram of $\infty $-categories

4.42
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square29} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{U} \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}

is a categorical pullback square if and only if the induced diagram of Kan complexes

4.43
\begin{equation} \begin{gathered}\label{equation:bellie} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{01} )^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{0} )^{\simeq } \ar [d] \\ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{1} )^{\simeq } \ar [r] & \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})^{\simeq } } \end{gathered} \end{equation}

is a homotopy pullback square.

Proof. We proceed as in the proof of Proposition 4.5.2.12. By definition, the diagram (4.42) is a categorical pullback square if and only if the induced map $\theta : \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is an equivalence of $\infty $-categories. Using the criterion of Theorem 4.5.7.1, we see that this is equivalent to the requirement that $\theta $ induces a homotopy equivalence of Kan complexes $\rho : \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1)^{\simeq }$. Using Remarks 4.5.2.5 and 4.5.2.5, we can identify $\rho $ with the map

\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}_{01} )^{\simeq } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_0)^{\simeq } \times ^{\mathrm{h}}_{\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})^{\simeq } } \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}_1)^{\simeq } \]

determined by the commutative diagram (4.43). The desired result now follows from the criterion of Corollary 3.4.1.6. $\square$