Kerodon

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 4.5.6.18). 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 1.1.4.12 supplies a pushout diagram

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 $\Delta ^1$). $\square$

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

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). $\square$

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

is a categorical equivalence of simplicial sets. By virtue of Corollary 4.5.7.2 (and Remark 1.1.4.4), 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 1.1.4.12 supplies a pushout diagram of simplicial sets

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

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 $f$. $\square$

Proof. We proceed as in the proof of Proposition 4.5.2.14. By definition, the diagram (4.43) 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 4.5.2.6 and 4.5.2.7, we can identify $\rho $ with the map

determined by the commutative diagram (4.44). The desired result now follows from the criterion of Corollary 3.4.1.6. $\square$