Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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 4.5.1.16, the implication $(2) \Rightarrow (3)$ from Remark 4.5.1.19, 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 $[F] \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{QCat}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ induces a bijection

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

so that $[F]$ is an isomorphism in the category $\mathrm{h} \mathit{\operatorname{QCat}}$ 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 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 3.4.1.10).

$(e)$

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