Kerodon

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3.3.5 Comparison of $X$ with $\operatorname{Ex}(X)$

The goal of this section is to prove the following variant of Proposition 3.3.4.8:

Theorem 3.3.5.1. Let $X$ be a simplicial set. Then the comparison map $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ of Construction 3.3.4.3 is a weak homotopy equivalence.

The proof of Theorem 3.3.5.1 will make use of the following fact, which we prove at the end of this section:

Proposition 3.3.5.2. Let $f: X \rightarrow Y$ be an anodyne morphism of simplicial sets. Then the induced map $\operatorname{Sd}(f): \operatorname{Sd}(X) \rightarrow \operatorname{Sd}(Y)$ is also anodyne.

Corollary 3.3.5.3. Let $f: X \rightarrow Y$ be a Kan fibration of simplicial sets. Then the induced map $\operatorname{Ex}(f): \operatorname{Ex}(X) \rightarrow \operatorname{Ex}(Y)$ is also a Kan fibration.

Proof. We must show that every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ \Lambda ^{n}_{i} \ar [r] \ar [d] & \operatorname{Ex}(X) \ar [d]^{ \operatorname{Ex}(f) } \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur] & \operatorname{Ex}(Y) } \]

admits a solution. This follows by applying Remark 3.1.6.6 to the associated lifting problem

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Sd}(\Lambda ^{n}_{i}) \ar [r] \ar [d] & X \ar [d]^{ f} \\ \operatorname{Sd}(\Delta ^{n}) \ar [r] \ar@ {-->}[ur] & Y, } \]

since the left vertical map is anodyne by virtue of Proposition 3.3.5.2. $\square$

Corollary 3.3.5.4. Let $X$ be a Kan complex. Then the simplicial set $\operatorname{Ex}(X)$ is also a Kan complex.

Proposition 3.3.5.5. Let $X$ and $Y$ be simplicial sets, where $Y$ is a Kan complex. Then the bijection

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}(X), Y) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X, \operatorname{Ex}(Y) ) \]

respects homotopy. That is, for every pair of maps $f,g: \operatorname{Sd}(X) \rightarrow Y$ having counterparts $f',g': X \rightarrow \operatorname{Ex}(Y)$, then $f$ is homotopic to $g$ if and only if $f'$ is homotopic to $g'$.

Proof. Assume first that $f$ and $g$ are homotopic, so that there exists a morphism of simplicial sets $h: \Delta ^1 \times \operatorname{Sd}(X) \rightarrow Y$ satisfying $h|_{ \{ 0\} \times \operatorname{Sd}(X) } = f$ and $h|_{ \{ 1\} \times \operatorname{Sd}(X) } = g$. The composite map

\[ \operatorname{Sd}( \Delta ^1 \times X) \rightarrow \operatorname{Sd}( \Delta ^1 ) \times \operatorname{Sd}(X) \xrightarrow { \lambda _{ \Delta ^1} \times \operatorname{id}} \Delta ^1 \times \operatorname{Sd}(X) \xrightarrow {h} Y \]

then determines a morphism of simplicial sets $h': \Delta ^1 \times X \rightarrow \operatorname{Ex}(Y)$, which is immediately seen to be a homotopy from $f'$ to $g'$.

Conversely, suppose that $f'$ and $g'$ are homotopic. Since $\operatorname{Ex}(Y)$ is a Kan complex (Corollary 3.3.5.4), we can choose a morphism of simplicial sets $h': \Delta ^1 \times X \rightarrow \operatorname{Ex}(Y)$ satisfying $h|_{ \{ 0\} \times X} = f'$ and $h'|_{ \{ 1\} \times X} = g'$, which we can identify with a map $u: \operatorname{Sd}( \Delta ^1 \times X) \rightarrow Y$. Let $v$ denote the composite map $\operatorname{Sd}( \Delta ^1 \times X) \rightarrow \operatorname{Sd}(X) \xrightarrow {f} Y$, so that $u$ and $v$ have the same restriction to $\operatorname{Sd}( \{ 0\} \times X)$. Note that the inclusion of simplicial sets $\{ 0\} \times X \hookrightarrow \Delta ^1 \times X$ is anodyne (Lemma 3.1.6.9), so the subdivision $\operatorname{Sd}( \{ 0\} \times X ) \hookrightarrow \operatorname{Sd}( \Delta ^1 \times X)$ is also anodyne (Proposition 3.3.5.2). It follows that the restriction map $\operatorname{Fun}( \operatorname{Sd}( \Delta ^1 \times X), Y) \rightarrow \operatorname{Fun}( \operatorname{Sd}( \{ 0\} \times X), Y)$ is a trivial Kan fibration, so that $u$ and $v$ belong to the same path component of $\operatorname{Fun}( \operatorname{Sd}( \Delta ^1 \times X), Y)$ and are therefore homotopic. It follows that $f = v|_{ \operatorname{Sd}( \{ 1\} \times X)}$ and $g = u|_{ \operatorname{Sd}( \{ 1\} \times X) }$ are also homotopic. $\square$

We can now prove a special case of Theorem 3.3.5.1.

Proposition 3.3.5.6. Let $Y$ be a Kan complex. Then the comparison map $\rho _{Y}: Y \rightarrow \operatorname{Ex}(Y)$ of Construction 3.3.4.3 is a homotopy equivalence.

Proof. Fix a simplicial set $X$. We wish to show that postcomposition with $\rho _{Y}$ induces a bijection

\[ \xymatrix { \{ \text{Maps of simplicial sets $X \rightarrow Y$} \} / \text{homotopy} \ar [d] \\ \{ \text{Maps of simplicial sets $X \rightarrow \operatorname{Ex}(Y)$} \} / \text{homotopy}. } \]

By virtue of Proposition 3.3.5.5, this is equivalent to the assertion that precomposition with the last vertex map $\lambda _{X}: \operatorname{Sd}(X) \rightarrow X$ induces a bijection

\[ \xymatrix { \{ \text{Maps of simplicial sets $X \rightarrow Y$} \} / \text{homotopy} \ar [d] \\ \{ \text{Maps of simplicial sets $\operatorname{Sd}(X) \rightarrow Y$} \} / \text{homotopy}, } \]

which follows from the fact that $\lambda _{X}$ is a weak homotopy equivalence (Proposition 3.3.4.8). $\square$

To deduce Theorem 3.3.5.1 from Proposition 3.3.5.6, we will need the following:

Proposition 3.3.5.7. Let $X$ be a simplicial set, and let $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ be the comparison map of Construction 3.3.4.3. Then the morphisms $\rho _{ \operatorname{Ex}(X) }, \operatorname{Ex}( \rho _{X} ): \operatorname{Ex}(X) \rightarrow \operatorname{Ex}( \operatorname{Ex}(X) )$ are homotopic.

Proof. Let $Q$ be a partially ordered set. Using Example 3.3.3.16, we can identify the subdivisions $\operatorname{Sd}( \operatorname{N}_{\bullet }(Q) )$ and $\operatorname{Sd}( \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) )$ with the nerves of partially ordered sets $\operatorname{Chain}(Q)$ and $\operatorname{Chain}( \operatorname{Chain}(Q) )$, respectively. Under this identification, the morphisms of simplicial sets

\[ \operatorname{Sd}( \lambda _{ \operatorname{N}_{\bullet }(Q) } ), \lambda _{ \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) )}: \operatorname{Sd}( \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) ) \rightarrow \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) \]

can be correspond to with nondecreasing functions $\operatorname{Chain}(\operatorname{Chain}(Q)) \rightarrow \operatorname{Chain}(Q)$, whose value on a linearly ordered subset $\vec{S} = (S_0 \subset S_1 \subset \cdots \subset S_ n)$ of $\operatorname{Chain}(Q)$ are given by

\[ \operatorname{Sd}( \lambda _{ \operatorname{N}_{\bullet }(Q) } )( \vec{S} ) = \{ \mathrm{Max}(S_0), \ldots , \mathrm{Max}(S_ n) \} \quad \quad \lambda _{ \operatorname{Sd}( \operatorname{N}_{\bullet }(Q))}( \vec{S} ) = S_ n. \]

Note that we always have an inclusion $\{ \mathrm{Max}(S_0), \ldots , \mathrm{Max}(S_ n) \} \subseteq S_{n}$. It follows that there is a unique map of simplicial sets

\[ h_{Q}: \Delta ^1 \times \operatorname{Sd}( \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) ) \rightarrow \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) \]

satisfying $h_{Q} |_{ \{ 0\} \times \operatorname{Sd}( \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) )} = \operatorname{Sd}( \lambda _{ \operatorname{N}_{\bullet }(Q) } )$ and $h_{Q} |_{ \{ 1\} \times \operatorname{Sd}( \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) ) ) } = \lambda _{ \operatorname{Sd}( \operatorname{N}_{\bullet }(Q) )}$, depending functorially on $Q$.

Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Ex}(X)$, which we identify with a map $\sigma : \operatorname{Sd}( \Delta ^ n ) \rightarrow X$. We let $f(\sigma )$ denote the composite map

\[ \Delta ^1 \times \operatorname{Sd}( \operatorname{Sd}( \Delta ^ n) ) \xrightarrow { h_{[n]} } \operatorname{Sd}( \Delta ^ n) \xrightarrow { \sigma } X, \]

which we will identify with an $n$-simplex of the simplicial set $\operatorname{Fun}( \Delta ^1, \operatorname{Ex}( \operatorname{Ex}(X) ) )$. The construction $\sigma \mapsto f(\sigma )$ then determines a morphism of simplicial sets $f: \operatorname{Ex}(X) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{Ex}( \operatorname{Ex}(X) ) )$, which we can identify with a map $\Delta ^1 \times \operatorname{Ex}(X) \rightarrow \operatorname{Ex}( \operatorname{Ex}(X) )$. By construction, this map is a homotopy from $\rho _{ \operatorname{Ex}(X) }$ to $\operatorname{Ex}( \rho _{X} )$. $\square$

Proof of Theorem 3.3.5.1. Let $X$ be a simplicial set. We wish to prove that the comparison map $\rho _{X}: X \rightarrow \operatorname{Ex}(X)$ is a weak homotopy equivalence. Fix a Kan complex $Y$; we must show that composition with $\rho _{X}$ induces a bijection $\pi _0( \operatorname{Fun}( \operatorname{Ex}(X), Y) ) \rightarrow \pi _0( \operatorname{Fun}(X,Y) )$. This map fits into a diagram

\[ \xymatrix@R =50pt@C=50pt{ \pi _0( \operatorname{Fun}( \operatorname{Ex}(X), Y) ) \ar [r]^-{ \circ \rho _ X} \ar [d]^{ \rho _{Y} \circ }_{\sim } & \pi _0( \operatorname{Fun}(X,Y) ) \ar [d]^{ \rho _{Y} \circ }_{\sim } \ar [dl]_{f \mapsto \operatorname{Ex}(f)} \\ \pi _0( \operatorname{Fun}(\operatorname{Ex}(X), \operatorname{Ex}(Y) )) \ar [r]^-{ \circ \rho _{X} } & \pi _0( \operatorname{Fun}(X, \operatorname{Ex}(Y)) ),} \]

where the vertical maps are bijective (Proposition 3.3.5.6) and the lower triangle commutes by the naturality of $\rho $. To show that the upper horizontal map is bijective, it will suffice to show that the upper triangle also commutes. Fix a map $f: \operatorname{Ex}(X) \rightarrow Y$. We then compute

\[ \operatorname{Ex}(f \circ \rho _{X} ) = \operatorname{Ex}(f) \circ \operatorname{Ex}(\rho _{X}) \sim \operatorname{Ex}(f) \circ \rho _{\operatorname{Ex}(X)} = \rho _{Y} \circ f \]

where the equality on the left follows from functoriality, the equality on the right from the naturality of $\rho $, and the homotopy in the middle is supplied by Proposition 3.3.5.7. $\square$

We close this section with the proof of Proposition 3.3.5.2.

Lemma 3.3.5.8. Let $J$ be a nonempty finite set, let $P(J)$ denote the collection of subsets of $J$ (partially ordered by inclusion), and set $P_{-}(J) = P(J) \setminus \{ J \} $. Then the inclusion of simplicial sets

\[ \theta : \operatorname{N}_{\bullet }( P_{-}(J) ) \hookrightarrow \operatorname{N}_{\bullet }( P(J) ) = \operatorname{\raise {0.1ex}{\square }}^{J} \]

is anodyne.

Proof. Fix an element $j \in J$ and set $I = J \setminus \{ j\} $, so that the simplicial cube $\operatorname{\raise {0.1ex}{\square }}^ J$ can be identified with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I} \simeq \Delta ^1 \times \operatorname{N}_{\bullet }( P(I) )$. Under this identification, $\theta $ corresponds to the inclusion map

\[ (\Delta ^1 \times \operatorname{N}_{\bullet }( P_{-}(I) )) \coprod _{ \{ 0\} \times \operatorname{N}_{\bullet }( P_{-}(I) )} ( \{ 0\} \times \operatorname{N}_{\bullet }( P(I) ) ) \hookrightarrow \Delta ^1 \times \operatorname{N}_{\bullet }( P(I) ), \]

which is anodyne by virtue of Lemma 3.1.6.9. $\square$

Proof of Proposition 3.3.5.2. Let $S$ be the collection of all morphisms of simplicial sets $f: X \rightarrow Y$ for which the induced map $\operatorname{Sd}(f): \operatorname{Sd}(X) \rightarrow \operatorname{Sd}(Y)$ is anodyne. Since the subdivision functor $\operatorname{Sd}$ preserves colimits, the collection $S$ is weakly saturated (in the sense of Definition 1.4.4.15). To prove Proposition 3.3.5.2, it will suffice to show that $S$ contains every horn inclusion. Fix a positive integer $n$ and another integer $0 \leq i \leq n$. We will complete the proof by showing that the inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ induces an anodyne map $\operatorname{Sd}( \Lambda ^{n}_{i} ) \rightarrow \operatorname{Sd}( \Delta ^ n )$.

Let $J = [n] \setminus \{ i\} $, let $P(J)$ denote the collection of all subsets of $J$, partially ordered by inclusion. Set $P_{-}(J) = P(J) \setminus \{ J \} $, $P_{+}(J) = P(J) \setminus \{ \emptyset \} $, and $P_{\pm }(J) = P(J) \setminus \{ \emptyset , J \} $. In what follows, we identify $\operatorname{Sd}( \Delta ^ n )$ with the nerve of the partially ordered set $\operatorname{Chain}[n]$ of nonempty subsets of $[n]$, and $\operatorname{Sd}( \Lambda ^{n}_{i} )$ with the nerve of the partially ordered subset of $\operatorname{Chain}[n]$ obtained by removing the elements $[n]$ and $J$ (Proposition 3.3.3.15). The construction $J_0 \mapsto J_0 \cup \{ i\} $ determines an inclusion of partially ordered sets $P(J) \rightarrow \operatorname{Chain}[n]$, hence a map of simplicial sets

\[ g: \operatorname{\raise {0.1ex}{\square }}^{J} = \operatorname{N}_{\bullet }( P(J) ) \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Chain}[n]) = \operatorname{Sd}( \Delta ^ n ). \]

Let $Z \subseteq \operatorname{Sd}( \Delta ^{n} )$ be the union of $\operatorname{Sd}( \Lambda ^{n}_{i} )$ with the image of $g$. An elementary calculation shows that the inverse image $g^{-1}( \operatorname{Sd}( \Lambda ^{n}_{i} ) )$ can be identified with the nerve of the subset $P_{-}(J) \subseteq P(J)$, so that we have a pushout diagram of simplicial sets

\[ \xymatrix { \operatorname{N}_{\bullet }( P_{-}(J) ) \ar [r] \ar [d] & \operatorname{Sd}( \Lambda ^{n}_{i} ) \ar [d] \\ \operatorname{N}_{\bullet }( P(J) ) \ar [r]^-{g} & Z. } \]

The left vertical map is anodyne by virtue of Lemma 3.3.5.8, so the right vertical map is anodyne as well. Let $h: [1] \times P_{+}(J) \rightarrow \operatorname{Chain}[n]$ be the map of partially ordered sets given $h(0, J_0) = J_0$ and $h(1, J_0) = J_0 \cup \{ i \} $. Then $h$ determines a map of simplicial sets $\Delta ^1 \times \operatorname{N}_{\bullet }( P_{+}(J) ) \rightarrow \operatorname{Sd}( \Delta ^ n )$. An elementary calculation shows that this map of simplicial sets fits into a pushout diagram

\[ \xymatrix { (\{ 1\} \times \operatorname{N}_{\bullet }( P_{+}(J) )) \coprod _{ \{ 1\} \times \operatorname{N}_{\bullet }( P_{\pm }(J) ) } ( \Delta ^1 \times \operatorname{N}_{\bullet }( P_{\pm }(J) )) \ar [r] \ar [d] & Z \ar [d] \\ \Delta ^1 \times \operatorname{N}_{\bullet }( P_{+}(J) ) \ar [r]^-{h} & \operatorname{Sd}( \Delta ^ n ). } \]

The left vertical map in this diagram is anodyne by virtue of Lemma 3.1.6.9, so the inclusion $Z \hookrightarrow \operatorname{Sd}( \Delta ^ n )$ is also anodyne. It follows that the composite map $\operatorname{Sd}( \Lambda ^{n}_{i} ) \hookrightarrow Z \hookrightarrow \operatorname{Sd}( \Delta ^ n )$ is anodyne, as desired. $\square$