Kerodon

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

3.2.7 Closure Properties of Homotopy Equivalences

We now apply Whitehead's theorem (Theorem 3.2.6.1) to establish some stability properties for the collection of homotopy equivalences between Kan complexes (and weak homotopy equivalences between arbitrary simplicial sets).

Proposition 3.2.7.1. Suppose we are given a commutative diagram of Kan complexes

\[ \xymatrix { X \ar [r]^-{g} \ar [d]^{f} & X' \ar [d]^{f'} \\ S \ar [r]^-{h} & S', } \]

where $f$ and $f'$ are Kan fibrations and $h$ is a homotopy equivalence. Then the following conditions are equivalent:

$(1)$

The morphism $g$ is a homotopy equivalence.

$(2)$

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

Remark 3.2.7.2. In the situation of Proposition 3.2.7.1, the assumption that $S$ and $S'$ are Kan complexes can be eliminated at the cost of working with weak homotopy equivalences in place of homotopy equivalences: see Proposition 3.3.7.1.

Proof of Proposition 3.2.7.1. Assume first that $(1)$ is satisfied. Let $s$ be a vertex of $S$ having image $s' = h(s)$ in $S'$; we wish to show that the induced map $g_{s}: X_{s} \rightarrow X'_{s'}$ is a homotopy equivalence. By virtue of Remark 3.1.5.5, it will suffice to show that for every simplicial set $W$, the induced map $\operatorname{Fun}(W,X_{s} ) \rightarrow \operatorname{Fun}( W, X'_{h(s)} )$ is bijective on connected components. Replacing $X$ by $\operatorname{Fun}(W,X)$ (and making similar replacements for $X'$, $S$, and $S'$), we may reduce to the problem of showing that $g_{s}$ induces a bijection $\pi _0( X_ s) \rightarrow \pi _{0}( X'_{s'} )$. Let us regard $\pi _0(X_ s)$ and $\pi _0( X'_{s'} )$ as endowed with actions of the fundamental groups $\pi _{1}(S,s)$ and $\pi _{1}(S',s')$, respectively (Variant 3.2.4.5). Using our assumption that $g$ and $h$ are homotopy equivalences, we conclude that the induced maps

\[ \pi _0(X) \rightarrow \pi _0(X') \quad \quad \pi _0(S) \rightarrow \pi _0(S') \quad \quad \pi _{1}(S,s) \rightarrow \pi _{1}(S',s') \]

are bijective. Applying Corollaries 3.2.5.3 and 3.2.5.5, we conclude that $g_{s}$ induces a bijection $G \backslash \pi _0(X_ s) \rightarrow G \backslash \pi _0( X'_{s'} )$. It will therefore suffice to show that, for every vertex $x \in X_{s}$, the stabilizer in $G$ of the connected component $[x] \in \pi _0(X_ s)$ is equal to the stabilizer of the connected component $[g(x)] \in \pi _0(X'_{s'} )$. This follows from Corollary 3.2.5.7, since $g$ induces an isomorphism $\pi _{1}(X,x) \rightarrow \pi _{1}(X', g(x) )$.

We now show that $(2) \Rightarrow (1)$. Assume that, for each vertex $s \in S$ having image $s' = h(s)$ in $S'$, the induced map $g_{s}: X_{s} \rightarrow X'_{s'}$ is a homotopy equivalence. We wish to show that $g$ is a homotopy equivalence. We first show that the map $\pi _0(g): \pi _0(X) \rightarrow \pi _0(X')$ is bijective. Our assumption that $h$ is a homotopy equivalence guarantees that the map $\pi _0(h): \pi _0(S) \rightarrow \pi _0(S')$ is bijective. It will therefore suffice to show that, for each vertex $s \in S$ having image $s' = h(s)$, the induced map $\pi _0(X) \times _{ \pi _0(S)} \{ [s] \} \rightarrow \pi _0( X' ) \times _{ \pi _0(S') } \{ [s'] \} $ is bijective. Using Corollaries 3.2.5.3 and 3.2.5.5, we can identify this with the map of quotients $(\pi _{1}(S,s) \backslash \pi _0(X_ s)) \rightarrow ( \pi _1(S',s') \backslash \pi _0(X'_{s'} ) )$. The desired result now follows from the bijectivity of the map $\pi _0(g_ s): \pi _0( X_ s ) \rightarrow \pi _0( X'_{s'} )$ and of the group homomorphism $\pi _{1}(S,s) \rightarrow \pi _{1}(S',s')$.

To complete the proof that $g$ is a homotopy equivalence, it will suffice (by virtue of Theorem 3.2.6.1) to show that for every vertex $x \in X$ having image $x' = g(x)$ and every positive integer $n$, the group homomorphism $\pi _{n}(X,x) \rightarrow \pi _{n}(X',x')$ is an isomorphism. Setting $s = f(x)$ and $s' = f(x')$, we have a commutative diagram of exact sequences

\[ \xymatrix { \pi _{n+1}(S,s) \ar [r] \ar [d]^{\sim } & \pi _{n}(X_ s, x) \ar [r] \ar [d]^{\sim } & \pi _{n}(X,x) \ar [r] \ar [d] & \pi _{n}(S,s) \ar [r] \ar [d]^{\sim } & \pi _{n-1}(X_ s,x) \ar [d]^{\sim } \\ \pi _{n+1}(S',s') \ar [r] & \pi _{n}(X'_{s'}, x') \ar [r] & \pi _{n}(X',x') \ar [r] & \pi _{n}(S',s') \ar [r] & \pi _{n-1}( X'_{s'}, x'). } \]

Our assumptions that $g_{s}$ and $h$ are homotopy equivalences guarantee that the outer vertical maps are bijective, and elementary diagram chase shows that that the middle vertical map is an isomorphism. $\square$

Proposition 3.2.7.3. 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 weak homotopy equivalences. Then $\operatorname{\mathcal{W}}$ is closed under the formation of filtered colimits in $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$.

Proof. Suppose we are given a filtered diagram $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ in $\operatorname{\mathcal{W}}$, so that each $f_{\alpha }$ is a weak homotopy equivalence of simplicial sets. We wish to show that the induced map $f: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha } )$ is also a weak homotopy equivalence. Using Proposition 3.1.7.1, we can choose a diagram of morphisms $\{ u_{\alpha }: Y_{\alpha } \hookrightarrow Y'_{\alpha } \} $ with the following properties:

  • Each of the maps $u_{\alpha }$ is anodyne, and the induced map $u: (\varinjlim _{\alpha } Y_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y'_{\alpha } )$ is anodyne.

  • Each of the simplicial sets $Y'_{\alpha }$ is a Kan complex, and (therefore) the colimit $\varinjlim _{\alpha } Y'_{\alpha }$ is also a Kan complex.

Since every anodyne morphism is a weak homotopy equivalence (Proposition 3.1.6.7), we can replace $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ by the diagram of composite maps $\{ (u_{\alpha } \circ f_{\alpha }): X_{\alpha } \rightarrow Y'_{\alpha } \} $, and therefore reduce to the case where each $Y_{\alpha }$ is a Kan complex.

Let us regard the system of morphisms $\{ f_{\alpha } \} $ as a morphism from the filtered diagram of simplicial sets $\{ X_{\alpha } \} $ to the filtered diagram $\{ Y_{\alpha } \} $. Applying Proposition 3.1.7.1 again, we see that this diagram admits a factorization $\{ X_{\alpha } \} \xrightarrow { \{ g_{\alpha } \} } \{ X'_{\alpha } \} \xrightarrow { \{ h_{\alpha } \} } \{ Y_{\alpha } \} $ with the following properties:

  • Each of the morphisms $g_{\alpha }$ is anodyne, and the induced map $g: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } X'_{\alpha } )$ is anodyne.

  • Each of the morphisms $h_{\alpha }$ is a Kan fibration, and (therefore) the induced map $( \varinjlim _{\alpha } X'_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha })$ is also a Kan fibration.

Arguing as before, we can replace $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ by the diagram of morphisms $\{ h_{\alpha }: X'_{\alpha } \rightarrow Y_{\alpha } \} $, and thereby reduce to the case where each $f_{\alpha }$ is a Kan fibration. In this case, Proposition 3.2.6.8 guarantees that each $f_{\alpha }$ is a trivial Kan fibration. It follows that the colimit map $f: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha } )$ is also a trivial Kan fibration, and therefore a (weak) homotopy equivalence by virtue of Proposition 3.1.5.9. $\square$