Kerodon

Proof of Proposition 3.2.8.1. 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 3.1.6.6, 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 3.2.5.5). Using our assumption that $g$ and $h$ are homotopy equivalences, we conclude that the induced maps

are bijective. Applying Corollaries 3.2.6.3 and 3.2.6.5, we conclude that $g_{s}$ induces a bijection $\pi _{1}(S,s) \backslash \pi _0(X_ s) \rightarrow \pi _{1}(S',s') \backslash \pi _0( X'_{s'} )$. It will therefore suffice to show that, for every vertex $x \in X_{s}$, the stabilizer in $\pi _{1}(S,s)$ of the connected component $[x] \in \pi _0(X_ s)$ maps isomorphically to the stabilizer in $\pi _{1}(S',s')$ of the connected component $[g(x)] \in \pi _0(X'_{s'} )$. This follows from Corollary 3.2.6.7, 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 3.2.6.3 and 3.2.6.5, 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 3.2.7.1) 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

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. $\square$

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 3.1.6.14), 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 3.2.7.2 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 3.1.6.10. $\square$

Proof. By virtue of Corollary 3.2.8.4, we may assume without loss of generality that $S$ is finite. In this case, there is an isomorphism $S \simeq [n]$ for some integer $n \geq 0$, so that $\operatorname{N}_{\bullet }(S)$ is isomorphic to the standard simplex $\Delta ^ n$. $\square$