Kerodon

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

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 3.1.6.1, 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 3.1.5.13), 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 3.1.6.1 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 3.1.5.10. $\square$