Kerodon

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

Corollary 4.4.2.4. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S, } \]

where $i$ is a monomorphism. Then:

  • If $q$ is either a left or right fibration, then the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ of Construction 3.1.3.7 is a Kan complex.

  • If $q$ is a left fibration and $i$ is left anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B, X)$ is contractible.

  • If $q$ is a right fibration and $i$ is right anodyne, then the Kan complex $\operatorname{Fun}_{A/ \, /S}(B, X)$ is contractible.

Proof. Without loss of generality, we may assume that $q$ is a left fibration. By virtue of Remark 3.1.3.11, the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with a fiber of the restriction map

\[ \theta : \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,X). \]

Proposition 4.2.5.1 asserts that $\theta $ is a left fibration of simplicial sets, so its fibers are Kan complexes (Corollary 4.4.2.3). If $i$ is left anodyne, then $\theta $ is a trivial Kan fibration (Proposition 4.2.5.4), so its fibers are contractible Kan complexes. $\square$