$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 3.5.2.1. Let $f: X \rightarrow Y$ be a Kan fibration of simplicial sets and let $n$ be an integer. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-connective.
- $(2)$
For every vertex $y \in Y$, the Kan complex $X_{y} = \{ y\} \times _{Y} X$ is $n$-connective.
- $(3)$
For every simplicial set $B$ of dimension $\leq n$ and every simplicial subset $A \subseteq B$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & Y } \]
admits a solution.
- $(4)$
For every integer $0 \leq m \leq n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & X \ar [d]^{f} \\ \Delta ^{m} \ar [r]^-{\sigma } \ar@ {-->}[ur] & Y } \]
admits a solution.
Proof.
The equivalence $(1) \Leftrightarrow (2)$ is Proposition 3.5.1.22, the implication $(3) \Rightarrow (4)$ is immediate, and the converse follows from Proposition 1.1.4.12. Note that any morphism $\sigma : \Delta ^{m} \rightarrow Y$ is homotopic to a constant map. Using the homotopy extension lifting property (Remark 3.1.5.3), we see that $(4)$ is equivalent to the following a priori weaker assertion:
- $(4')$
For every integer $0 \leq m \leq n$ and every vertex $y \in Y$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & X_{y} \ar [d]^{f_ y} \\ \Delta ^{m} \ar [r]^-{\sigma } \ar@ {-->}[ur] & \{ y\} } \]
admits a solution.
The equivalence of $(2)$ and $(4')$ now follows from Proposition 3.5.1.12.
$\square$