$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 3.5.1.30. Let $f: X \rightarrow Y$ be a morphism of Kan complexes which is surjective on connected components and let $n \geq 0$ be an integer. The following conditions are equivalent:
- $(1)$
The morphism $f$ is $n$-connective.
- $(2)$
The induced map
\[ \theta : \operatorname{Fun}( \Delta ^1, X ) = X \times ^{\mathrm{h}}_{X} X \rightarrow X \times ^{\mathrm{h}}_{Y} X \]
is $(n-1)$-connective.
- $(3)$
For every pair of vertices $x,x' \in X$, the map of path spaces
\[ \{ x\} \times ^{\mathrm{h}}_{X} \{ x'\} \rightarrow \{ f(x) \} \times ^{\mathrm{h}}_{Y} \{ f(x') \} \]
is $(n-1)$-connective.
Proof.
Using Proposition 3.1.7.1, we can factor $f$ as a composition $X \xrightarrow {i} \overline{X} \xrightarrow {\overline{f}} Y$, where $\overline{f}$ is a Kan fibration and $i$ is a homotopy equivalence. Replacing $\overline{X}$ by a full simplicial subset if necessary, we may further assume that $i$ is surjective on vertices. It follows from Remark 3.5.1.14 (and Proposition 3.4.0.9) that conditions $(1)$, $(2)$, or $(3)$ is satisfied by $f$ if and only if it is satisfied by $\overline{f}$. Consequently, we may replace $f$ by $\overline{f}$ and thereby reduce to proving Variant 3.5.1.30 in the special case where $f$ is a Kan fibration. In this case, we have a commutative diagram
\[ \xymatrix { X \ar [r] \ar [d]^{\delta _{X/Y}} & \operatorname{Fun}( \Delta ^1, X ) \ar [d]^{\theta } \\ X \times _{Y} X \ar [r] & X \times ^{\mathrm{h}}_{Y} X} \]
where the horizontal maps are homotopy equivalences (Proposition 3.4.0.7), so the equivalence of $(1)$ and $(2)$ follows from Corollary 3.5.1.29 (together with Remark 3.5.1.14). Since $\theta $ is a Kan fibration (Theorem 3.1.3.1), the equivalence of $(2)$ and $(3)$ follows from Proposition 3.5.1.22.
$\square$